written by Michael Keller

Fifth Edition July 2020

Introduction

A Microhistory of Twisting Puzzles

A Quick Account of my Cubing History

A Brief Glossary of Terms Used in Twisting Puzzles

Solving the 3x3x3 Cube -- an Intermediate Solution

Picture Cube

Other Coloring Patterns

Mirror Cube

Shape Modifications

Fisher Cube

Twist 3x3x3

Super Skewb

Master Pyramorphix

Fluctuation Angle Cube (Axis Cube)

Mixup 3x3x3Twist 3x3x3

Super Skewb

Master Pyramorphix

Fluctuation Angle Cube (Axis Cube)

Magic Domino

Cuboid Puzzles (Larger Dominos)

Floppy Cube

A Tactile CubeFloppy Cube

The Pocket Cube (2x2x2)

Rubik's Insanity -- a Scrambling Question

Rubik's Revenge (4x4x4 Cube)

Pyraminx in 27 Moves

The Skewb

The Orb

ImpossiBall and Kilominx

Ivy Cube

Dino Cube

Redi Cube

More Solutions Coming

The Professor's Cube (5x5x5)

Larger cubes

The 7x7x7 Cube

The 8x8x8 and 9x9x9 Cubes

Siamese Cube

Square-1

Square-2

Megaminx

Alexander's Star

The Missing Link

Vertex-Turning Octahedron

Puzzle RoundupLarger cubes

The 7x7x7 Cube

The 8x8x8 and 9x9x9 Cubes

Siamese Cube

Square-1

Square-2

Megaminx

Alexander's Star

The Missing Link

Vertex-Turning Octahedron

Engel's Puzzles

Top-Spin and Back-Spin

Smart Alex and Galaxy Lock

Masterball

Rubik's Clock

Gripple

New Shapes for Rubik's Snake

Where to Buy PuzzlesTop-Spin and Back-Spin

Smart Alex and Galaxy Lock

Masterball

Rubik's Clock

Gripple

New Shapes for Rubik's Snake

Annotated Bibliography

Encyclopedias

A Survey of Rubik's Cube Maneuver Catalogs

Mastering the Cube Literature: A Short Guide to Books on the Magic Cube of Ernő Rubik

Sequels to the Cube: Books on Big and Little Cubes, Snakes, Pyramids, and Chains

Who's Who in CubologyA Survey of Rubik's Cube Maneuver Catalogs

Mastering the Cube Literature: A Short Guide to Books on the Magic Cube of Ernő Rubik

Sequels to the Cube: Books on Big and Little Cubes, Snakes, Pyramids, and Chains

Appendices

Appendix 1 -- Varasano method for
speedsolving 2x2x2

Appendix 2 -- MES and xyz notation

Appendix 2 -- MES and xyz notation

Some of the material in this booklet appeared in WGR (numbers 1-12) between 1982 and 1994. Portions of the bibliography appeared in

Most of the puzzles discussed herein are protected (or may have been in the past) by patent, trademark, or both.

In 1974, Ernő Rubik, a Hungarian professor of interior architecture, conceived of a new puzzle design, a 3x3x3 cube with six solid-colored sides, where each of the six outer layers could turn independently. He named it

Before the end of 1981, Rubik's Cube was followed up by a number of puzzles, each of which might be considered to be a sequel to it in some sense. Rubik himself created the Magic Snake, made out of 24 triangular prisms joined by springs, which allow the snake to be twisted into a large variety of shapes. Ideal Toy Corporation devised a puzzle called The Missing Link, an interesting modification of the Fifteen Puzzle popularized by Sam Loyd. Tomy manufactured a tetrahedral puzzle called Pyraminx, which was actually conceived by Uwe Mèffert prior to the cube. In 1982, Ideal brought out a smaller version of the Magic Cube called Rubik's Pocket Cube (a 2x2x2 cube) and a larger version called Rubik's Revenge (4x4x4). A number of books appeared in order to help the enthusiast with this new torrent of puzzles. The Cube was a worldwide craze for a few years, spawning dozens of related and unrelated puzzle designs, several bestselling books, at least three newsletters, and huge media publicity. It won numerous awards, including the 1980 Spiel des Jahres for Game of the Year. A World Championship for speed solvers was held in June of 1982, but not much more than a year later, the bubble had burst and the craze was over, and it would be 11 years before a second World Championship was held.

More than a decade after the original Cube boom ended, another fertile period of puzzle development began, fueled by a number of companies which entered the puzzle field or expanded their operations, several magazines (most notably

I bought my first Cube in 1981. I was not able to make any headway with it until I read Douglas Hofstadter's first article in

The Cube fad collapsed during 1983, and a number of solutions I had written up by hand never appeared in WGR. A sequel I wrote to the

I had probably not solved a cube in five years or so when I saw descriptions of the Mirror Cube on the Internet in late 2008. I ordered one from an eBay dealer in Japan, and while I was waiting, I started relearning the details of my Cube solution, which I had mostly forgotten. The arrival of the Mirror Cube rekindled my interest in twisting puzzles in general, and I started work on this booklet in mid-2009.

For speedcubers who'd like to laugh at me, my personal records at the moment are: 2x2x2 11.50s (but a fluke 9.94s), 3x3x3 47.77s, Icon 1:52.10, 4x4x4 2:20.24, 5x5x5 4:22.61, 6x6x6 8:43.53, 7x7x7 14:01.40, 8x8x8 23:09.22, 9x9x9 40:19.63, combined solves [2x2x2 through 7x7x7] 34:15.37, [2x2x2 through 8x8x8] 1:00.15.35, one-handed 2x2x2 50.03, one-handed 3x3x3 2:31.50, Mirror 2:43.81, Twist 3x3x3 2:06.98, Mixup 3:17.60, Tactile 6:04.41, cube pattern replication 3:48.72, Shepherd's Cube 11:29.53, Pyraminx 15.41s, Skewb 18.34s (fluke 16.83), Redi Cube 35.43, Master Pyraminx 1:48.65, Kilominx 1:48.87, Orb 1:58.62, ImpossiBall 5:00.91, face-turning Octahedron 7:06.59, Megaminx 7:31.57, Professor Pyraminx 11:56.53. Fewest moves (including slice and multiple turns): Redi Cube 23 turns, 3x3x3 40 turns, Twist 3x3x3 76 turns, 4x4x4 128 turns, 5x5x5 252 turns, Megaminx 265 turns. Most of these were timed by hand with a stopwatch or online timer, and all of them were solved with no preexamination time.

The following terms will be used throughout the various puzzle solutions:

antislice move -- a turn of two opposite layers in opposite directions.

baryon -- a sequence of moves which twists three corners in place 120 degrees in the same direction

center -- a piece located in the interior of a turning layer, usually with one colored facet. The 3x3x3 cube has six centers, which are fixed to the central spindle of the puzzle, and can rotate but not move to other faces. In cubes with pictures or other patterns on the faces, the orientation of the centers matters. Larger cubes have more than one center piece per face; they can freely move to different faces via slice moves. In shapeshifting 3x3x3 puzzles, a center is the middle of a turning layer, and can change orientation but not position. It has four orientations, and may have one, two, or four colored facets. Some puzzles, like ImpossiBall, Pyraminx, Square One, Dino Cube, and Redi Cube, have no centers.

Christman Cross -- a pretty pattern for the Cube in which a cross appears on all six faces, with the colors grouped in threes.

commutator -- a sequence of moves of the form ABab, where two moves (or sequences) are performed, then the inverse of both are performed. Longer commutators are possible; e.g. ABCbac or ABCacb (see, for example, the Pyraminx solution).

conjugation -- a modification of a sequence by adding one or more setup turns to place pieces in certain locations. After performing the main sequence, the setup turns are performed in reverse order and reverse direction. This is described in more detail later.

corner -- a piece which is part of three mutually perpendicular turning layers, and which can be twisted in 120 degree increments, so that it has three possible orientations. On a normal cube it has three colored facets, and cubes (of any size from 2x2x2 up) each have eight corners. In shapeshifting puzzles, a corner may have one, two, or three colored facets, and on some puzzles (e.g. Master Pyramorphix), the orientations may be visually indistinguishable.

deep move -- a turn of an outer layer and one or more adjacent parallel layers at the same time

Dots -- a Cube pretty pattern in which each side shows a center of one color surrounded by eight facets of a second color (e.g.

double bicycle -- a sequence of moves which swaps the locations of two pairs of pieces

double edge flip -- a sequence of moves which reverses the orientation of two edges at once

double meson -- a sequence of moves which twists four pieces, two in each direction

edge -- a piece which is part of two perpendicular turning layers, and usually has two orientations. On most puzzles it has two colored facets which meet along a line segment. The 3x3x3 Cube has 12 edges, Pyraminx has 6, Square One 8, Skewb and Pocket Cube zero. On order-3 puzzles edges can be reversed in place (flipped). Larger cubes have multiples of 12 edges: 24 for 4x4x4, 36 for 5x5x5, 48 for 6x6x6, etc. On puzzles with four layers, edges cannot be flipped in place; their orientation is determined by their position (e.g. Master Pyraminx; see the diagram in that section). On the Mixup Cube, edges can be twisted 90 degrees via half-slice moves. In shapeshifting 3x3x3 puzzles, edges may have up to three colored facets, but still only two orientations. In some puzzles (e.g. Fisher Cube and octagonal barrel variant cube), some or all edges have only one color and both orientations look the same. In others (Master Pyramorphix), edges have one color but are asymmetric, so flipped Edges stick out from the normal puzzle shape.

equator (Alexander's Star, ImpossiBall) -- the band of pieces lying between two opposite faces

face -- a flat visible surface formed by a group of pieces (usually) sharing the same color. There are six faces on a Cube of any size, four on Pyraminx, etc. Occasionally the face is not flat (as in ImpossiBall). In normal cubes, the face is part of a layer and turns as a unit (but see Skewb and Square-1 as exceptions).

facet -- an individual segment (of one color) of a center, edge, or corner piece. A center usually has one facet, an edge two, and a corner three. Each facet will have a sticker or tile of (or be painted) a particular color.

flip -- to reverse the orientation (and colors) of an edge while leaving it in the same location (a 180 degree rotation). In most puzzles, every sequence of moves flips an even number of edges (perhaps zero).

inverse -- a sequence which performs the exact opposite of another sequence. It is derived from the original sequence by reversing the order and direction of the turns. For example, the inverse of

layer -- a group of pieces which turn as a unit (9 in the 3x3x3 Cube)

meson -- a sequence of moves which twists two corners in place 120 degrees in opposite directions

middle -- centrally placed (used as an adjective while center is used as a noun). In particular, the middle layer of the 3x3x3 cube is the layer in between the first and last layers to be solved (typically the Up and Down layers), and middle edges are the four edges in that layer. On 5x5x5 (and larger odd) cubes, a middle edge is the centermost of a set of three edge pieces.

orient -- to turn a correctly positioned piece so that its colors match those of correctly positioned adjacent pieces

parity -- a mathematical principle which limits the possible configurations which can occur in a puzzle. For example, the number of edges which are flipped in a scrambled 3x3x3 Cube is always an even number.

piece -- an outer unit of a puzzle which is connected to the central mechanism. It usually has one, two, or three facets, each a different color.

place -- to put a piece in the correct location and orientation at the same time; usually done one piece at a time

Plummer Cross -- a pretty pattern for the Cube in which a cross appears on all six faces, with the colors grouped in pairs

position -- to put a piece in the correct location without regard to orientation; usually done three or four pieces at a time

Pons Asinorum -- the simplest Cube pretty pattern, formed by half turns of all six outer layers (

pretty pattern -- a sequence of moves which place a previously unscrambled cube or other twisting puzzle into an attractive arrangement, usually symmetric

quark -- a one-third (120 degree) twist of a corner

shapeshifter -- a puzzle which changes shape from turn to turn (e.g. the Square One and Pyramorphix families; Mirror, Fisher, Twist, Axis, and Mixup cubes)

slice -- the subcubes lying between two opposite layers (4 edges and 4 centers in each slice in the 3x3x3 Cube)

speedcubing -- competition solving against the clock, which generally requires very complex algorithms, fast reflexes, and a well-lubricated cube

Start -- the unmixed goal position of a twisting puzzle; normally each face of the puzzle is one color. Always capitalized when used as a noun in this sense.

tip -- a three-colored piece on tetrahedral puzzles (the Pyraminx family) connected only to one larger corner. It is trivially fixed by aligning its colors with the attached corner.

tricycle -- a sequence which cyclically exchanges the locations of three pieces

twist -- to change the orientation of a corner without changing its location

twisterflipper -- a sequence which performs flips and twists simultaneously. Coined by M. Razid Black and Herbert Taylor in their book

wing -- a two-colored piece on larger Pyraminxes where the two facets meet only at a point.

wring -- a simultaneous 90 degree turn of an outside and middle layer together, followed by an additional 90 degree turn of the outside layer alone, done in one motion.

What is an intermediate solution anyway?

Beginner's solutions to the Cube often use pictograms instead of notation to show each move in a sequence (somewhat like trying to teach someone music without teaching them the basics of music notation). They are usually slow and inefficient, frequently requiring a sequence to be applied twice (or more!) rather than give an inverse version. They rarely use slice moves, and generally require 100 moves (turns) or more to complete a solution. Nearly all of them make the mistake of solving the entire top layer before proceeding to the middle layer.

Speedcubing solutions to the Cube normally require the solver to memorize more than 100 different sequences to handle dozens of individual positions in particular situations, and the individual sequences are often long and hard to remember (using turns of only a few faces, and few repetitive patterns). They frequently use a mixture of three different notations for regular, slice, and whole cube turns, and often require a reorientation in the middle of a long sequence. A common speedcubing method needs 57 different sequences just to orient the last layer; we will see later how this can be done only a little less efficiently with three variants of a single sequence.

The solution we present here is somewhere in between a beginner's solution and a speedcubing solution. I assume you do not want to learn 100 or more move sequences in order to solve the cube, but want to learn to solve the cube in well under two minutes, usually taking no more than 80 moves, sometimes considerably less. I am currently under 1 minute about 40 percent of the time (recent average 63 seconds) using this method with an ordinary (not speed) cube, and occasionally have been under 60 moves. The best time I have recorded since I started working on this booklet is 47.77 seconds. The move sequences I use are the ones which I feel are the easiest to memorize, as they are either relatively short or have a strong pattern to them.

I count, as some analysts do, both 180 degree turns and slice turns as single turns. Others insist on counting by outer layer quarter turns, so that a 180 degree outer layer turn or a 90 degree slice turn counts as two turns, and a 180 degree slice turn counts as four. Even by this more restrictive counting system, I can usually get a solution in under 100 quarter turns. [There's another system which counts outer layer turns, but both 90 and 180 degrees count as one; any slice turn counts as two.]

We use the standard Singmaster (English language) notation for the Cube, where the six faces/layers are designated Front, Back, Left, Right, Up, and Down, and clockwise 90-degree turns of each layer are designated by capital letters. We designate anticlockwise turns using lower-case letters (rather than using a prime or minus sign); I think this makes the notation easier to read. 180-degree turns are designated by the capital letter followed by a 2 (these may be made in the clockwise or anticlockwise direction, whichever you find more convenient; R2 and r2 have the same effect). We also designate slice turns (of one of the three middle layers) with an asterisk indicating a 90-degree clockwise turn of the adjacent middle layer in the same direction: R* means to move the slice in-between R and L in the same direction as an R turn. (This is much easier to remember for non-mathematicians than the MES notation used elsewhere). The same turn in the opposite direction is L* (so R*2 and L*2 have the same effect).

Individual pieces are designated, where necessary, by a single letter for centers, two letters for edges (UF is the edge shared by the Up and Front faces), and three letters for corners (BLD is the corner shared by the Back, Left, and Down faces).

In the rare cases where we show a specific reorientation of the whole cube in the middle of a sequence, we designate a turn of the entire cube in brackets, again avoiding the unnecessarily complex notation used elsewhere, with whole-cube turns designated with xyz notation. [F] means to turn the whole cube in the direction of a Front turn, so that the Front face stays the same and the Left face becomes the new Up face (this has the same effect as FF*b). We will often add spaces to notation to make it easier to read and to clarify the structure of the sequence. The six main clockwise turns, and an example of a slice and an anticlockwise turn, are shown above in a variety of diagrams and photos.

Diagrams show the color pattern which Ideal quickly made standard on its cubes (shown in the four small photos above right): blue is opposite white, red is opposite orange, and green is opposite yellow.

Apparently some cubists do not like using slice turns, because they are slower for speed cubists to perform, and because they complicate the notation. But slice moves are used in speedcubing, sometimes in disguise, and they are essential for solving larger cubes (4x4x4 and up). They also make it much easier to understand and remember many sequences. For example, a routine to flip all 12 edges can be notated very concisely as

Some sequences also use an

A faster equivalent of the antislice move is a move is sometimes called a

Useful sequences can be found using a number of techniques. One basic kind of move which is widely used in solving the first layer of a puzzle is an

Another way of generating sequences is by combining two sequences in a way that some of the moves at the end of the first sequence are reversed at the beginning of the second. For example, I found the Upside-Down Insertion (F2D)2F2D2F2 (described later) by combining two well-known sequences: the I-Swap F2DF2dF2, and the sequence (F2D2)3, which swaps FL with FR and LD with RD. Joining these, we get F2DF2dF2F2D2F2D2F2D2. The middle section dF2F2D2 cancels out and becomes D, and we drop the last D2 because we don't care about the Down layer yet. The second half unswaps FL and FR which were swapped by the first half, and the result is a pure insertion from the Down layer which has no effect on the middle layer. This sequence is also useful in solving the 3x3x2 and 3x3x4 cuboid puzzles we will see later.

Rubik himself discovered the double edge flip (R*U)2R*U2(L*U)2L*U2. He may have discovered the Spratt Wrench too, and realized that he could join it to an almost-mirror image of itself, (UL*)4. If you just join those you get (R*U)4(UL*)4, which flips UL and UR once and three middle-slice edges (UB DB DF) twice, resulting in a pure double-edge flip of 15 moves. But what happens when you stick U2 in the middle? Now the original UL is going to be flipped a second time (and UR not at all), and instead of flipping the original UB a second time, we 're swapping the original UF instead. Now it becomes R*UR*UR*UR*U(U2)UL*UL*UL*UL*, and the underlined section mostly vanishes: the four consecutive U's become nothing, R* and L* cancel out, and what's left is just U2. Adding another U2 at the end to undo the middle one, we have Rubik's discovery.

Another kind of sequence alternates turns of two faces, turning one face the same direction each time (sometimes with double turns) and the other in alternating directions. The best known of these is the Sune, RUrURU2rU2, named by Lars Petrus as part of his 3x3x3 solution. It twists three corners in the same direction and cycles three edges (on the 2x2x2 it acts as a pure baryon). Other similar sequences are useful in solving the Megaminx, Redi Cube, and other puzzles.

Another way of finding sequences is by adapting sequences from one puzzle to another. The 3x3x3 corner tricycle lURuLUru which can be used on any size cube, can be adapted to dodecahedral puzzles like Kilominx and Megaminx, where it becomes lU2Ru2LU2ru2.

The solution we describe here has six stages:

(1) Place the four edges in the Up layer -- these are done one at a time, but with each edge we place, we try to eliminate any problems with edges yet to be placed.

(2) Swap any middle edges which are already in the middle layer into their correct locations, regardless of orientation.

(3) Place the four Up corners and place and fix the remaining middle edges. We use a group of short sequences (3-8 turns) with dual purposes, with each sequence trying to fix an Up corner and a middle edge at the same time whenever possible.

(4) Flip the cube over so that the unsolved layer is now Up. Place the corners in the correct positions, regardless of orientation.

(5) Place the edges in the correct positions, regardless of orientation.

(6) Orient all of the edges and corners simultaneously, using one or two applications of Benson's Twisterflipper.

The first stage in solving the Cube is to place the four Up edges in their correct positions around the Up center. To do this efficiently, we want to get as many of the edges into good positions (either in the Down layer with the Up colored facet facing down, or anywhere in the middle layer), so that we can put them into the correct Up position with a single turn of a side layer. The bad positions, in contrast, are in either the Up layer or Down layer with the Up color facet on one of the side faces. An edge in the Up layer with the Up facet facing up is good (shown as beige, above center) if there is only one, or if by chance there are more than one in the correct positions. A properly mixed cube shouldn't have any edges matching the center color when you begin solving, but if you do have such an edge, you can consider it already correctly placed as the first edge.

Otherwise the first edge can be placed anywhere in the Up layer; the remaining edges must be placed in the correct positions

If there are any edges in the middle layer which need to go to the Up layer, we should begin with one of these, but make sure turning it into the Up layer will not make another middle layer edge into a bad edge by pushing it into the Down layer). Consider the example above right: it is tempting to just turn Front anticlockwise, putting the white-yellow edge into the Up layer and pushing out the bad green-white edge. But doing this will make the red-white edge bad by pushing it into the Down layer flipped the wrong way. Instead we want to place the red-white edge first by turning Left anticlockwise, putting it in the Up layer with the white facet Up. But before we do that, we turn Up clockwise to put the bad green-white edge into the target location, and also turn Down clockwise so that the bad orange-white edge is underneath the target location. Now when we turn Left anticlockwise, we have placed the first edge and converted both of the bad edges into good ones in only three turns. [It will actually only take four more turns (

The diagram above shows the first edge (white-orange) already placed, although the orange side facet is not aligned with the orange center (we will not align the Up layer with the middle edge centers until the first two layers are completely finished). The small white and yellow squares show the target location for the white-yellow edge. We want to get the white-yellow edge into one of the three positions shown, so that a turn of F or f or F2 will put it in the correct spot. If the white facet of the white-yellow edge is in a good location, we can turn either the Down layer or the Down and middle layers together (a deep move) to put it in one of the three locations shown. If the white-yellow edge is in the Up layer with its white facet on a side face, or with its white facet Up but in the wrong position relative to the first (white-orange) edge, we can first turn the side layer it is in (F, L, or B) so that it goes into the middle layer. If it is in a bad position in the Down layer, we can again turn the side layer it is in to put it in the middle layer. but we must not displace an Up edge already placed, so if it is underneath one of the edges already placed, we must turn the Down layer to put it in a different side layer first, before turning that side layer.

Below are some more examples of placing two bad edges at the same time. In the first two examples, U* is not a slice move, but part of the deep move dU*, in which the bottom two layers are turned together anticlockwise.

Let's now look at a complete example of placing all four edges when all of them are bad at the beginning. In the position above left, we have two white edges in the Up layer but flipped, and two in the Down layer with their colored facets Down. We begin by turning

Some utility routines for fixing troublesome edges

If you haven't pushed out all of the bad edges by the time you have placed the first two or three edges, you may find the last edge or two in a troublesome position. The routines shown above should help: the first two show how to fix the last edge when it is in the correct location but flipped, or in the Down layer in a bad position. The third routine lets you move an edge in the wrong place (from UF to UR), and the fourth lets you switch two adjacent edges.

Stage 2 requires the solver to find all middle edges which are already in the middle layer, even if in the wrong position or orientation. These will be edges without the color of the Down face (in our example, blue). The simple sequences shown below allow any edges already in the middle layer to be swapped into their correct locations (relative to the four side centers). At this point their orientation is not important -- we are going to flip edges during stage 3. The bottom diagrams show an X-ray view of the middle layer looking down through the Up layer.

The 4th sequence, the I-Swap, also moves a corner from DRF to URF. This should be used as often as possible; it can be used to get a headstart on Stage 3 by inserting a corner whose white face is in the Down layer from the Down layer to its Up position, or pushing out an Up corner which is in the wrong place (occasionally both at once). An alternate version of the I-Swap, which moves a corner from DLF to URF, is faster, but dislodges ULF (actually putting DRF there). If you want to swap two middle edges without disrupting any Up corners,

Most of the time you will only find one or two edges which belong in the middle layer and are already there. A single edge can almost always be moved to its correct location with a V- or I-Swap: turn the whole cube so any correct edges are at LB, or LB and RB (the only tricky case is a single or double swap when LB and RF are correct: this requires a V-Swap followed by an I-Swap; see the two examples below, where correct edges are shown as solid black). Two edges can usually both be swapped to their correct locations with one of the four routines above (except in the second case shown below). More complex swaps of three or four edges need a V or X swap to put two of the edges in their correct spots, followed by an I-swap to fix the remaining edge or edges.

An important point to understand is that although we are going to be trying to place (or fix) Up corners and middle edges simultaneously with each sequence we carry out in Stage 3,

Since Stage 2 often puts middle edges in the correct position but flipped, we first look at move sequences which insert a corner from the Down layer into the Up layer, while flipping a middle edge already in the correct location. In the examples pictured below, we want to flip the orange-green middle edge, while inserting the white-yellow-red corner into its correct Up position. We first turn the whole cube so that the flipped edge is at RF, turn the Up layer so that the white-yellow-red target location is at URF, and turn the Down layer so that the white-yellow-red corner is in the correct location for the selected sequence. If the corner has its Up facet (white in our examples) on the Down face, it should begin at DRF. If its Up facet is on any side face, it should begin diagonally opposite, at DLB. Now perform the indicated routine, depending on the position of the white facet. The Column Flip is actually two Drop Insertions joined together. The other two routines, which are mirror images of each other, each bring a corner from the diagonally opposite location, by reorienting the corner in the Down layer and putting it in position for a Drop Insertion.

The Column Flip should be used whenever possible; i.e. anytime there is both a flipped middle edge and a corner in the Down layer with its Up facet on the Down face. If there are no such corners, but more than one flipped middle edge, use one of the other two routines if possible.

If you have an Up corner in the correct position, but twisted, there is another set of routines to twist an Up corner in place (clockwise or anticlockwise), while optionally putting a new middle edge in place. Turn the bottom two layers together so that RF is either the target for a middle edge, or at least an incorrect location (remember that we don't want to put the last middle edge in place unless we are placing or fixing the third Up corner at least). Then (if you are placing a middle edge) turn the Down layer so that the edge being placed is in the correct spot (there is one valid location for any edge, depending on which way it is flipped and in which direction we need to twist the corner). The first and third twists are well-known routines, and we will use them again when solving Rubik's Pocket Cube. The second and fourth routines are the same as the third and first routines, except that the Down turns are 180 degrees.

Now we use the four routines below to move any of the corners which belong in the Up layer, but are currently in the Down layer with their Up-colored facet on one of the side faces, into their correct positions, including orientation. If the edge at RF is correct (or flipped), turn the whole cube so that the edge at RF is not a correct one (flipped or not). If an edge in either of the two corresponding Down layer positions also belongs in the middle layer, we will also

We will continue using these routines to put in Up corners (and middle edges at the same time if possible), jumping back to use the Edge-flipping insertions as needed, trying to get at least three Up corners correct.

You can actually do Drop Insertions from other positions; e.g. if the green-orange edge is at FL with its green facet on the left face, you can insert it via rD2R. In fact, although we will not show all eight possible positions, when you are placing edges alone, any middle edge still in the Down layer will either already be in position for a Push Insertion, or can be inserted from where it is via a variant Drop Insertion (turn whichever of the Front or Right faces is the color of the edge facet not on the Down face, then turn the Down layer so that the edge moves into the side layer you just turned, and undo the side layer turn. So the Drop insertions may also be rDR, rD2R, Fdf, or FD2f.

Occasionally the last Up corner and the last middle edge can be put in correctly at the same time using one of the four Corner/Edge Insertions, but usually either they are not aligned properly or the edge is flipped the wrong way. In this event, it is quicker to put the middle edge in by itself, but flipped, unless you still have another middle edge flipped. If all of the other three middle edges are correct, and you want to put the last edge in backwards, follow the same procedure as the previous paragraph, but consider the orientation of the edge reversed. Finally put the last corner in using the Edge-flipping Insertions we learned at the beginning of the phase, which should both flip the last middle edge and place the last Up corner.

If you make a mistake and accidentally insert the last corner in the wrong orientation while a middle edge is flipped, you can fix both using a routine we are going to learn soon, Benson's Twisterflipper. Turn the whole cube so that the almost complete Up layer becomes the Left layer, and the flipped middle edge is at UF (also turn the new Left layer if needed so that the twisted corner is at ULF). Now do

If you prefer, you can insert the last middle edge correctly, and use one of the first two routines shown above to put in the last Up corner without disrupting the middle layer. These routines also may be used when you make a mistake, or otherwise end up in a situation where all edges are correctly placed and oriented while an upper corner still needs to be placed (you may do this intentionally, if the sequence which inserts the third Up corner also inserts the last middle edge correctly). The third routine, which flips a middle edge without affecting the Up layer, can also be used to fix a mistake, or when all of the edges are in place (two of them flipped) with only one Up corner left to insert. In the latter case, put the last corner in using an edge-flipping insertion, and then use the Middle Edge Monoflip on the last middle edge. The Upside-Down Insertion will also be useful in solving the 3x3x4 Cuboid puzzle we will see later.

Turn the Down face so that the last corner comes to DFR, while its target location is UFR. If the last corner is upside-down, we can insert it with a special sequence (above left) which combines two well-known sequences (removing an unnecessary move): the I-Swap we learned at the beginning of the second stage, and the simple double-edge swap

Most speedcubing methods require dozens of sequences to be learned in order to solve the last layer (the CFOP system has 57 sequences to orient the last layer and 21 more to permute the last layer). This is extremely fast and efficient, but at a huge cost in memorization. Our system requires only 11 sequences (three of which are inverses and two minor variants) instead of 78, once you understand conjugation and learn how to use Benson's Twisterflipper effectively.

Stage 4 is an easy one and requires performing only one sequence of seven or eight turns. Pick one of the Up corners (one with its blue facet on the Up face if there is one) and turn the Up layer so that that corner goes to its correct location, then turn the whole cube ([U], [u], or [U2]) so the correct corner is at URF. Now either two, three, or four corners are correct (disregarding orientation). If four are correct, you can proceed to Stage 5.

If three are correct, the corners need to be cycled clockwise (the corner at ULF needs to go to ULB), or anticlockwise (ULF needs to go to URB). Use the Corner Tricycle below far left, or its anticlockwise inverse, below second left), to put the remaining corners in the correct position. The Corner Tricycle is one of the best-known and easiest routines to learn, alternating left/right turns with up turns in alternating directions (we just saw this in Stage 3 as a method of inserting the last Up corner without disrupting middle edges). Proceed to Stage 5.

If two adjacent corners are correct, turn the whole cube so that the correct layers are at ULF and ULB (i.e, if URF and URB are correct, turn [U2]; if URF and ULF are correct, turn [U]), and do the Adjacent Corner Swap (third below), which is identical to the Corner Tricycle except that the last turn is a half turn. Proceed to Stage 5. If two diagonally opposite corners (i.e. URF and ULB) are correct, do the Diagonal Corner Swap (below right) and proceed to Stage 5.

There are five possible cases after the corners are in the correct positions. In one case out of 12, the four edges are all in their correct positions (though perhaps either two or all four are flipped). Two-thirds of the time, one edge is correct and the other three need to be cycled, either clockwise or anticlockwise. In this case, turn the whole cube (as in Stage 4, keeping the Up face in place and rotating around the vertical axis) so that the good edge is at UR. Now use the Edge Tricycle or its inverse (below left) to fix the other three edges. The anticlockwise tricycle is identical except that the two single U turns are anticlockwise. Speedcubers prefer a longer but faster edge tricycle which uses only Up and Right turns: R2URU(ru)2rUr, which cycles UL=>UR=>UF.

Only the Tricycle is strictly necessary, since double swaps can be done by doing a tricycle twice, but it will speed up solving to use the two double swaps, which are easy to learn because of their strong repetitive patterns. The Edge Cross-Swap (an elegant palindrome), which occurs in one case out of 12, is considered fast enough to be used in speedcubing methods (e.g.

One of the most powerful maneuvers in the world of cubology is a 9-move twisterflipper discovered by David Benson,

You can also perform the sequence as a set of the same three turns performed three times, with a 120 degree rotation around a diagonal axis in between: (ruR2[BU])3 and its inverse (U2BU[LD])3.

Below each color picture is a skeleton diagram of the top face showing which corners need to be twisted (+ = clockwise, - = anticlockwise) and which edges need to be flipped (+). This maneuver, the shortest known twisterflipper, is flexible enough to provide the basis for a complete method of orienting the up layer. All of the 35 twisterflippers can be produced using one or two repetitions of this move and/or its inverse or variant. Usually the first repetition will be the standard version, which we will designate @, but in a few cases we will need the inverse or variant.

The basic idea is to put ourselves, regardless of what configuration of twists and flips we began in, into a configuration with three corners twisted (they will all be twisted in the same direction) and two edges flipped. We'll call this a

There are six positions of the two flipped edges relative to the three twisted corners, one of which is the standard shape above left and above middle. The other five positions are shown below. These will require conjugation: one or two setup moves to put the edges and corners into the correct shape, then undoing the setup moves in reverse order after performing the Benson sequence. The first two positions require two turns before performing the Benson sequence, though in the first sequence (if the twists are clockwise) the second setup turn (underlined) combines with the first move of Benson to form an R2 turn (I usually do it anticlockwise anyway, so the sequence becomes

Now what do we do if there are four, two, or zero corners twisted, or zero or four edges flipped? We want to hold the cube (rotating the whole cube around the vertical axis while keeping the Up layer up) so that the first Benson Twisterflipper flips half of the edges which need to be flipped (one out of two or two out of four), and twists one corner correctly and two incorrectly, putting us into a three-two position. It would be possible to show a diagram of every single case, but it's better to understand the concept rather than memorizing 57 individual cases.

If all four corners, or none, are twisted (which happens a little over 1/4 of the time), any of the four positions will work as far as the corners are concerned. If all four edges, or none, are flipped, or two opposite edges are flipped (which happens half the time), again, any position will work as far as the edges are concerned. If we're in one of the easy positions, we can then perform the regular or inverse Benson, choosing whichever one fixes one of the three corners which we are twisting: if two of them need to be twisted anticlockwise and one clockwise, we use the normal Benson, if two clockwise and one anticlockwise, we use the inverse (see example 1 below). If none of the corners are twisted, we can use either one. In any event, we end up in one of the three-two positions already described above. [There are three cases where it is preferable to use the variant Benson instead of the normal Benson, to avoid the worst conjugation cases -- see examples 1, 2, and 6 below.]

So the only positions we have to think harder about are three corners twisted in the same direction, two corners twisted in opposite directions, and two adjacent edges flipped. If all four corners, or none, are twisted, and two adjacent edges need to be flipped, we hold the cube so that the edges to be flipped are either at UF and UL, or UB and UR, and do the normal or inverse Benson as described in the previous paragraph (see example 3 below).

If three corners are twisted, we're going to hold them in the normal position so that the untwisted corner is at ULB. If two edges are flipped, we're already in one of the three-two positions we've already learned above. If no edges or all four edges are flipped, we're going to do something odd: we're going to do a Benson Twisterflipper in what seems to be the wrong direction (the normal Benson if the corners need to be twisted anticlockwise, or the inverse if they need to be twisted clockwise). This will

If two corners are twisted in opposite directions, we need to hold the corner which needs to be twisted anticlockwise at ULB, so the first Benson will twist the other one and the two untwisted corners all clockwise, leaving us with three corners which need to be twisted anticlockwise. Again, this might seem like we're messing the cube up more, but remember that we want to get to a position with three corners twisted in the same direction. As long as the edges are not in one of two troublesome positions, we do the standard Benson, once again ending up in a three/two position we know how to finish. See examples 7 and 8 above.

The only two edge positions which cause a problem are the two positions where two opposite corners are twisted in opposite directions, and two edges, which are adjacent to one of those corners, are flipped. Any way we hold the cube will either flip both edges or twist both corners, neither of which we want to do. The solution is to hold the cube in the same way we have just described, with the corner needing to twist anticlockwise at ULB. The two edges to be flipped are either at UB and UL, or at UF and UR. Now we use the variant Benson (that's the second and more important reason we want to learn it),

The orientation phase should take no more than 23 moves, even in the worst cases, and the cube will be solved.

[It is possible, by the way, to reverse the order of stages 5 and 6, doing all of the orientations first and then fixing the edge positions. I usually do stage 6 first if the pieces which need to be oriented are already in the standard V-shaped Benson configuration.]

If you want a little more efficiency at the cost of learning some extra sequences, you can learn some more sequences which do corner rotations alone or edge flips alone at the end. The most useful of these are two which were discovered by Ernő Rubik himself. These can actually be used as an alternative to the Benson method, though you will still need to understand how to conjugate the two Rubik routines.

Rubik's Meson (above left) twists the front-right corner clockwise and the back-left corner anticlockwise on the Up layer. Note that the first six turns are simply repeated to complete the sequence, and that the six moves consist of three moves and a sort of mirror image of those moves. This is a pretty easy sequence to learn, despite being 12 moves long, since it is a six-move sequence repeated twice. If the corner which needs to be twisted anticlockwise is the front-left corner, do an

Rubik's Flip flips the front and back edges on the Up layer. Note again the pattern: the first three slice moves are in one direction, and the last three are in the opposite direction. The Up turns are all in the same direction, with the third and last being 180 degree turns. If the two edges to be flipped are adjacent, hold them at the front and left, do

If you need to flip all four edges, you can do Rubik's Flip twice, or use the Quadruple Edge Flip shown below, center. The Quadruple Edge Flip also has a strong pattern, but note that the first and second half end with quarter turns rather than half turns. The Quadruple Edge Flip unfortunately does disrupt centers somewhat if you use it on larger cubes to flip partial edges; we will later show a routine based on the Spratt Wrench which can flip four partial edges without disrupting centers.

An alternate sequence for double edge flips is:

A corresponding technique for corner twists is a

A shorter monotwist pair, rDRFDf for clockwise twists and FdfrdR for anticlockwise twists, must be used in opposite pairs to keep the lower two layers correct. This has the disadvantage, in the case of a baryon, of requiring that one of the three corners must be twisted the wrong way twice.

There are hundreds of these; we will show only a few of the most elegant ones. The crosses were first discovered by David C. Plummer and David Christman respectively.

Almost since its inception, many versions of the Cube have been made with pictures on the stickers. Solving such a cube is essentially identical to the standard cube except that you may have centers which are rotated at the end. The photograph above shows a cube with numbers in the form of a 3x3x3 magic square (nowadays you can also find Sudoku Cubes). The orange 5 needs to be rotated 180 degrees and the purple and green 90 degrees clockwise and anticlockwise respectively. We will notate this as {U2Fr}, indicating that the Up center needs a double turn, the Front center a clockwise turn, and the Right center an anticlockwise turn. So we need routines to rotate centers independently. Christoph Bandelow's book

Cubes with full pictures on each face add another layer of complexity, as it is necessary to figure out which pieces go on which layers, like solving six separate jigsaw puzzles.

Shape Modifications

A good number of variant puzzles have been designed by altering the shape of the outer pieces, often by making cuts through the cube at angles different from 90 degrees.

Originally known as the Bump Cube, it was invented by Hidetoshi Takeji, and submitted to the Puzzle Design Competition for the 2006 International Puzzle Party (shockingly, it did not win a prize, being beaten out by the much less interesting Floppy Cube among others). The Mirror Cube is essentially a 3x3x3 Cube, but each of the six outer layers is a different thickness. When the faces are turned, the cube turns into a jumble of differently sized rectangular prisms. Three of the photos above show the gold version I bought recently (there are also pointless six-colored versions, designed by someone who didn't understand the concept of the puzzle). The Mirror Cube can be solved by most of the usual methods, but it is hard to see quickly where each piece goes (push insertions work better than drop insertions). There should be a lot of scope for making interesting pretty patterns with the Mirror Cube (the third photo shows a Giant Meson, on the original silver model I bought in 2008). It is possible to solve the Mirror by touch, as with the Tactile cube shown below, but it is much harder, because it is difficult to get the oddly shaped layers aligned exactly without looking (my first try took nearly half an hour).

Fisher Cube

The first true shapeshifting puzzle was invented by Tony Fisher in the 1980's. By cutting through the cube at 45 degree angles in two directions, he created a puzzle in which edges, corners, and centers interchange their functions. The Fisher Cube has 12 pieces which appear to be edges, but eight of them (those in the Up and Down layers) act as the eight corners of a normal cube. The other four (in the middle layer) act as centers for the four layers which turn at 45 degree angles. The Up and Down layers turn normally; their centers are true centers. The eight house-shaped three-colored pieces act as eight of the 12 edges; the one-colored pieces on the side faces (which look like centers) act as the other four edges. Keeping in mind that edges and corners on the Up and Down layers swap roles, the Fisher Cube can be solved by basically the same method as the normal 3x3x3, with the addition of the Picture sequences above which turn centers. The first two layers can be solved as usual; if any middle layer centers are twisted, leave them for the end, since some last-layer sequences disrupt middle layer centers. For example, Rubik's Flip, (R*U)2R*U2(L*U)2L*U2, twists the the Front and Back centers 180 degrees, while the standard quadruple edge flip twists the Back center 90 degrees clockwise.

It is possible to end up with one center twisted 90 degrees, which looks like an impossible position, but it can be fixed by simultaneously turning either a yellow or white center 90 degrees in the opposite direction using the same sequence

Twist 3x3x3

First made by Eitan Cher, this is a standard 3x3x3 cube distorted in shape by twisting the top half of the cube 90 degrees clockwise. The Up and Down layers are solidly colored, and the four side layers are bicolored: the middle centers have two colors and the middle edges only one. This makes the puzzle a (partial) picture cube; the photo above shows a puzzle laying on its side with one center twisted 90 degrees, exactly analogous to the Fisher cube. Like Fisher, you can use the standard solving sequence, leaving rotation of the four middle centers for last. There are also positions which appear to have an odd number of edges flipped, but the middle edges have no visible orientation: a Spratt Wrench ((R*U)4) with the puzzle held sideways (with the middle layer running around the F/U/B/D faces) flips LF and also invisibly flips three middle edges (it also turns all four middle centers 90 degrees clockwise).

Super Skewb

Although this puzzle (a rhombic dodecahedron with 12 diamond-shaped faces) looks like an expanded Skewb with 12 faces instead of 6, it is once again a cleverly disguised 3x3x3 Picture Cube. The three-colored corners are still corners, but what appear to be centers are actually edges (with no orientation: flipping them has no effect), and the four-colored corners are actually face centers. It can be solved in the same way as the Picture Cube, using one center to place four edges correctly, and keeping in mind that opposite faces are always the same color. But it's much harder to visualize than any other 3x3x3 variant, because of the rotating centers and the fact that there are two identical edges of each color, and two corners of each color pattern in reverse order. You may end up with a position where it appears that two edges need to be swapped: this is fixed by doing a conjugated edge tricycle with two edges of the same color.

Master Pyramorphix

Yet another disguised 3x3x3 Picture Cube is the four-colored, three-layer Master Pyramorphix (also called Mastermorphix). Like many larger puzzles, the faces are not flat, but slightly rounded, as if the puzzle had been inflated slightly (such puzzles are usually called pillow-shaped). The photos above show an unscrambled puzzle, the Pons Asinorum pretty pattern (still retaining its original shape, but showing all four colors on each face), and scrambled (it is a shapeshifting puzzle). The two-colored pieces which appear to be edges actually act as the centers of the six oblong outer layers. The four corners are true corners, while the triangular face centers also act as corners (but without visible orientation). The twelve trapezoidal pieces act as edges. Although they are one-colored, they have orientation (i.e., they can be flipped) because of their shape. Despite being a shapeshifting puzzle, it is somewhat easier to solve than the Super Skewb: it is not necessary (nor even helpful) to restore it to its original shape before finishing the solution. Having only four colors makes it easier to visualize than the six colors in pairs of the Super Skewb. Because there are three trapezoidal edges of each color, you can end up, as with the Super Skewb, with two edges which apparently need to be swapped. You can also end up with a single corner twisted (or even two in the same direction), which can be fixed by twisting a corner and a triangular face center (pseudocorner) in opposite directions (or three in the same direction).

Fluctuation Angle Cube (Axis Cube)

This puzzle has six cuts through the cube at 30 degree angles, dividing each colored face into seven pieces (above left). This produces six small two-colored pieces which are the actual centers of the six turning layers. Two of the corners are normal; the other four corners are the six one-colored isosceles triangles. Six of the twelve edges are elongated two-colored pieces; the other six are one-colored trapezoids. When the puzzle is scrambled, it becomes an irregular shape (above second left). The third photo shows a partial solution with one layer left to fix. The fourth photo shows a center twisted 90 degrees clockwise (leftmost) and another twisted 180 degrees (rightmost); there is a hidden center twisted 90 degrees anticlockwise. If you have solved some of the other 3x3x3 shapeshifters, this one shouldn't be too hard and solves basically the same way, but it takes some practice to visualize what pieces go in each slot.

Mixup 3x3x3

In 1985, Sergey Makarov conceived and patented a shapeshifting 3x3x3 variant in which the edges are slightly elongated so that their cross-section equals their length. Oskar Deventer later built a more robust design, and WitEden mass-produced it. The Mixup Cube allows slice turns of 45 degrees, and thereby allows edges and centers to be swapped. Making eighth-slice turns changes the shape of the cube. We will denote a 45 degree turn of a slice with the suffix & (R& turns the left-right slice 45 degrees in the direction of an R turn; R* is a 90-degree slice turn as usual). Solving the Mixup involves two problems: returning the cube to its original shape, and then completing the solution. Restoring the cube shape is not difficult: the simple algorithm D&RurU& swaps a vertical edge currently in the center of the F face with a center currently at UF.

Restoring the original color pattern is a little harder, since the centers (even after restoring the cube shape) can be swapped, and several parity problems can happen. The sequence R&U2L&U2 swaps the F and U centers (and also the UF and UB edges, flipping the latter). [(R&U2L&U2)2 is a short double edge flip for UF and UB).] Often an odd number of edges are flipped: a single edge at UF can be flipped via L&D*R2U* R&D*R2U*. The eighth-slice moves allow other odd-looking maneuvers, even edge twists: L&D*RU* R&D*rU* twists the edge at UF 90 degrees clockwise (plus a usually-invisible turn of the F center 90 degrees anticlockwise).

The hardest problem to fix is a single pair of edges swapped: the long sequence (R&RUruR&URur)4 swaps UB and UR: note the half-wring at the start of each repetition. This is the longest sequence in this book, and it is very easy to make mistakes while doing it.

Another invention of Rubik himself (patented in 1983), this is a 2x3x3 reduction of the standard cube, with a similar mechanism. The bottom half, concealed in the first photo above, has white pieces numbered identically to the black pieces directly above them (so that the unscrambled bottom is the mirror image of the top). The Up and Down layers can be turned normally; F/B/L/R can only be turned 180 degrees at a time. It's a much easier puzzle than the 3x3x3 Cube, since edges cannot be flipped and corners cannot be twisted: as long as a piece is in the correct layer, it is automatically oriented correctly. The parity laws are different on the Magic Domino: since there are imaginary middle edges, a single pair of edges or corners can be swapped. You can also swap the Up and Down centers by (R*2F2)2D2R*2D2. The total number of positions is (8!)^2/4 = 406,425,600.

It is almost trivial to solve the Up half: putting a edge which is in the wrong layer under its proper position in the top layer and turning that side layer 180 degrees allows the four top edges to be placed around the 5. The I-Swap we learned earlier,

A modified routine using two UDIs creates an edge tricycle:

The Domino is no longer in production (you can find them readily on eBay, but they're somewhat pricey), but you can buy 2x3x3 Cubes with the standard six colors instead of domino markings (third and fourth pictures above). These are solved in essentially the same way as the Domino, though if you don't have the positions of the colors memorized you may put in two initial Up edges reversed. You will see this when you try to insert the first Up corner. If you have UF and UB backwards, you can swap them as described above. If you don't want to buy a new puzzle, you can simulate it on a regular 3x3x3 cube by mixing carefully, only making double turns of the outside (L/B/R/F) layers. (You'll usually have to fix the middle layer when you're done.)

Cuboid Puzzles (Larger Dominos)

Some non-cubic twisting puzzles have been created by essentially stacking Dominos on top of each other. A 3x3x4 puzzle, for example (photo above), is a double Rubik's Domino, with four layers which act like layers of the Domino. It has double centers on the F/B/L/R faces, which are once again restricted to half turns. The two outer layers can be solved exactly like the Domino. The inner layers can also be solved like the Domino, but the inner layer edges act like outer layer corners and the inner centers act like outer layer edges. You can insert an edge from the third layer to the second layer using the Upside Down Insertion (not the I-Swap, which disrupts the Up and Down layers)

There are some parity issues which can be solved by applying (R2 12U2)3, where 12U2 means to turn the top two layers together a half turn. If you need to swap one double pair of middle edges, 12D2 R2 F2 2D2 F2 R2 12D2 swaps FL and FR in both middle layers (this is a relative of the Single Edge Swap we will see later on the 4x4x4 cube).

Some layered domino puzzles are also being made in cube shapes by slicing some of the layers thinner: a 3x3x4 has normal Up and Down layers and two half-thickness middle layers; a 3x3x5 has a normal middle (third of five) layer and four half-thickness layers, including the Up and Down layers. They go up at least as far as 3x3x9. I don't have any of these yet, but you can see pictures of them by doing an image search, and buy them from sites that sell twisting puzzles. The particularly interesting feature of these variations is that quarter turns can once again be made in any direction, and the thin layers will inevitably be split up and turned at different angles, adding a complex new problem to be solved.

A 1x3x3 cube has been developed by Katsuhiko Okamoto. This won First Prize (which is actually second prize, since there is a Grand Prize) in the 2006 Puzzle Design Competition. It seems prizes are awarded in these competitions based on the ingenuity of the mechanical design rather than how interesting a puzzle it makes: the Floppy Cube beat out the much more interesting Bump Cube in the 2006 competition. The Floppy Cube has only four possible turns, equivalent to F2, R2, B2, and L2 on the standard cube, and only 192 possible positions. The hardest position (a sort of quadruple edge flip with a checkerboard pattern) takes 8 turns to solve (

I made a rough mockup of a textured 3x3x3 Cube, to see if it could be solved by touch alone. This version has raised rubber bumps, sandpaper, vinyl floor tile, soft felt squares, hard felt circles, and glossy contact paper. It's not an ideal selection, but served well enough as a proof of concept. It took an hour of shopping at a home supply store and a crafts store, and another two hours or so to strip the old stickers off of a battered 1981 vintage Ideal cube and cut and glue the new stickers (everything had sticky backing except the sandpaper, which I had to superglue on). My first try at solving without looking at it was successful, but took 18 minutes. I got a little faster each of the next few times I did it. I'm now consistently getting under 10 minutes, so far getting as fast as 6:04.41. The superglue didn't work very well: several sandpaper squares have fallen off and I had to reattach them with double sided cellophane tape.

The solution described here is in three stages, and takes a maximum of 32 moves and an average of about 24 moves (38 and 27 measured in quarter turns). In Stage 1, the four Up corners are placed to complete half of the puzzle. Since there are also no centers, we can start by considering any corner to be correct, and holding it at ULF. We then place the corner which belongs at UFR. If that corner is in the upper layer in the wrong position (at URB or UBL) or at UFR but twisted, turn either R, r, B, or b, to put it in the Down layer. Turn the Down layer so that its Up-colored facet (blue in the diagrams) is at one of the positions shown (bottom of DRB, or left or front of DRF), and use the move or moves shown below to place it. The first two turn the edge directly into its correct location without disturbing ULF or ULB. The third is the same sequence we learned for the same situation when solving the 3x3x3 Cube.

Turn the whole cube so that the correctly placed corners are at ULF and ULB, and place the third corner at URF in exactly the same way (if the corner is at UBR, or at URF but twisted, turn R2 to put it in the Down layer).

The fourth Up corner requires slightly different handling. If it is in the Down layer, turn the Down layer so the last Up corner is at blue facet is in one of the three positions shown, and perform the maneuver shown. The last two are the same routines used in the 3x3x3 Cube (and the third is identical to the 2nd and 3rd corner case). The first sequence is the same one we use in solving the Magic Domino.

If the last Up corner is already at URF but twisted, we need to use one of the corner twists we learned for the 3x3x3 cube to fix it. Don't worry about the arrows right now: they show which Down corners are being swapped, for a trick you can learn later.

In Stage 2, the cube is flipped over and the completed layer becomes the bottom half. Turn the new Up layer if necessary so that the correct corner goes to URF, and use the standard 3x3x3 routines to position the rest of the new Up corners, not worrying about orientation yet.

In Stage 3, the corners could be oriented using the standard 3x3x3 routines you already know (e.g. Benson's Twisterflipper works fine as a baryon), but we suggest using faster routines which take advantage of the lack of corners, such as the lightning-fast baryon now known as the Sune, which some speedcubers can perform in under a second (Singmaster credits Katalin Fried with the original discovery of this sequence; the name was coined by Lars Petrus). All of the sequences have been selected to use as many Right and Up turns as possible; only the first double meson uses turns of any other faces. Note that if you seem to be in a meson position which is the vertical mirror image of the first one shown below, just reorient the cube so that the Right and Up faces trade places (that is, [BU2]), and you will be in the correct position. Note also that the second meson has the same moves as the first, except that the sixth and last turns are 180 degree turns (instead of 90 degree turns) of the Up face.

When the fourth Up corner is already in place but twisted, it is possible to bypass the second stage by simultaneously twisting that corner in place and positioning the Down corners. Two-thirds of the time, the Down corners are in a configuration where swapping two adjacent corners will put the corners in the correct relative positions. If the last corner needs to be twisted clockwise, turn the Down layer so that the corners which need to be swapped are at FDL and FDR, and do the normal twist rDRFDf (below far left). If the corner needs to be twisted anticlockwise, put the corners to be swapped at FDR and BDR, and do the twist FdfrdR (below second left). Even if you only learn this part of the trick, you will save time whenever this case occurs (in fact, when the fourth corner has its blue facet on the Down face, or in one of the other positions which take more than three moves, you can deliberately put the fourth corner in wrong, then fix it using this trick).

In one case out of six, two diagonal corners need to be swapped (and it doesn't matter which!). In this case, do the special twisting sequence

In one case out of six, the Down corners are already in the correct relative positions, and you can perform a skewed double meson which twists the last Up corner and three of the Down corners. Turn the whole cube via [U] so the corner which needs to be twisted is at ULF. The routine

Once you understand the basic method here, if you want to learn to speedcube the 2x2x2, you can graduate to the Varasano method, a version of which is described in Appendix 1. Unlike speedcubing methods for the 3x3x3 Cube which usually have over 100 sequences to learn, the Varasano method needs only 12 sequences to solve the 2x2x2. The basic outline is to put the four Up corners in the Up layer with the correct colored facet up (but not necessarily in the correct positions), turn the cube over, orient the new Up corners (again regardless of position), and then position the top and bottom layers simultaneously, using one of a set of only five routines. There are much more complicated methods, including the Guimond and Stern-Sun methods.

The diagrams below show sequences as short as possible for placing each of the Up corners, depending on the location of the Up-colored facet (in the examples here, dark blue). Light blue shows the target location of the next blue facet. For example, if the third corner to be placed has its dark blue facet on the right side of the URB corner, the correct sequence to place it is RdR. Becoming familiar with as many of these as possible will speed up your solutions to the first half of the puzzle. Note that many of the 2nd and 3rd corner sequences are the same for corresponding positions (and a few of the 3rd and 4th).

Rubik's Insanity -- a Scrambling Question

The well-known puzzle Instant Insanity consists of four cubes with colored sides, the object being to arrange them in a row of four so that every row has four different colors. We can do this with the Pocket Cube: what is the shortest routine to get the 2x2x2 Cube from its Start position to a position in which every face shows four different colors? Ideal's solution booklet gives a simple five move answer:

We can do a six-color version with any larger cube: what is the shortest routine to get the 3x3x3 Cube from its Start position to a well mixed-up state in which every face shows all six colors, with no edge or corner facets matching the centers? When I first wrote this section in 2010, I mixed up a cube until I had a valid position, and used Herbert Kociemba's Cube Explorer to find a 19-move solution:

Rubik's Revenge (photos above left and (scrambled) above second left) is one of the original names for the 4x4x4 cube, using a mechanism designed by Péter Sebestény. In Europe it was originally marketed as Rubik's Master Cube. There are now alternate designs made by various Asian manufacturers. East Sheen Industrial Company of Taiwan makes cubes of much lighter plastic: their 4x4x4 (second right) weighs about 110 grams, compared to 133 g for a 3x3x3 Deluxe Rubik's Cube, 187 g for a new 4x4x4 from Winning Moves and 230 g for a vintage Rubik's Revenge. Eastsheen also has new designs for the 2x2x2 and 5x5x5 which, like their 4x4x4, allow for smoother turning and are preferred by some speedcubers, but the internal edges are sharp and the stickers wear quickly.

Like most cubes made today, the Eastsheen cube is built with stickers or tiles on a white base instead of a black one, and uses a different coloring scheme from the old Ideal standard: instead of white opposite blue and green opposite yellow, white cubes use yellow opposite black and green opposite blue (sometimes white stickers are still used, with white opposite yellow). In the colored diagrams below, we use white/yellow and blue/green.

Shown far right is a new cube made by Mo Fang Ge. It has no tiles or stickers: the pieces are molded of plastic, and each piece appears to be solid. They must be glued together, because I can't see any evidence that the sides are painted with different color combinations, but the gluing is skillful and seamless. This is the wave of the future: light, smooth-turning, and durable, and is my current favorite 4x4x4 and 5x5x5. The pattern shown, with each face having a 2x2 center of the opposite color, is impossible on a 3x3x3 (on a 5x5x5 you can make equal signs). The sequence (23R2 B2)2 (23R2 12B2)2 swaps the 2x2 centers of the U and D faces; you can then reach the pattern shown using an extended version of Dots (23R 23B 23R2 23F 23R), or by repeating the first sequence on the other two pairs of faces.

Although we prefer diagrams, when we need to designate individual pieces on the 4x4x4 cube, we use three letters for each piece. A corner is designated by three capital letters, indicating the three faces it lies in. An edge is designated by two capital letters, indicating the two faces it lies in, and a lowercase letter showing which half of the cube it lies in (the edge DRf is the edge in the front half of the cube, on the intersection of the Down and Right faces). A center is designated by one capital letter, and two lower case letters (the center Ful is on the front face, in the upper left quadrant). See the diagram above for examples.

We want a notation which is relatively easy to read, does not require subscripts or superscripts, resembles the 3x3x3 notation as closely as possible, and is extendable to larger cubes later on. The problem is that the moves we want to make involve a mixture of outer face turns and inner slice turns, sometimes at the same time (it is much faster to turn an outer face and the adjacent inner slice together than to turn the inner slice alone, so we will do so whenever possible). One way we are are going to make the notation a little cleaner is to eliminate designations of anticlockwise turns entirely: instead of using r to designate an anticlockwise turn of the Right layer, we are going to say 4L, indicating that the fourth Left layer is turned clockwise. Each turn will consist of an optional number prefix indicating which of the four slices is being turned clockwise (omitting 1 in the case of a clockwise outer layer turn), a capital letter indicating the layer(s) involved, and an optional 2 suffix if the turn is 180 degrees. Since turns may now begin and end in numbers, we will put a space after every turn.

Since we already know how to solve the 3x3x3 cube, if we can get the centers of the 4x4x4 correctly arranged, and the edges paired up, we can then finish solving the cube as if it is a 3x3x3, except for a couple of potential

The centers are fixed in three easy stages. The first stage is complete a 2x2 center by putting all four centers of a chosen color (red in the diagrams below, but any color will do) into position on the Up face. We will use half-cube turns (

The fourth center is placed using a simple drop insertion. Turn the whole cube so that the missing center needs to go to Urf, and turn the bottom three layers to bring the last red center to the Right face, then turn the Right face if necessary to put it at Rfd (diagram below left). Then the drop insertion

The second stage of solving the centers is to complete half of the four side faces. Turn the whole cube so the completed center is on the left face, and turn any one of the side faces so that one of the red corners comes to the Left layer with a red facet on the Left face (always possible with a single turn). Turn the whole cube so that corner is at ULF, which shows you which color of center goes on each of the other five faces (in the example above, diagram third left, the Up face must be yellow, Front must be green, and the other three faces the correct opposite colors (Down -- white, Back -- blue, Right -- orange). The goal of the second phase is to put two centers of the correct color in the left half of the Up, Front, Down, and Back faces (diagram above center).

We do this using the same idea we used to align the first two red centers. To put the two yellow centers in the left half of the Up face, find a yellow center. If it is already on the Up face, turn the Up face if necessary to put it in the left half. Find a second yellow center, putting it in the right half of the cube if necessary by turning the layer it is in (Front, Down, or Back). Turn the right half of the cube until it comes to the Front face, turn Up if necessary to put the two yellow centers in the correct relative position, align them by turning 12R, and turn Up anticlockwise to store them safely in the left half (so that later turns will not dislodge them). Turn the whole cube so that the yellow centers go to the Back face, and repeat the process with the new Front face, putting two green centers there in the same way. Do this twice more, putting two white centers opposite yellow and two blue centers opposite green.

Sometimes you will find two centers of the desired color already adjacent. If they are already on the correct face, you can immediately turn them into the left half. Otherwise, turn the face they are in so they both go into the right half, and bring them together to the correct face (still turning the right half of the cube), then store them in the left half again. Rarely, three or four of the desired color are stuck in the Right face. If so, turn the Right face so that two of them are in the back half of the Right face, turn the Back half clockwise to bring them into the Up face, turn the Up face a half turn, turn the Back half anticlockwise (to restore the Left face), then turn the Up face clockwise to store them as normal (see above right).

Once all of the faces except the Right are half filled with centers of the correct color, the second of the three center phases is complete. Turn the right half of the cube to get a third center correct on at least one or two of the U/F/D/B faces, to get a small headstart on the third phase. Turn the entire cube so that the red face is Down, then turn each of the four side faces if necessary so that the completed centers are in the right half of their face (as the green faces in the diagram above, far left). Turn any faces with three complete centers so that the missing center spot is at Ful (third diagram above). Now we want to swap centers between the Up face and the Front face, until all four side faces are completed, which will also complete the Up face (orange in our example).

Whenever the Up face has two adjacent centers of the same color, as in the first diagram above, you want to choose that color next. Turn the Up face so that the double center is at Urf and Urb, turn the whole cube so that the target is the Front face, and put them both in with a single maneuver

If two centers of the same color are diagonally adjacent, as in the fourth diagram above, choose a different color if possible (yellow in our example) and do it in the normal way. This will bring the two green centers together so you can do them as in the first diagram. You should use the maneuvers in the fourth and first diagrams as often as possible, enabling you to put in two centers at a time. If it is not possible to do a different color (usually because the other two colors are the sixth color, as in the fifth diagram above), you will need to put them both in using a double version of the basic maneuver,

Usually seven swaps should complete all six center quartets. Occasionally you will find that the four orange centers are in place, but some of the side centers are still out of place. You can swap centers between two adjacent side faces by turning the whole cube so that the faces you want to swap become the Up and Front faces. Make sure the centers are aligned as in one of the diagrams above, and do the usual swap. The ugliest case is when centers need to swap between opposite faces (e.g. one green center is on the mostly blue face, and vice versa. The easiest way to fix this is to push the green center from the blue face into the Up face (using the third routine above), place the green center on its correct face (pushing the fourth blue center to the Up face), and then placing the blue center (which refixes the orange face).

The general idea is to match up two pairs of edges at a time (it's too slow to match only one at a time). Dave Baum's solution page has a neat eight-move tricycle of edges,

Since it takes time to search for the correct edge and bring it to the correct spot, we can always bring an edge to FRd which is the partner of UFr, so that the tricycle will match up UF and UR (and probably not FD). If the partner of UFr is at FRu instead of FRd (first diagram below), we do the usual adjustment to put the pieces in the correct position (second diagram), then do the same tricycle, which results again in two matched edges (third diagram). Note that the edge at UFr (blue-red in the diagrams below) appears to flip over as it rotates 90 degrees clockwise from UFr to FRu. Once you have UFl and FDr in position, look for the partners of UFr and FDl at the same time and bring the first one you see to FR, then adjust if needed and do the tricycle.

Frequently we are going to end up with two pairs of edges left to match up on the last iteration (sometimes more than one such pair may even occur). The first diagram below shows a typical case where one two edge pairs remain to match up. If they are in position so that the edge tricycle would fix one, we want to do our usual three move adjustment, which switches the relative positions of the two Front-Down edges (second diagram below). Now we do the routine shown below center, 12R U 4B L 4D F 34L, which matches up the pairs correctly as shown in the third diagram (it also moves some complete edges and corners around, but we don't care about those yet).

At this point the 4x4x4 cube has been reduced to a 3x3x3 cube, and can be solved by the 3x3x3 methods we have already learned (make only single outer layer and double inner slice turns, treating each pair of double inner layers as a single unit). There are two potential edge problems which may come up near the very end (each occurs independently about half the time, so about one time in four there are no problems and one time in four we need both fixes). If we see that an odd number of edges need to be flipped, apply the Single Edge Flip to flip the UF edge. This takes a little work to memorize, but there is a nice pattern to the moves. If we see that the edges cannot all be placed using our normal 3x3x3 edge tricycle or double swap, it must be the case that two edges need to be swapped. The Single Edge Swap (another nice palindrome) is quick and easy to learn, and swaps UF and UB. Once the parity problems have been fixed, any 3x3x3 method for finishing the last layer will work. These routines are widely known; I learned them from Dave Baum's page. If you need to swap two adjacent edges, swap one of them with the opposite edge using the Single Edge Swap, then do the standard edge tricycle. [An alternate version of the Single Edge Swap,

A version of the Single Edge Flip will be used on the 5x5x5 and larger cubes; it can be modified to flip any symmetric parts of a single edge. The Single Edge Swap will be used on 6x6x6 and larger cubes of even order only; it is not needed on 5x5x5 and actually has a side effect there of swapping two opposite centers.

I learned to solve most of the older twisting puzzles back in the 1980's before speedcubing was commonplace. I never worked with a highly lubricated cube until recently, and have not learned any of the finger tricks that experts use nowadays. I tend to turn hard, and sometimes cut corners. On a Rubik's Cube, which has a very robust design, this is no big deal, since the worst that is likely to happen is that some pieces will pop out and perhaps the whole cube will come apart. On rare occasions I have twisted a single corner by accident, but in almost 40 years I have never broken a 3x3x3 cube. For a long time I could not found a 3x3x3 designed for speedcubing that I liked all that much; my best results were with lubricated Deluxe Rubik's Cubes (the tiled model made by Ideal in 1982). Now there are a number of good stickerless 3x3x3's.

Other puzzles are a different story. I did all right with the V-Cubes: the 5x5x5 is excellent and I have never had any problems, and have only once or twice had a center pop out of a 7x7x7. The 6x6x6, which tends to stick along the center axes (which must have something to do with having even order), has come apart many times, and it is not easy to reassemble -- the internal mechanism is quite complicated -- but I have not broken it. I had a tiled 5x5x5 become too loose to use and could never figure out how to get it tightened again. I now prefer the stickerless cubes: I have a 4x4x4 and 5x5x5 made by Mo Fang Ge and a 7x7x7 by 55Cube.

I have broken an Ideal 2x2x2 or two (I prefer the larger 2x2x2's by V-Cube and others), but my worst luck has been with 4x4x4's. I tried at least eight different models of 4x4x4's, from the original Ideal Rubik's Revenge to models by various Asian manufacturers. I have broken several of these, some of which have corners which are glued onto the interior mechanism (though these can be glued back on). Even worse, some have eight interior face plates (you might call these

[This was my first published solution, which appeared in the inaugural issue of WGR in November of 1983. It has been considerably edited for this booklet.]

Pyraminx is a tetrahedral twisting puzzle devised by a German inventor, Uwe Mèffert, and originally manufactured by Tomy Corporation. Though conceived earlier, it was only manufactured in the wake of Rubik's Cube. Pyraminx is composed of 14 visible parts: four small corners (sometimes called

The diagram above is a flattened aerial view (with the Down face hidden), showing the various parts of the Pyraminx. The large corners are shown in darker shades. Note that, from a vertical view, you can see all three facets of the Up corner and two of the three facets of the Left, Back, and Right corners. You can see both facets of the Left-Right, Front-Left, and Front-Right edges, and only one facet of the Front-Down, Left-Down, and Right-Down faces.

Shortly after the puzzle was manufactured, published solutions began to appear, some extremely difficult or tedious. The most efficient (claiming 28 moves) of these was by Benjamin L. Schwartz (Pyraminx - An Improved Solution,

I believe that the solution we present here is reasonably efficient, and is fairly easy to learn and use. Two tricks, explained at the end, add some complexity, but bring the maximum number of turns required down to 27. We will use the notation above. The four large corners (from now on simply called corners) are called up, left, right, and back. Edge locations are named by the capitalized initials of the two faces they lie on. Edges are named by the two faces they should lie on when Pyraminx is solved, in lower case initials (fd is the edge which belongs at location FD). The faces are called Front, Left, Right, and Down. Clockwise turns of the four corners are named by capital letters U, L, R, and B. We will also use a turn of the base (the part of the Pyraminx which is not part of the Up corner), holding the Up corner in place. This is called D (Down). The five corresponding anticlockwise turns are named by lower case letters (u, l, r, b, and d). We only use the D and d turns during Phase 2, to keep the position of the Up corner (and its two adjacent edges) fixed. We will only need 8 turns in Phases 3 through 5.

We will solve Pyraminx in five phases. Phase 1 turns the small corners so that their colors match the corners they are attached to. Phase 2 puts the two front edges fl and fr in place. Phase 3 turns the left and right corners, and Phase 4 finishes the front face by placing the edge fd. Phase 5 reorients the Pyraminx so the front face becomes the down face, turns the up corner, and finishes the solution by simultaneously placing the fl, fr, and lr edges.

Phase 1 is the easiest one. Simply turn each tip (small corner), if necessary, in the correct direction, so that its three colors match those of the corner it is connected to. After all four small corners are turned correctly, Phase 1 is finished, and we will not need to turn the small corners again. Phase 1 can take as many as four turns. (I usually don't bother twisting the small corners when scrambling the Pyraminx, since this phase is trivial anyway).

In Phase 2, we place the correct edges at FL and FR so that their colors match the up corner. We will assume that the Up corner is correctly placed. The color of the Up corner which shows on the Front face will be called the front color. For the rest of the solution, we will assume that the front color is blue, but of course you may choose any of the four colors in an actual solution. First find fl, the edge which belongs at FL (check the colors of the Up corner on the Front and Left faces -- blue and orange in our example -- and find the edge which has those colors). If fl is already at FL and it shows blue on the Front face, continue with the edge fr. If not, we still have work to do.

We need to get fl to one of the two target positions (above left) from which it can be placed correctly by turning the Left corner. We need to get fl either to FD with its blue side on the Down face (from where an l turn puts it in place), or to LD with its blue side on the left face (from where an L turn puts it in place). See the diagram above. If fl is at LR or FR, or already at FL but with the colors flipped, make a turn (B from LR, r from FR, or L from FL) which puts it at FD or LD. Half of the time, B from LR or r from FR will put it in a good position directly. Once fl is in the bottom layer, turn the base if needed so that fl goes into one of the two good positions described above. The base turn is always necessary always when fl starts at FR, and is also needed when fl is at FD or LD, but with blue on the wrong face. Once fl is in the correct position, make the Left turn that places fl in FL. The process of placing fl may take as many as three turns. The edge fr is now placed in the same way. The two key positions to aim for (above right) are FD with its blue side on the Down face (R now puts fr in place), and RD with its blue side on the Right face (r now puts fr in place). If fr is at LR, turn r to put it at FR (where either r or dr puts it in place). If it is at FR but flipped, rdR fixes it. If it is already in the Down layer, turn that layer if needed to put it in a good position and then turn R or r. Placing fr can also take up to three turns. Phase 2 has now been completed, taking a maximum of six turns. The Front face is now all blue except for the bottom row.

In Phase 3, we turn the Left and Right corners so that they also show blue on the Front face. But we must find which face is really the Front face. Look at the Left, Right, and Back corners. Two of these will have blue on them. The face these two corners share is the

First let us handle the case where the current front face is correct. Now we need to turn the left and right corners so that they show blue on the front face, but without displacing the edges from FL and FR. We don't care yet which edge is the third edge on the Up corner; we show this edge in gray in the examples above. We must turn the Up corner so that this gray edge is on the Left or Right corner which is being fixed, so that turning the Left or Right corner does not disrupt one of the two edges already placed correctly on the Up corner. If the Right corner needs to be turned, make turn U first (getting fr out of the way), and turn R or r (whichever brings the blue side of the right corner to the front side). Now if the left corner needs to be turned, turn U again (getting fr out of the way again), turn L or l, and turn U a third time to finish phase 3. The diagram above left shows an example. If only one (Left or Right) corner originally needed to be turned, only two turns of the Up corner are needed -- the first away from the corner needing to be fixed, and the second in the opposite direction (uLU or ulU to fix the Left corner, Uru or URu to fix the Right corner). [When you need to fix both corners, you can actually fix either one first; the upper left diagram can also be solved by the sequence uluru.] Now we handle the case where the Left or Right face should really be the front face. In either of these cases, we turn the whole Pyraminx so that the other two blue corners are Left and Right, regardless of whether their blue facets are on the Front face or not. Let us assume that it is the Left face. After we turn the whole puzzle, the blue portion of the Up corner is on the Right face. We should fix the Left corner first if necessary, since the Up corner is already in the correct position to do so.

If the Right corner needs to be fixed, turn u, fix the Right corner, then turn u again to finish. If Right does not need to be fixed, turn U to finish. See an example in the above center diagram. Possible sequences are LU, lU, Luru, LuRu, uru, or uRu (you don't need to memorize these, just understand the process).

If the Right Face is the correct Front face, turn the whole puzzle again, putting the blue portion of the Up corner on the Left face. Now we should fix the Right corner first if necessary. If Left needs to be fixed, turn U, fix Left, and U again. If Left is correct, turn u to finish. Possible sequences are Ru, ru, RUlU, RULU, UlU, or ULU.

Summarizing the procedure for Phase 3: find the correct Front face, hold the pyramid so that the correct face is on the Front, turn Left and Right corners to show the Front color (blue in our example) on that face, moving the Up corner out of the way as necessary. The Front face should have eight (or nine if we are lucky) of its facets showing the same color. Phase 3 can take up to five turns. In one case out of eight, we are lucky and FD is correct too, and we can go directly to the last step, Phase 5. If not, we go to Phase 4 to place fd at FD.

In Phase 4, we need to find and place edge fd. If fd is at LD or RD with blue on the Down face, turn b or B to get it to LR. If fd is at LD with blue on the Left face, place it with the sequence of turns rBR. If fd is at RD with blue on the Right face, place it with sequence Lbl (the mirror image of rBR). If fd started at LR, or you moved it there as instructed, you can place it with LBl if its blue side is on the Left face, or with rbR if its blue side is on the Right face.

If fd is at FD but flipped (i.e., with blue on the Down face), fix it by turning rBRLBl. The diagram above summarizes these sequences. Phase 4 is completed in a maximum of six turns.

Finally we come to Phase 5. The Front face is all blue, so we are finished with it. Turn the entire Pyraminx so that the blue face becomes the new Down face. Now turn the new Up corner so that its colors match those of the base. We are faced with four possibilities. We may be finished solving (an 11-to-1 longshot). If so, mix it up and solve it again -- practice makes perfect. If we are not so lucky, we need to find out whether we need to move the last three edges (FL, FR, LR) cyclically, and whether two of the edges need to be flipped. The easiest way to tell whether an edge needs to be flipped is to observe whether either of its facets match the color of the adjacent Up corner facet.

If one of the three edges is completely correct, the other two need to be flipped (this happens about a quarter of the time). If none of the three edges is correct, then a cyclic exchange of the three edges is needed (a tricycle, which happens about 1 time in 6). Decide which direction this cycle must be in by visualizing whether the clockwise turn U or the anticlockwise turn u will put at least one of the edges in the correct place with respect to the base. If it is U, you need a clockwise tricycle. If it is u, you need an anticlockwise one. If all three edges are put in the right place by your trial turn, you do not need to flip any of them. If only one edge is put right, you need to flip the other two. If two of the edges need to be flipped (whether you need a tricycle or not), hold the Pyraminx so that they are at FL and FR.

Now we need five sequences. The first sequence flips the edges at FL and FR without a tricycle. This sequence is rUluLuRU. This is a somewhat longer sequence than we have seen so far, but it is not hard to memorize, since it has a natural rhythm similar to the corner tricycle used in solving Rubik's Cube. It is actually another example of an monoflip. The second and third sequences perform tricycles without flipping edges. The clockwise tricycle is ruRuruR. The anticlockwise one is rURUrUR. Note that these sequences are identical except for the direction of the Up turn, and you don't even have to remember which is which; just make the first Up turn in the direction which will cause the second Right turn to put an edge in its correct slot, and keep turning the Up corner in the same direction; each Right turn (after the first one) will place an edge correctly. These three crucial sequences are shown in the diagrams above. In fact, you now have enough sequences to solve the whole Pyraminx: the rest of the solution consists of shortcuts.

The fourth and fifth sequences are not absolutely necessary, since they handle the cases which can be be solved using a tricycle followed by a double edge flip. But they are short and not too hard to remember, and considerably shorten the solution when you need to perform a tricycle and flip the edges FL and FR simultaneously (which happens about half the time). When you are in one of these situations, hold the Pyraminx so that the edge which does not need to be flipped is at LR (note again that the edges which need to be flipped will each have one of their facets the same color as the face it is on). The clockwise tricycle is

If you wish to shave a few more moves off of 30, try these two shortcuts. They were found by looking at the worst situations in Phases 2 and 4, and looking for shorter ways to handle them. The first one is fairly easy, and cuts Phase 2 from six turns to five. There are three positions for fl which take three turns to get to FL correctly. The first is FL, flipped (Ldl). The second is FR with the front color on the right face (rdL). The third is LR with the front color on the left side (BDl). Similar positions exist for fr. Phase 2 takes six moves when both fl and fr are in bad positions. There are three such situations. If fl and fr are in the correct locations, but flipped, RLdRL will put them both into the correct positions. In the other two situations, one of the two is at LR and badly oriented, and the other is in its correct location but flipped. If it is fl which is at LR and fr flipped, use the sequence rbDrL. If fr is at LR and fl is flipped, use LBdLr (this is the mirror image of the previous sequence).

All three sequences (shown above) are based on the idea of moving both edges to the base and making only one down turn instead of two. All other positions can be handled in the usual way. If fl and fr are swapped and badly oriented, fixing fl will push fr into a good position, so fixing fr only takes two turns. If fl is at FR and fr is at LR, and both are oriented badly, fix fr first, thereby pushing fl into the base where it can be fixed in two turns. If fl is at LR and fr is at FL, fix fl first (you would do this anyway). Phase 2 can always be done in five turns or less.

The second shortcut, much more difficult, is based on the fact that Phase 4 takes six moves

In Phase 5A, there are seven possible positions. Turn the entire Pyraminx so that the completed portion (except for the flipped fd) is the base. Make the turn of the Up corner described in Phase 5, making its colors match the base. Use the method described for Phase 5 to determine whether a tricycle is needed and which edges have to be flipped (now there will either two or four, since we know fd must be flipped, and the total edges to be flipped must be an even number for parity reasons). We need seven sequences.

The first sequence (not shown) we need is the standard double edge flip we've already seen in the standard Phase 5, which we will use here whenever two adjacent edges need to be flipped. Simply hold the Pyraminx so that the edges which need to be flipped are at FL and FR. If two edges need to be flipped which are not adjacent, we need an alternate sequence, luLuRUrU (above left), which flips edges at FD and LR.

A third sequence, the Tetraflip (above second left), flips all four remaining edges (LD, RD, FD, and LR) without a tricycle: the moves are ULuRuLuRLuR (note that this is held in sort of the inverse position of the standard double edge flip: the two edges not to be flipped are where the two edges flipped by the standard double-edge flip would be). This is a little trickier to memorize, but notice the first Up turn is clockwise and the rest are anticlockwise; the clockwise left and right turns alternate, with an Up turn in between each except for the little hiccup near the end, where the Up turn skips a beat.

The Tetraflip was derived by combining a modified double-edge flip with the Hexaflip (above third left), a pretty pattern with a nice sequence, which flips all six edges (we don't need Hexaflip directly in our solution method). Since the double-edge flip is symmetric, we can use its mirror image; since its effect is its own inverse, we can uses its inverse sequence: if we take the mirror image of rUluLuRU, we get LuRUrUlu. Reversing it gives ULuRurUl; appending the Hexaflip gives ULuRurUlLuRLuRLuR; the six underlined moves undo themselves and can be eliminated, giving the 11-move sequence shown. [There are shorter 10-move tetraflips, but they are harder to remember and perform quickly. A tetraflip position in which the two unflipped edges are not adjacent is also possible, but we don't need it in our method.]

Another pretty pattern (above right) twists all four large corners and rotates three of them around the Front face. This, along with the Hexaflip, might be useful in starting to mix up the Pyraminx before solving, just as Pons Asinorum is with Rubik's Cube.

The remaining four sequences perform tricycles and flip two or four edges simultaneously. When you have determined in which direction the tricycle must be made, and how many edges need to be flipped, select the correct sequence.

The four sequences, diagrammed above, are :

(4) clockwise tricycle, flip fd fl fr lr -

(5) anticlockwise tricycle, flip fd fl fr lr -

(6) clockwise tricycle, flip fd and fl -

(7) anticlockwise tricycle, flip fd and fr -

Note that when using the last two sequences, you may have to turn the up corner before using the sequence, in order to get the edge which needs to be flipped to either fl or fr, as required. If so, the last Up turn may be unnecessary, or in the opposite direction from that indicated (it will be obvious in which direction to make the last turn) in order to finish the solution.

This solution, with practice, will allow you to solve Pyraminx in well under 1 minute. The 30 move version is probably better for speed, until you become experienced with the shortcuts.

[The original version of this solution was published in WGR2 in 1984. I later received a copy of Dr. Kurt Endl's book

The

This method for solving the Skewb will take a maximum of 40 moves. Phase 1 takes up to 13 moves, phase 2 up to 7, and phase 3 up to 20 (when five diamonds need to be swapped). The average length is about 27 moves. The Skewb can be twisted into 4!4!6!3^8/2*2*2*9*12, or 3,149,280 possible positions. Although it has more non-trivial positions and a longer solution method than the Pyraminx, the Skewb is actually slightly easier to solve, because there is less to memorize. The bulk of the moves are repetitions and combinations of four-move commutator sequences. When I'm in practice I can get times usually under 30 seconds.

Turns will be designated by the lower corner being turned, using a capital letter for a clockwise turn or a lower case letter for an anticlockwise turn. Like most twisting puzzles, the colors of the Skewb are not standardized. I have three Skewbs with different color arrangements. The pattern we will be showing here has yellow opposite white, blue opposite green, and orange opposite pink.

Because it is much easier to do Front and Right turns, we will use them almost exclusively. We add one more notation, using % to designate a 180 degree rotation of the entire cube around the Up/Down axis (picture a globe spinning on its axis), so that the Right face exchanges with the Left and the Front face with the Back (this is equivalent to [U2] on Rubik's Cube). The solution proceeds in three phases. Phase 1 places the four upper corners (i.e. correctly oriented, with the proper facets on the up face): see the diagram above left; gray areas are the portions we don't care about yet. Phase 2 orients the four lower corners, which were automatically put in the correct positions by Phase 1. Phase 3 makes any necessary exchanges of the diamond-shaped centers. Many of the move sequences will be built from the powerful four-move commutator fRFr.

Look at the color of the diamond on the up face (you may turn the Skewb so that any color is face up, so let's say that it is white). We want to put all four corners which have white facets (we will call these white corners) in the Up layer so that their other facets match in color. If any of the Up corners already have a white facet on the Up face, we can consider that corner to be already correct, and turn the whole Skewb so that corner is at the ULB (Up Left Back) position and proceed to the second corner (next section). Otherwise, we need to find a corner which contains white, preferably one in the Down layer but whose white facet is not on the Down face. If we have a white corner in the Down layer with its white facet on one of the side faces, hold it at FRD. If its white facet is on the Front face, turn R (clockwise) to put it at URB; if its white facet is on the Right face, turn l (anticlockwise) to put it at ULF. Proceed to the second corner.

Otherwise, if all of the corners are either in the Up layer but with their white facets not Up, or in the Down layer with their white facets Down, we can twist any of the four lower corners which contains a white corner to bring it into a usable position. Specifically, if a white corner is at URF, turn F if the white facet is on the front, or f (anticlockwise) if the white facet is on the right. If a white corner is at FRD with its white facet down, turn F. Now we have a good corner, which we can hold at URF (again turning the whole Skewb as required, keeping white Up), and turn as already described (R to URB or l to ULF). At worst the first corner of Phase 1 should be placed in two turns.

Above is a diagram showing every possible position and orientation of a corner to be placed in the second, third, and fourth steps of Phase 1. Each location shows the correct move to be made and an arrow pointing to the resulting position. The goal position is in the middle of the second column. The two easy positions which can be brought to the goal position are directly above and below it. The key positions to remember, however, are to the left and right of the goal. Any position which is not already correct (or an easy single turn) can be brought to one of these two positions in one move. Then at most three turns will put the corner in the goal position and restore any momentarily displaced corners.

Reorient the entire Skewb, keeping the white diamond up, so that the correctly placed corner is at ULB.

The second corner needs to be placed at URF, diagonally across from the first corner. It has a white facet and two facets of different colors than the first correctly placed corner. This corner can only be at one of three positions: URF already, FLD, or RBD. Unless the white facet of the corner we want is on the Up, Left or Back face, we want to get it to one of two positions: on the front of FLD or the right of RBD (second row of the diagram above).

(A) If the white facet is in one of the two positions (left of FLD or back of RBD) where a single turn will put it in place, turn F or f to do so, and advance to the third corner. If the white facet is on the Down side of either FLD or RBD, turn the Left corner clockwise or the Right corner anticlockwise, respectively, so it goes into one of the single-turn positions just mentioned; this is a shortcut only usable for the second corner, and is shown with red arrows on the diagram above.

(B) In the event that it is at URF with its white facet Up, it is already correctly positioned and you can advance to placing the third corner. If it is at URF with its white facet on the Front face, turn F to put it on the right of RBD. If it is at URF with its white facet on the Right face, turn f to put it on the front of FLD. Continue to Case (C) in either case.

(C) The white facet either was already in one of the two key positions, or you have made one turn to get it there. Now if it is on the front of FLD, turn lF to put it in place (the extra L turn shown in the diagram is not needed when placing the second corner, since the anticlockwise l turn did not displace a corner). If it is on the right of RBD, turn Rf to place it (again the extra r is not needed for the second corner). The shorter sequences for the second corner only are shown in red.

Placing the second corner will take at most three turns. (It's not absolutely necessary to do place the diagonally opposite corner second: if you can see that one of the adjacent corners can be placed in one move, you can go ahead and place it. This may also shorten the third corner procedure by one move, if it goes to the key position where a two-move turn can be used.)

Orient the Skewb again so that the two corners already placed are at UFL and UBR. (This can be done in two ways; with experience you will be able to spot which of the two remaining corners is better placed and do it as the third corner, orienting so its target location is UFR). Find the corner which belongs at UFR (its colors will be those of the up diamond (white), the front facet of the upper left corner, and the right facet of the upper right corner). Occasionally it will already have been placed correctly at UFR, and you can proceed to the fourth corner. Otherwise, with two exceptions, follow the same procedure as for the second corner, making a single turn either to place it directly in the correct position or to reach one of the two key positions. Once it is in a key position, use the full three-move turn of lFL or Rfr to place it correctly, and proceed to the fourth corner.

If the white facet is on the Down side of either FLD or RBD, turn the Front corner so it goes to the opposite position, so that the white facet is in one of the two key positions, on the front of FLD or the right of RBD. Finish with the corresponding three-move sequence, and proceed to the fourth corner. If the corner is at ULB, and its white facet is on the Left or Back face, it takes only two turns to get it in the correct position. Turn the Back corner clockwise if the white facet is on the back, or anticlockwise if it is on the left. This puts it in one of the single-turn positions we have already seen. The worst case is if the white facet in on the Up face. In this case, turn the Back face in either direction to place it in one of the two key positions, and place it as usual.

Placing the third corner will take at most four turns.

Orient the Skewb once more so that the Up corner yet to be placed belongs at UFR, and find that corner at either the upper front or the lower left or right. Follow the same procedure used to place the third corner, and you will have completed phase 1.

Placing the fourth corner will take at most four turns, for a total of 2 + 3 + 4 + 4 = 13 turns for Phase 1.

Phase 2 is the shortest and easiest, requiring a single maneuver of either four or seven turns. First determine the color which belongs on the Down face. This is the color common to all four of the lower corners (which have been placed, but not necessarily oriented, by phase 1). It is also the color opposite the Up face color -- memorizing the colors which are opposite each other on your unscrambled Skewb is helpful (on mine, they are blue-green, yellow-white, pink-orange). In our example, the Down color is yellow. Note that the diamond on the down face is usually not the correct color. Nevertheless, determine the direction in which each lower corner must be twisted to put its yellow facet on the down face. There are three cases, one of which is trivial (in case (a), none of the corners need to be twisted, so phase 2 is finished). You can flip the Skewb over to see the Down face more easily, but it is important to remember, once you are ready to perform either of the sequences shown, to hold the Skewb so that the corners are all still in the Down layer (this allows us to continue turning the lower corners only, in this case F and R). The diagrams below show a view from underneath the Skewb.

In case (b), all four corners need to be twisted, two clockwise and two anticlockwise. Orient the Skewb so that the corners needing to be twisted clockwise are at the Down Left Front and Back, and those needing to be twisted anticlockwise are at the Down Right Front and Back. Learn to recognize the pattern of the four yellow facets: note that two of the yellow facets start on the Front face (diagram below left). Now do

In case (c), two of the corners do not need to be twisted. The other two (diagonally opposite each other) need to be twisted in opposite directions. In this case, orient the Skewb so that the corner which needs to be twisted clockwise is at BDR, and the one which needs to be twisted anticlockwise is at the FDL. Note again the pattern of the yellow facets: two are on the Down layer, one on the left, and one on the Back (diagram below center). Complete phase 2 by doing the maneuver

Phase 3 -- Swapping the remaining face centers (diamonds)

Now all eight corners are in place, and their colors determine where each diamond belongs. At least one diamond (the white one) is in the correct place, but three, four, or five may need to be swapped to complete phase 3. Again we will consider the possibilities one case at a time. In case (a), none of the diamonds need to be swapped (a 1/60 chance!), and the Skewb is solved. In cases (b), (c), and (d), three, four, or all five diamonds need to be swapped. In case (b) or (c) a single sequence will complete the solution. In case (d), we use the first sequence for case (b) to fix one or two of the diamonds (depending on the exact configuration) and put us in case (b) or (c), where a second sequence completes the solution. Phase 3 may take as many as 20 moves in case (d). The diagrams below are shown from the same point of view as the mesons, with the Down face in the middle and the Front face on the top.

In case (b), three diamonds need to be swapped. There are three subcases. In (b1), the three diamonds are side by side, and the Skewb can be oriented so that the left diamond needs to go to the down face, the down diamond needs to go to the right, and the right diamond needs to go to the left. Now do (fRFr%)2, an important set of moves we call the Center Tricycle, shown in the diagram below left).

In (b2) and (b3), the three diamonds are positioned on mutually adjacent faces, and need to be cycled either clockwise or anticlockwise. We are going to fix either case with a conjugation of the Center Tricycle. Orient the Skewb so that the Down diamond is wrong, and the face it needs to go to is Left. The third wrong face will either be Front or Back. If it is Front, do the Center Tricycle except that the first turn is clockwise instead of anticlockwise, then finish with another F turn (this is equivalent to the conjugation f(fRFr%)2F, but the two initial f turns become an F turn). This is shown below middle.

If the third face is Back, do the conjugation R(fRFr%)2r (below right). Note that in either of the last two cases, it will be obvious how to make the final turn to complete the solution.

In case (c), four diamonds need to be swapped, in pairs. There are five subcases. If the two good diamonds are on adjacent faces, orient the whole Skewb so that the white face is the Up face and the other correct face is Left. Depending on where the Front diamond needs to go (Back, Right, or Down), perform the correct sequence to fix the remaining diamonds. The Front-Back swap is a threefold repetition of the commutator we have already used in mesons and tricycles. The Front-Right swap uses a different commutator where the first two moves of each quartet are clockwise. In the Front-Down swap the turns are backwards (Right first), and the first two moves are anticlockwise. Note that in four-diamond swaps, we are repeating the commutator without reorienting the Skewb in between repetitions.

If the two good diamonds are on opposite faces (white and yellow), then orient the whole Skewb so that the Front diamond needs to swap with either Back or Right. Now we need to conjugate the second Double Bicycle sequence (FRfr)3, by making one preliminary turn to put the four diamonds into the shape we want (shown in the diagrams below). If the pairs which need to be swapped are opposite, do a clockwise Left turn first and an anticlockwise Left turn at the end. If the pairs are adjacent, do an anticlockwise Back turn first and a clockwise Back turn at the end. Once again, the final turn will be obvious.

In case (d), five diamonds need to be swapped. There are actually six different possibilities here, but we will simplify the problem by doing it in two steps. Orient the Skewb so that the white diamond (which is in the correct place) is on the up face, and the diamond on the Down face needs to be moved to the Right face. Now do the Center Tricycle you already know, which corrects one or two diamonds, and puts you into case (b) (three diamonds to swap) or (c) (four diamonds to swap). After doing (b) or (c), the Skewb will be solved.

Here are a few more sequences. The first two were used in our original 1984 solution, which twisted corners in Phase 2 without disrupting centers. The meson twists two opposite corners on the Up face (I would now do this as (FR)3[U](rf)3). The double meson twists all four corners on the Down face; it's actually two-thirds of a Double Bicycle. The third sequence is a center pentacycle from one of David Joyner's web pages. It moves the Front diamond to the Down face, the Down diamond to the Left face, and the other three diamonds around the sides in order. I looked at whether this could be used with conjugations to handle case (d) of Phase 3 directly, but it is slow and unwieldy to perform (requiring the solver to switch hands constantly), and only a tiny bit shorter than the two-stage solution, even if unconjugated.

The Orb is a puzzle invented by British designer Chris Wiggs and manufactured in the U.S. by Parker Brothers (it is now out of production). The Orb looks somewhat like a world globe, with four circular channels cut in it at roughly the locations of the Arctic and Antarctic Circles and the Tropics of Cancer and Capricorn (photo above left). The upper and lower circles contain eight beads each, and the two middle circles twenty each. The Orb is divided into two hemispheres, divided by a great circle which runs through both poles. The West and East hemispheres can rotate relative to each other in such a way that the ends of the four circles can match up in eight different ways: Two ways (including the start position) produce four tracks of 8, 8, 20, and 20 beads (above far left); two ways produce two tracks of 28 beads (above center left); and four ways produce one track of 56 beads (above center right). In the Start position, each of the four tracks contain beads of one color (e.g. blue and green on the top and bottom tracks, red and yellow on the middle tracks). The Orb is mixed up by turning the halves in various positions and pushing the rows of beads along the tracks (the photo above far right shows a well-mixed Orb). The object of the puzzle, as usual, is to unscramble the puzzle, getting all of the beads separated into four tracks, one of each color. The total number of positions is 56!/8!8!20!20!, or 73,888,773,475,012,113,089,523,051,000. This is a much larger number than Rubik's Cube, but the Orb is nevertheless quite a bit easier to solve.

There are four phases to the solution. In Phase 1, one of the colors of which there are eight beads (we will use blue) are brought one by one to the left half of the upper middle track. In Phase 2, the Orb is flipped over so that the blue beads are now in the lower middle track, and the other set of eight (green) beads are brought to the left half of the upper middle track. In Phase 3, the green and blue beads are put into the upper and lower tracks respectively. In Phase 4, the remaining red and yellow beads (sets of 20) are exchanged until one middle circle is entirely red and the other is all yellow. Hereafter,

Phase 1: We will move (usually) one blue bead at a time to the centermost position on the left half of the UM, each time sliding the blue beads already there one position towards the back. We want to collect all eight blue beads together, so start by finding the largest group of consecutive blue beads (there may not be more than one if the Orb is well-mixed) in either middle track (if they are in LM, flip the Orb over so they are in UM), and slide the UM track so they are immediately to the left of the great circle at the front. Now find another blue bead somewhere on the Orb. The easiest case is when a blue bead is on LM, so you should do those first, until there are none left. Slide LM until the blue bead is just to the right of the great circle, on the

You should eventually have no blue beads in the two middle tracks except for the continuous group you have collected so far. There may still be beads in the U and L tracks. To move a bead from U, slide that track until the blue bead is just to the right of the great circle, on the

To move a bead from L, slide L so that it is immediately right of the great circle on the

Phase 2 is a little more complicated, since we can no longer freely slide LM without disrupting the blue beads. Flip the entire Orb over so what used to be the UM is now LM, and vice versa. All of the blue beads should now be on the left half of LM, running from the back of the great circle, to within two beads of the front of the great circle. We are going to repeat Phase 1, substituting green for blue, but we are going to transfer all of the green beads from U or L to the left half of UM. Once again we should find the largest group of adjacent green beads in UM, and slide them just to the left of the great circle. Now any green beads already in U or L can be moved just as we did in Phase 1, but after every transfer of a green bead,

If all of the unconnected green beads are now in the two middle tracks, we need to get them one or two at a time into U. If a green bead is in the right half of UM or LM, twist 90 degrees (two clicks) clockwise, and slide the combined track until the bead goes into the left half of U, then twist 90 degrees anticlockwise to return to normal four-track position. Now move from U to UM as normal.

An extra maneuver is needed to get a green bead out of the left half of LM or UM. If a green bead is one of the first two on the left half of LM, slide LM to the right so that the green bead crosses into the right half of LM. Twist 90 degrees clockwise and slide the green bead into the left half of U, twist 90 degrees anticlockwise, and return the blue beads to their correct LM position. If a green bead is in the left half of UM but separated from the main group of green beads collected so far, slide UM so that the green bead crosses the great circle onto the right half. Now twist 90 degrees anticlockwise, slide the green bead into the left half of U, twist 90 degrees clockwise, and slide the green beads back into the correct position in UM. When all eight green beads have been placed in UM, running from the front of the great circle to within two beads of the back of the great circle, Phase 2 is finished.

Phase 3: Twist the Orb two clicks anticlockwise (90 degrees) so that the right half of U is aligned with the left halves of UM and LM. Slide four green beads into the right half of U, which automatically also slides four blue beads into half of L. Twist 90 degrees clockwise to return to normal position, and slide both U and L four positions, putting four green beads into the left half of U and four blue beads into the left half of L. Repeat the procedure to put the remaining four greens and blue into U and L (it's not necessary to slide U and L a second time). You will find that the green beads now occupy the entire U track, and the blues occupy the L track. The photos below show the sequence of moves.

[There is an alternate way to move all eight greens into U and all eight blues into L at once, but it requires putting all 56 beads into a single track. Twist one click anticlockwise, slide the track 8 positions so that the green beads all go into the left and right halves of U (see photograph below), and twist the Orb back into standard position. You may find this method faster if your Orb slides more smoothly than mine.]

Phase 4: Beads will now be exchanged between UM and LM, without messing up U or L. We will move the red beads to UM, and the yellow beads to LM. (If the majority of UM is yellow, it is better to do the opposite.) We will do a series of exchanges, trading a red bead in LM for a yellow in UM. Rotate UM so that one yellow bead lies directly to the right of the great circle on the front. Rotate LM so that one red bead lies directly to the left of the great circle on the front. Twist the Orb 180 degrees, rotate LM one position to the right, and twist the Orb 180 degrees again. This exchanges the two beads. Yellow and red beads can also be exchanged more than one at a time, by placing equal numbers of yellow beads in UM and red beads in LM at the correct positions, and shifting LM that number of positions to the right between the two 180 degree twists (see photos below). After a number of repetitions of this maneuver, the Orb will be completed. With a little practice, you should be able to solve the Orb in under 2 minutes.

The ImpossiBall is a twisting puzzle invented by William O. Gustafson in 1981, and also known as IncrediBall. The ImpossiBall is a rounded icosahedron, consisting of 20 triangular pieces fitted together into a spherical shape. Each triangle has three different colors on its three corners. There are six colors in all (e.g., red, orange, green, blue, yellow, and white). The pieces turn five at a time, with a flexing motion. There are twelve groups of pieces (called faces) which can be turned, like the twelve faces of a dodecahedron. In the Start position, the circle in the middle of each face is a single solid color. Pairs of opposite faces have circles of the same color. As usual, as the faces are turned at random, the colors become mixed up, and the object of the puzzle is to return a scrambled ball to its original position. For each of the 10 combinations of three colors, there are two pieces with those colors -- one with the colors in clockwise order on its corners, the other with the colors in anticlockwise order (so antipodal pairs are not interchangeable). The total number of combinations possible on the ImpossiBall is 20!*(3^20)/(6*60), or 23,563,902,142,421,896,679,424,000. The ImpossiBall was one of the first twisting puzzles without stickered pieces.

The solution will use 72 degree turns, clockwise and anticlockwise, of seven faces, called Up, Down, Front, Left, Right, West, and East. For completeness, the five remaining faces can be labelled Back, Europe (P), Africa (I), Asia (A), and Australia (S), but we do not use turns of any of these faces in the sequences used in the solution. These twelve faces are located as indicated in Diagram 1. Clockwise turns are indicated by the capitalized initial letter of the face being turned; anticlockwise turns are similarly indicated by lower case letters. A 2 following a letter indicates that the face should be turned 144 degrees in the direction indicated. A number following a group of turns in parentheses indicates that the group of turns should be performed that many times in succession. For conciseness, certain turn sequences are designated by the single letters C, H, M, X, and Y (C, M and Y have inverse sequences, designated c, m and y). A piece will be designated by the initials of the three faces it lies in (e.g. FLW).

The solution proceeds in five stages. Stage 1 places the five pieces of the front face, and turns that face to become the down face. Stage 2 places the five pieces adjacent to the down face. Stage 3 places the remaining five pieces of the equator, and turns the ball so that the face yet to be done is the front face. Stage 4 positions the five pieces of the front face. Stage 5 orients the five pieces of the front face to complete the solution. This solution relies heavily on the various diagrams below, so study them carefully.

Begin stage 1 with the scrambled ball in any orientation. Consider the piece at FLW to be correct, and use it as a guide to place the other four pieces of the front face. Look for the piece which can be placed at FWE (location shown in purple above). One of its edges must match the right edge of FLW, and its third color must not appear on FLW. Maneuver this piece to DWE without disturbing the front face. Check the Stage 1 diagram (above left) to see which of the three orientations it is in, and use the indicated sequence to put it in place. Now turn the entire ball, keeping the same face front, so that the piece just placed becomes the new FLW. The next piece to be placed must match the free edge of FLW, and its third color cannot appear on any piece already placed. Maneuver this piece to DWE without disturbing pieces already placed, use the sequence in the Stage 2 diagram (above right) matching its orientation, and turn the whole ball as before so that the newly placed piece is at FLW. Use the same process to place the last two pieces of the front face (the last piece to be placed must match the edges of FLW and FRE, and will require a slightly longer sequence, shown in red, for the first two cases).

Before beginning stage 2, turn the ball so that the solved face is now the down face. Now find the piece which belongs at FWE. It matches the top edge of DWE, and its third color is the same as the part of the back which lies on the down face. Maneuver this piece to FUL without disturbing the down face or any pieces already placed in stage 2. Check the orientation of the piece on Diagram 3, and use the indicated sequence to place it correctly. Now turn the entire ball, keeping the same face down, until the face at FWE is incorrect. Use the same procedure to place the remaining four pieces adjacent to the down face.

Stage 3 is somewhat similar to stage 1, but uses different sequences. Turn the entire ball upside down, so that the up face is now the correct one. Find a piece on the down face which does not have a corner the same color as the on the up face. (If no such piece exists, turn the ball so that the piece at FWE is wrong, and use the sequence eDE to bring this incorrect piece to the down face). Turn the down face so this piece is under its correct location, and turn the whole ball, keeping the same face up, so that the piece is at DWE and belongs at FWE. Check the orientation of the piece in the Stage 3 diagram (above), and use the indicated sequence to place the piece. Again use the same procedure four more times (if necessary) to complete the equator.

Before performing stage 4, turn the entire ball so that the face still to be solved is the front face. Turn the front face so that at least one of its pieces is in the correct location (regardless of orientation). Now check to see how many pieces are incorrectly positioned. There will be none, three, or four. (If there are three incorrectly placed, there is a way to turn the front face so that either H or X is reached (see the last four diagrams above)). Determine how the incorrectly positioned pieces must be swapped to make them all correct, and find a picture in the Stage 4 diagram which matches the situation (turning the entire ball as necessary). Use the indicated move sequence to place all of the pieces correctly. Note that the double bicycle sequences H and X form the basis for all of the sequences. Now only one more stage is required to finish the solution.

In stage 5, check to see which of the five front pieces need to be twisted, and in which direction, in order to orient them correctly. Find a picture in the diagram which matches the situation (again turning the entire ball as necessary). A plus sign (+) indicates a piece which needs to be turned clockwise; a minus sign (-) indicates a piece which needs to be turned anticlockwise. Note that there are three basic sequences: the meson M (and its inverse m), the baryon Y (and its inverse y), and the double meson C (and its inverse c). The baryon is actually Benson's Twisterflipper transported to the ImpossiBall. Other sequences are built from these six sequences. When the correct picture has been located, and the indicated sequence performed, the ImpossiBall will be solved. This solution takes a maximum of 21, 28, 40, 9, and 20 turns for the successive stages, and can thus solve any scrambled ball in 118 turns or fewer.

There is also a somewhat easier version of ImpossiBall with 12 different colors. This is solved in exactly the same way, but the pieces are easier to pick out since there are no mirror-image equivalents.

Even easier is the Kilominx, a new dodecahedral version, which has 12 colors and miniature centers in the middle of each face; it is essentially an edgeless Megaminx. The Kilominx also turns much more smoothly than the original Impossiball.

The Ivy Cube, designed by Eitan Cher, is essentially a Skewb with restricted movement: each face has a football-shaped center, and corner pieces are missing where three footballs meet. Only four corners can turn: Left and Right cut through the Up face, Front and Back cut through the Down face. The puzzle can be solved quickly: turn the Left and Right corners so their colors match on the Up face, and Front and Back so their colors match on the Down face. Only three center tricycles are needed to swap all the centers correctly: the Flower (second diagram above) rotates three centers anticlockwise via RlrL (clockwise lRLr); the Hexagon (third diagram, rotated 90 degrees to show the Right corner frontmost) rotates anticlockwise via LFRfrl (clockwise LRFrfl); the Wraparound (fourth diagram, Left face is green with blue center) swaps R==>U==>L==>R via RLrl (turn the puzzle 180 degrees if you need the reverse).

Dino Cube

The Dino Cube is a small and easy cubical puzzle with no corner or center pieces (photo above, scrambled). It consists of twelve edges, which can be turned three at a time around each of the cube's corners. I have a six-color version, which is easy to solve since the edges cannot be flipped: an edge in the correct place is automatically oriented correctly. There are two mirror-image solutions (e.g., if blue is Up and red is Front, Right can be either yellow or white). You can switch from one solution to its mirror image (swapping the Front and Back colors) by the elegant ten-turn sequence (FrFrF[F2])2, where F is a clockwise turn around the URF corner, r is an anticlockwise turn around the UBR corner, and [F2] is a half turn of the whole cube keeping the Front face fixed. Several listings on Amazon have the notation: (difficulty 8 of 10), which is absurd: this is one of the easiest twisting puzzles there is, ideal for beginners (probably only the Ivy Cube is easier).

Redi Cube

Another invention by Oskar Deventer is the Redi cube, in which the corners rotate, rather than the face layers. There are eight corners and 12 edges, but no centers. The corners are restricted to their original positions, and can only rotate. The edges can be freely interchanged (subject to the usual parity law which prevents a single pair of edges from being swapped), but their orientation is fixed by their position: an edge in the correct position is automatically flipped correctly. The Redi Cube is barely more difficult than the Ivy Cube or Dino Cube, since orienting corners is trivial, and edges can be drop-inserted (to finish two or three perpendicular layers) and swapped by simple three- and four-move tricycles (to place the last five edges).

Pyramorphix

What appears to be a trivial two-layer Pyraminx is actually a shapeshifting, disguised 2x2x2 cube, which turns in halves along four axes. We saw the larger three-layer version of this earlier among the Picture Cube variants. Pyramorphix, however, is easier to solve as a tetrahedral puzzle rather than by using a 2x2x2 method. The solution is in four easy stages:

(1) Return to its original shape

(2) Put the corners in their correct relative positions

(3) Twist corners as needed

(4) Exchange the centers

Master Pyraminx

The Master Pyraminx (above left, unscrambled) is the next largest member of the Pyraminx family. It has four centers, four (trivial) small corners, six wings, and twelve edges. Wings can be flipped in pairs: the photo above center shows the puzzle unscrambled except for two wings. Unlike its smaller cousin, the Master Pyraminx has two edges of each color combination, and edges cannot be flipped in place: the orientation of every edge is determined by its position, as shown in the diagram below, which shows how the colors of a single edge appear in each of the edge positions (each edge has a twin which would appear reversed in each position). The photo above right appears to show all twelve edges flipped, but actually each pair of same-colored edges has been swapped.

Except where indicated, diagrams are shown from above, and give a panoramic view of the Front, Left, and Right faces.

Professor Pyraminx

This is the five-layer version of the Pyraminx, invented by Timur Evbatyrov and produced by Meffert. It is not easy to find: I bought mine at a reasonable price (about $40 US including shipping) from Puzzle Master in Saskatchewan, Canada. It is pillow-shaped like the 7x7x7 V-Cube. There are 54 visible pieces: 4 corners, 4 trivial tips, 12 wings (6 pairs in each combination of colors), 6 inner edges, 12 outer edges (in pairs like the wings), 4 inner centers, and 12 outer centers. The design is not flawless: edges have a tendency to pop out, but can easily be pushed back in.

It can be solved along the same lines as the Master Pyraminx, with variations of the same routines (e.g. Tilted Edge Tricycles are varied slightly to produce Tilted Wing Tricycles).

There are also 6x6x6 (Royal Pyraminx) and 7x7x7 (Emperor Pyraminx) versions of this puzzle.

Master Skewb

Face-Turning Octahedron

The original design for the 5x5x5 Cube, also known as Rubik's Wahn (Illusion), was by Udo Krell. Like the original 4x4x4, this is a somewhat fragile mechanism and breaks easily; I bought a couple, made by Ideal, from collectors in about 1983. Both eventually lost some stickers and other stickers became badly worn (for some unexplained reason, all of the missing and worn stickers are orange). I also have a 5x5x5 made by Meffert (photo far left) which has tiles (like a Deluxe Rubik's Cube) instead of plastic stickers. This is somewhat heavier (346g versus 309g) but turned more smoothly; however, it eventually came out of alignment somehow and became unusable. Eastsheen has an alternate 5x5x5 design for speedcubing, similar to its 2x2x2 and 4x4x4 cubes. The 5x5x5 cube can be solved by combining techniques used on the 3x3x3 and 4x4x4 cubes; in fact it is actually conceptually a little easier to solve than the 4x4x4 because the centers are fixed (though the solution still takes longer because of the larger number of pieces).

The photo above, second left, is a 1983 cube, showing Pons Asinorum (checkerboard pattern on all six faces); this can be done quickly by

The 5x5x5 cube can be solved using the same basic technique as the 4x4x4, with the addition of a few extra routines. The basic outline is:

(1) Solve the 8 outer centers on one face.

(2) Solve two-thirds of the centers on each of the four side faces by attaching 1x2 blocks of center pieces, assembling the 1x2 blocks when necessary.

(3) Solve the remaining center-corners on the remaining 5 faces using the same techniques as the 4x4x4.

(3) Solve the remaining center-edges using the two swaps (actually tricycles) shown below. The sequence below left swaps a center-edge from the Up face (Ur) with a center-edge from the Right face (Rd). The red starred center moves to the location shown by a circle, that piece moves on the same face to where the yellow starred center is, and the yellow starred center moves to where the red starred piece started. If two center-edges on opposite faces need to be exchanged, the routine shown below center is used instead (orange star goes to circle, circle to red star, red star to orange star).

(4) Match up the outer edges with their corresponding middle edges using the Outer Edge Tricycle (the same routine used in the 4x4x4 solution). But don't break up any matching outer edge pairs, instead placing the correct middle edge using the Middle Edge Tricycle (a similar five-move cycle, except that the last move is a slice move rather than a deep move). As in the 4x4x4, you are trying to match two pairs of edges at a time.

(5) Once you have 12 solid edges, you have no more parity problems and you can finish as if it is a 3x3x3 cube, turning only the outer layers.

The original mechanisms for 2x2x2 through 5x5x5 do not allow for larger order cubes, as the corner pieces of a 6x6x6 would fall out when a layer was turned 45 degrees. Panagiotis Verdes invented a new mechanism which allows for cubes from 2x2x2 through at least 11x11x11, and his company Verdes Innovations SA currently manufactures cubes from 5x5x5 through 7x7x7, as well as 2x2x2. The V-Cube 5 and V-Cube 6 are normal cubical shapes, while the V-Cube 7 and larger cubes have a slightly spheroidal shape. Shown above left is a 6x6x6, showing all 6 colors; this is about the same size (about 69 mm) as an Ideal 5x5x5, and actually a bit smaller than the tiled Meffert 5x5x5. It's difficult to see in the photo, but the edge pieces of the 6x6x6 have slightly rectangular facets (and the corners are slightly larger than the centers). Shown above right is the 7x7x7, in a checkerboard pattern showing all six colors.

The 6x6x6 cube is solved in a manner similar to the 5x5x5, through the centers now take an extra phase. The edges are also matched in two phases: first the inner edges are paired up, then the outer edges are matched to the inner pairs. The Single Edge Swap for the 6x6x6 is

I got the V-Cube 7 (shown further above in checkerboarded Pons Asinorum pattern) on October 27, 2010. My first solve was 50:09.38, with lots of mistakes. My second try was 39:26.44, and I've since occasionally gotten under 15 minutes. The solution is similar to the 6x6x6 (though the centers take even longer), with two phases to matching the edges. Like the 5x5x5, parity is fixed at the end of the edge-matching phase, and the solution finishes exactly like a 3x3x3.

The 8x8x8 and 9x9x9 Cubes

The Shengshou Company in China is now making 8x8x8 and 9x9x9 cubes, which are cubical, rather than the pillow shape of the V-Cube 7x7x7. I have the 8x8x8 and 9x9x9 cubes (available from Amazon; see the 8x8x8 photos above). My first try on the 8x8x8 was full of mistakes again, and took 49:17, but within a few tries I got it under half an hour, and I am now consistently under that even when I make mistakes, though most of my times have plateaued around 25 minutes. The centers, for me, take about 60% of the solving time. The solution is similar to the 6x6x6, except that the center and edge-matching stages each have one extra phase.

My first try on the 9x9x9 was mistake free until I made a wrong turn during the single edge flip in the parity phase. This cost me several minutes, but I still finished in 52:08. It is remarkable how well the solution method for the 7x7x7 scales up in solving the 9x9x9.

This is a mixed-up Rubik's Zwillinge (twins) made in 1983 by Arxon, the German division of Ideal (mine is actually colored wrong). This was also sold in English-speaking countries as Rubik's Mate. This is actually two conjoined copies of the same puzzle, the Bandaged Rubik's Cube suggested by Tony Fisher: a Rubik's cube in which turns along one axis (say F/f, F*/B*, and B/b) are disabled because one column of cubes is taped or fused together. Sadly, mine is now broken.

Of dozens of twisting puzzles to appear since Rubik's Cube, perhaps the most intriguing is Irwin Toys' 1991 puzzle

describes should apply.

Several solutions have appeared (see Bibliography); the most concise and efficient is Edward Hordern's. Richard Snyder has produced a comprehensive booklet which includes a complete solution with plenty of useful sequences, as well as dozens of pretty patterns. Clarence Hewlett has shown that every shape can be returned to a cubical shape with no more than seven Right turns (cutting through all three slices). Square-1 originally sold for about $10; it is still widely available from puzzle dealers on eBay and elsewhere.

Square-2

Square-2 is a modified version of Square-1, in which all of the pieces are cut in 30 degree segments. This actually makes the puzzle much easier, despite a far larger number of possible shapes. In solving Square-2, it is not necessary to bring the puzzle back to a cubic shape until the end of the solution. Half of the bottom layer can be saved by bringing pieces one at a time from the top layer (just to the right of the division) to the bottom layer (just to the left of the division) with the simple 180 degree Right turn. A simple routine, RuRuRUR, then allows the other six pieces to be added to the bottom layer one by one. Once the bottom layer is completed, a second basic routine, RuRUDRdR, is used to swap adjacent pairs of top layer pieces. A parity error can occur at the end, with two adjacent pieces swapped (above right). This can be fixed by swapping one of the pieces all the way around the top layer in the opposite direction (e.g. in the second diagram, the piece in the middle of UF is swapped with the piece to its left, then successively clockwise around the top layer).

The easiest method of flipping edges on the last layer of Megaminx is by using monoflips in inverse pairs. To flip the edge at UF, turn the Left and Right layers away from the Front face, turn the Front face two turns in the same direction, turn the West and East faces (below the Front face) in the same direction, make two more Front turns to return the edge to FU (flipped), and reverse the Left and Right turns. Turn the Up layer anticlockwise one or two turns to bring the second edge to be flipped to UF, and make the same set of turns as before, but in reverse, swinging the edge around the front face in the opposite direction.

Corners can be twisted in a similar manner: twist a corner at URF clockwise via rERFEf or anticlockwise via FefreR, turn U to bring another corner twisted in the opposite direction from the first to URF, and perform the opposite sequence.

Alexander's Star

In 1982 Ideal Toys put out another twisting puzzle, called Alexander's Star, invented by an American mathematician, Adam Alexander. This failed to make a big splash, perhaps due to a fragile mechanism: it is hard to turn quickly and breaks easily. It also has an unusual coloring pattern: each flat pentagon is a solid color when unscrambled, while the raised five-pointed stars are of mixed colors. The puzzle is shaped roughly like a dodecahedron, consisting of 12 overlapping five-pointed stars, each able to rotate. (Note: to avoid confusion between the name of the puzzle and the term for a five-pointed star, the puzzle will always be capitalized, while the uncapitalized word star will refer to a five-pointed star.) There are two stars each of six different colors (red, orange, green, blue, yellow, and white). Each arm of each star is divided into two colors, and can move as part of two different stars. There are a total of 30 arms (hereafter called edges). There are two edges with each of the 15 different pairs of colors; each pair of identical edges start at antipodal positions. When the puzzle is in the Start position, each star has points containing five different colors, and a solid background of the sixth color. This background color will be called the color of that face. Faces on opposite sides of the star have the same color. The object of the puzzle is to return a scrambled Star to its original position. One difficult feature is that any pair of edges which have the same two colors are indistinguishable from each other, and it is necessary (about half the time) in the last stages to exchange two of these indistinguishable edges, because only half of the 30! permutations of edges are possible. The total number of combinations possible is 30!*(2^30)/(4*60*(2^15)), which is equal to 36,215,857,126,357,819,205,810,651,136,000,000. We use basically the same notation as for ImpossiBall. The Star has twelve faces: Front, Up, Left, Right, West, East, Down, Back, Asia, auStralia, euroPe, and afrIca. The last five names are given for completeness: only the faces F,L,R,W,E,U, and D will be used in the sequences. Edges are designated by the two faces they lie on. There are five basic sequences: one double-edge flip (called G); one quadruple-edge flip (called Q); two double bicycles, which exchange two pairs of edges (called X and H); and one tricycle, which cyclically exchanges the locations of three edges (called T).

The solution proceeds in seven stages:

Stage 1 - place the five edges of the front star

Stage 2 - place the background of the front face

Stage 3 - place the equator edges which slant to the left

Stage 4 - place the equator edges which slant to the right

Stage 5 - place the background of the (new) front face

Stage 6 - position the five edges of the (new) front star

Stage 7 - flip any disoriented edges on the front star

[Details and diagrams to follow...]

Georges Helm was kind enough to send me an octahedron, a puzzle in the shape of two square pyramids connected base to base. This can be solved in a variety of different ways, due to its similarity to both Pyraminx and the standard cube. It is actually equivalent to a cube with no corners, but with centers whose rotations are distinguishable (like a picture cube). Therefore any 3x3x3 cube solution can be adapted to solving it (with the addition of face center sequences). However, I prefer to solve it in a manner similar to the Pyraminx (some of the sequences used on the Pyraminx are similar or identical on the octahedron).

Engel's Enigma is a twisting puzzle consisting of two overlapping circles, each composed of 12 separate pieces. The photograph above shows a hand-painted example from my collection. Three pieces are shared by the circles, making 21 pieces in all. The pieces are of two kinds: roughly triangular pieces Engel calls stones, and roughly rectangular pieces called bones. The puzzle is colored, using six different colors, so that each bone has two colors and each stone has three. The two circles can be rotated independently at 60 degree intervals, and when a series of random rotations is made, the pieces become scrambled like those of other twisting puzzles. The Enigma was introduced to the world by A. K. Dewdney in

Unlike earlier puzzles of the same family, such as the Hungarian Rings (photos above, described in David Singmaster's

Top-Spin and Back-Spin

Top-Spin (Ferdinand Lammertink, $11, 1989) and Back-Spin ($12, 1991), both published by Binary Arts (now ThinkFun)

In 1987, Binary Arts (now ThinkFun) published Spin Out, an exceptionally well-made variant of the classic Chinese rings puzzle. Two later puzzles which maintain the 'spin' motif are

A simple notation is:

L -- shift the loop one space anticlockwise and rotate the spinner 180 degrees

[1 2 3 4] 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ==> [5 4 3 2] 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

L*10 -- do L (one anticlockwise and rotate) 10 times

[1 2 3 4] 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ==> [14 2 3 4] 15 16 17 18 19 20 1 5 6 7 8 9 10 11 12 13

L10 -- shift the loop 10 spaces anticlockwise and rotate

[1 2 3 4] 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ==> [14 13 12 11] 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10

R -- shift the loop one space clockwise and rotate the spinner 180 degrees

[1 2 3 4] 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ==> [3 2 1 20] 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

A simple way to reverse 1 and 2 (or any two numbers starting in the first two spots in the spinner) is

Smart Alex and other sandwich puzzles

Smart Alex

The following two sequences are sufficient for solving the remainder:

(1)

(2)

Another puzzle of this type, with six positions instead of four, is

The following routines should prove useful:

{

Somewhat lost in the shuffle after Rubik's Magic (reviewed in WGR7) was

Ernő Rubik followed up his Magic Cube with another of his own inventions, the Magic Snake, a toy consisting of twenty-four triangular prisms, hooked up in a line with joints which allow the prisms to be turned in relation to their neighbors (either 45 or 90 degrees), and which give the snake a good bit of flexibility. The snake can be formed into an astronomical number of figures. Figures which use only 90 degree turns will lie flat, while figures using 45 degree turns will be three-dimensional. There are, to my knowledge, four good collections of Rubik's Snake figures (see the Bibliography). Of the figures in these books, my favorite is the cobra pictured on the cover of Fiore's book (and listed inside as figure 90). Rubik's Snake is still being marketed as Rubik's Twist, and you can buy larger ones with 36 segments.

Here are a few original figures. Of them, my favorite is the heart. It shows how it is possible to make curves using 45 degree turns. The safety pin is a very simple figure, but actually duplicates the locking action of a real safety pin (how did I photograph this?). The rattlesnake and giraffe are two more shapes from the enormous family of animals.

Rubik's Snake figures can be notated by lettering the 23 connections between pieces from A to W, starting at either end (the end you start at determines the coloring of each figure). Lower case letters then denote 45 degree left turns (above left), upper case letters denote 45 degree left turns (above center), and capital letters followed by a 2 denote 90 degree double turns (above right). The figures shown above may be notated as follows (note the shortcut for planar (flat) figures):

The Heart bdEGL2qsTV

The Safety Pin A2B2M2N2 = (ABMN)2

The Rattlesnake aegjM2NS2uoP

The Giraffe (AMNPSUV)2

The Spiral Staircase bcefhiklnoqrtuw

Also note that the notation is used to indicate the order in which turns should be made in complex figures, as in the Rattlesnake (to avoid pieces getting in each other's way). For example, the Rhombicuboctahedron, the shape in which the Snake is usually packaged (and pictured on the cover of Balfour's book), can be notated ABcDfeGhJIkLnmOpRTsQvWu. Fiore's Cobra is notated A2F2KmNO2qpR2Stv.

Many of the puzzles here are out of production, but those still being made can be bought from the manufacturers or through retail dealers. Most of the books are also out of print. Out of print books and puzzles from individual dealers range from dirt-cheap (less than a dollar before shipping) to ridiculously expensive.

eBay is a good place to find a variety of puzzles, including older ones which are out of production. Books can sometimes be found too, particularly the more common ones.

Amazon has fast mail order for standard cubes (2x2x2 through 9x9x9), and dozens of variant puzzles. It also sells books, of course, both in and out of print (via a network of used book dealers).

Rubik's brand cubes, sold by Winning Moves, are widely available in retail outlets.

Meffert sells a wide variety of puzzles, many of its own manufacture. Airmail delivery from Hong Kong is often free and quite fast.

V-Cubes sells various designs for 5x5x5, 6x6x6, and 7x7x7 cubes by direct mail order.

Bookfinder is a good search engine for finding used copies of out-of-print books from a wide variety of dealers. As with Amazon, prices vary wildly.

Pictured books are those we particularly recommend

Although hundreds of books on Rubik's Cube are simply descriptions of a solving method, books have also appeared on the mathematics of Rubik's Cube, on variants and relatives of the cube (Rubik's Revenge, Alexander's Star, etc.), and on pretty patterns and specialized move sequences which rectify a particular situation. Several general books on the cube also contain fairly extensive catalogs of cube sequences and pretty patterns. To distinguish between pretty patterns and sequences intended to help solve the cube, we will call the latter constructive sequences, though the distinction is not always clear. For example, variations of the pretty pattern known as

There are several major categories of constructive moves. Numbers in parentheses indicate the total number of sequences, excluding inverses and mirror images.

(1) Corner sequences (50) -- place and orient up corners simultaneously (may affect edges also). These sequences are especially useful for solving Rubik's Pocket Cube (the 2x2x2 cube, which has no edges or centers).

(2) Edge sequences (24) -- place and orient up edges simultaneously (may affect corners also).

(3) Permutations (14) -- place up corners and up edges simultaneously, without regard to orientation.

(4) Twisterflippers (35) -- orient up corners and up edges simultaneously without changing their positions. The name comes from the book

(5) Miscellaneous up layer sequences

(6) Middle edge sequences (20) -- place and orient (flip) middle layer edges.

(7) Face center sequences -- turn face centers without affecting any edges or corners. These are designed for solving picture cubes.

(8) Insertion sequences (up corner, up edge, and middle edge insertions)

(9) Whole-cube sequences -- perform tricycles and other sequences on subcubes which do not lie in the same layer.

Starting in 1980, a number of books and articles appeared in the United States and elsewhere on one of the most popular puzzles of all time, Ernő Rubik's Magic Cube. The beginner is likely to be at a loss to know where to start, and more experienced cubists may wish to know what information is available beyond step-by-step solutions. This is intended as a survey of the most easily available sources of information on the cube. I will try to give an idea of how each solution works, and indicate how easy and efficient each one is, without going into too much detail on move sequences.

Most of the books on the cube are primarily solutions to the problem of returning the cube to Start (the position in which every side is a single color). This is generally done by working on one group of cubes at a time, e.g. top layer edges, and proceeding to another group when the previous group has been correctly placed (put in their proper locations) and oriented (turned so the colors of the faces match the sides they are on). A group of cubes is worked on by means of a sequence of moves which place and orient those cubes, while leaving undisturbed any cubes placed during earlier sequences. Generally there are from four to eight main stages to a solution, depending on how the cubes are divided into groups. It is much easier to follow the progress of a solution when the cubes you are trying to place are in the top and middle layers. Several of the solutions described here work on the final layer (the most difficult) with the cube positioned so that the final layer is on the bottom. The solver must therefore turn the cube over repeatedly to look at the bottom, or else rewrite the move sequences from the point of view of the bottom face, a difficult task for a beginner. Almost half of the solutions described here follow a common pattern: one layer is done first (edges before corners), the four edges of the middle layer are done next, and finally the last layer is done (often in four stages in which edges and corners are placed and then oriented). Solutions of this type will be referred to here as orthodox solutions.

Many books feature a section showing how to produce colorful arrangements of the cube from the Start position (these arrangements are usually called pretty patterns). A few books also discuss how to solve cubes with pictures on them and cubes with different shapes or coloring patterns. Other features may include a brief history of the cube, details on the cube's physical mechanism, instructions for cube maintenance, mathematics, and catalogs of move sequences.

Adams, Jeffrey --

The most unusual presentation of a solution is Jeffrey Adams'

Angevine, James --

Logical Games, the other U.S. company (besides Ideal) which made the cube in the 1980's, distributed a solution written by James Angevine. The solution is similar to Bossert's, but gives no instructions for the top layer, instead assuming that the solver can finish the first layer unaided (this is a bad assumption to make if the solver is a beginner).

Bandelow, Christoph --

Comprehensive guide to the theory behind Rubik's Cube. Although some of the mathematics is somewhat heavy, it covers a basic strategy, pretty patterns (including photos in full color), and brief notes on solving a few related puzzles (Picture Cube, Magic Domino, 2x2x2 Cube, 15 Puzzle, and Pyraminx). There is a chapter on how to construct your own sequences, and a chapter on using computers to analyze and solve the cube. The book's most notable feature is a very extensive catalog of maneuvers (over 200 sequences, pages 106-119), compiled by Bandelow and his colleague Peter Klering. The maneuvers include the seven most useful face center sequences and nineteen out of the twenty middle edge sequences. Many of these sequences feature true slice moves: turns of the three layers running through the center. Slice moves are very powerful, and notation using them is often much more concise than normal fixed-center notation. For example, using our notation D* to designate a turn of the middle layer lying between U and D 90 degrees in the direction of a D turn, we can notate the powerful quadruple edge flip (D*F)4, rather than the fixed-center version UdL UdB UdR UdF. The powerful slice moves enabled Bandelow and Klering to find many sequences shorter than any others known. Now out of print and expensive where you can find it, but worth getting if you can. My own copy is rather battered by now.

Benjamin, Arthur T. -- The Mathematics of Games and Puzzles: From Cards to Sudoku, The Great Courses, 2013, 3 DVD's and accompanying 130 page booklet, ISBN 159803961-X

Lecture 9 (45:53), Mastering Rubik's Cube, gives some general information on the cube and an inefficient layer-by-layer elementary solution.

Berlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. --

The best book ever written on mathematical games and puzzles in general (reviewed in WGR1), originally in two volumes by Academic Press (1982), Pages 760-768 in Volume 2 contain a detailed mathematical look at Rubik's Cube, including a sketch of a solution. This is the book that brought David Benson's Twisterflipper to my attention. There is also an appendix (pp. 808-809) giving a complete table of 23 Up face processes (including mirror images) to position edges and corners simultaneously in 13 moves or less, and 57 twisterflippers (including inverses) to orient edges and corners simultaneously in 16 moves or less. Diagrams help the cubist pick out the two sequences needed to solve any scrambled up layer. This is somewhat similar to the CFOP system used today, although CFOP does orientations first (most of which disrupt positions). Republished in four volumes by A. K. Peters (2001-2004); pages 868-876 and 916-917 in the revised volume 4 cover Rubik's Cube.

Black, M. Razid and Taylor, Herbert --

A good, well illustrated guide to move sequences (cleverly shown using skeleton diagrams), with 32 pretty patterns (24 of them shown in full color). Many of the move sequences given were not found in other contemporary sources (even Jackson and Singmaster), mainly twisterflippers, corner sequences, and edge sequences. Some of the sequences operate on corners or edges which do not lie in the same layer. The solution, devised by Black, solves only three Up corners before placing all four middle edges: this completely changed my thinking on solution methods. This trick, even today, is never found in elementary solutions, although it will greatly speed up your solving with little additional knowledge. This book was also a best seller.

Bossert, Patrick --

For collectors only. A slow and poorly conceived solution to Rubik's Cube, written by a 13-year-old British schoolboy. This solution is quite long, and does not use the easy layer-by-layer approach. Instead it solves the top layer first (doing corners before edges), followed by the bottom corners. Then it solves the remaining eight edges (on the bottom and middle layers) with a series of cycles of three edges. This sequence makes for a very tedious solution. It is better to do the top edges before the top corners, since the edges are much easier to place when the corners are not already in place, while the top corners are placed in the same way whether they are placed before or after the top edges. The solver is also forced to learn several basic move sequences in a number of different positions, and the relationships between these sequences are not explained. A good deal of experience is necessary to be able to choose efficient sequences. Rather than Singmaster's notation, Bossert uses a pictorial notation, and is not always consistent in the way he describes moves.

Cairns, C. and Griffiths, D. -- Teach Yourself Cube-Bashing, September 1979, self-published, 7 pp.

One of the very first published solutions. It uses an unorthodox notation, and only three routines (with inverses and reflections): a corner tricycle which disrupts edges, a Sune (baryon for twisting corners), and an edge-tricycle with one edge in the middle layer. Incomplete, inefficient, and strange.

Eidswick, Jack --

Jack Eidswick, a professor of mathematics at the University of Nebraska, wrote

Frey, Alexander Jr., and Singmaster, David --

Detailed guide to the mathematics behind the cube. Describes a solution method that is easy to understand and remember (though not especially efficient). Singmaster's constructive sequence catalog is updated and expanded (again with discoverers credited) in a six-page appendix.

Hammond, Nicolas --

The four pages of Chapter Three contain a complete set of sequences for solving the Up layer -- first by positioning and flipping the edges simultaneously (edges should be done first, since shorter versions of many of the routines disrupt corners), then by positioning and twisting the corners simultaneously. This approach gives the cubist a little less to memorize (24 edge sequences and 50 corner sequences) than the

Harris, Dan --

This is a detailed description of the CFOP system of speedcubing to solve Rubik's Cube. It starts out with a mediocre basic method for solving the 3x3x3 cube, then goes into the powerful CFOP method. It includes lots of details on tricks and shortcuts, and plenty of illustrations in full color. The chapter on solving the 2x2x2, however, is a waste of space. There are at least three established speedcubing methods for the Pocket Cube, and surprisingly Harris does not describe or even mention any of them, giving instead a very slow beginner's solution, in his words "a simple method that allows even the most inexperienced puzzle solver...". Wait, isn't this supposed to be a book for experts? It also contains chapters on reduction methods for solving the 4x4x4 and 5x5x5, but these are slow and hard to learn. Some of the material included in the book is pointless (Felix Zemdegs alone rendered the list of world records on pages 5-8 obsolete: none of the times listed there are even in the top 100 anymore; some of the websites listed under Further Resources have already disappeared). There are quite a few errors in the book. Harris compiled a list of these and posted them on his website, but his site is no longer up and you will need a web archive service like the Internet Wayback Machine to find the list of errors, which is incomplete: the first formula on page 131 (Table 8.9) starts with (U u') instead of the correct (U' u'). I can't recommend this book, since it isn't very useful except for 3x3x3 methods, and there is already a lot of material on the Internet on speedcubing methods. It's probably only worth picking up for collectors.

Hofstadter, Douglas --

A very good article on the cube. Hofstadter gives an interesting history of the cube, and discusses some of the contributions made by cubists from various countries, including the United States (giving information which Singmaster lacks). Hofstadter does not give an actual solution, but explains some of the mathematics behind the cube, and discusses how to work out your own solution.

Ideal Toy Co. --

The Ideal Toy Corporation, one of the two U.S. companies which had rights to produce the cube in the 1980's, produced its own solution. This is a well illustrated booklet with full color diagrams, and gives an interesting, but difficult method. The solution follows an plan somewhat similar to Thai or Varasano, solving the top and bottom layers first, then the middle layer, but it is fairly difficult to visualize. The notation used is not Singmaster's, but a pictorial notation with captions (only the captions clearly show the difference between half and quarter turns).

Jackson, 3-D --

A very valuable source of information, which unfortunately exists only in handwritten manuscript form. It consists of several pages of background information, a step-by-step solution to the cube (requiring the solver to remember only a few fairly short sequences), and a 16-page catalog of more than 200 move sequences which can be used to build an extremely powerful solution. Included are corner and edge sequences, middle edge and up corner insertions, and face center sequences (useful for solving picture cubes). The sequences are not as well organized as those in some other catalogs, but there is a lot of good material here for anyone willing to put some effort into it.

Kenzed, Inc. --

A short pamphlet with long and tedious move sequences, some described verbally, some in a bizarre and inconsistent notation. The information provided is not even sufficient for a beginner to solve the cube.

Kolve, Donald I. --

A short booklet giving a solution which has only a few sequences to remember, but is not very efficient. Kolve uses his own notation, a peculiar one, rather than Singmaster's.

Kosniowski, Czes --

Another orthodox solution is presented by Czes Kosniowski in an attractive full color booklet,

Last, Bridget --

Strange approach using tricycles of corners and edges to place everything. I haven't tried to work through this solution, frankly. It doesn't seem to have any routines for flipping edges.

Morales, Phillip -- Gott'cha: Rubik's Cube, 1981, paperback, 60 pp., T/D Publications, $2.95 [reprinted in larger format 2010, 42 pp, $9.95, ISBN 978-1453687116]

An attempt to build a solution to the cube based on four basic routines. I confess I couldn't make heads or tails out of this. The author uses an unorthodox notation, spelling out the face turns in all caps (and using TOP and BOTTOM instead of U and D), and reckoning clockwise from the point of view of the FRU faces, so his BACK, for example, is actually our anticlockwise

Nourse, James G. --

This is an orthodox solution, but many of the individual move sequences used are quite long, being built on repetitive sequences of moves. Nourse does mention a method of combining the top corner and middle edge sequences, but gives no details. He uses his own notation, which differs slightly from Singmaster's. This book sold more copies in the U.S. than any other book on the cube (over a million copies in print), and for several weeks around the end of 1981 was the number one selling mass market paperback. Surprisingly, this book is harder to find cheap on the used book market than many others (including Nourse's much better follow-up on other puzzles).

Östrop, Cyril --

A small booklet written by a Swedish cubist, Cyril Östrop, translated into English and published in the U.S. at the budget price of one dollar. It contains an orthodox solution which is awkward in certain places, and rather inefficient. But the appendix explains an interesting method of partially combining the top corner and middle edge steps, which alone was worth the price of the book to me: I expanded his ideas into part of the method described in this guide.

Parks, Tom, ed. --

One of several newsletters published in the cube's heyday. Only four issues of this eight-page foldout were published between May 1982 and August 1983, available only to members of the Rubik's Cube Club. It covered news of competitions, new releases from Ideal, and club activity, but contained little technical material.

Peelle, Howard A. --

Howard A. Peelle, a professor at the University of Massachusetts, has written a very good book for beginners. The solution is similar in its basic outline to Varasano's, but is easier to learn. Dr. Peelle, a well known educator, explains his solution simply and clearly, and then gives shortcuts to enable the reader to solve the cube more efficiently. He places a great deal of emphasis on visualization, an important part of solving which is ignored or glossed over in many other books on the cube.

Quittendon, R. --

A slow layer-by-layer solution; the author does Up corners first, then inserts Up edges with slice moves. Notable for the 5-move Up corner insertion FLD2lf, where most solutions use a 6-move sequence. The author oddly refers to edges as "centers".

Rubik, Ernő; Tamas Varga, Gerszon Keri, Gyorgy Marx, and Tamas Vekerdy --

This is the fourth volume to appear in the series

In Chapter 3, Gerzson Keri presents "Restoration Methods and Tables of Processes" -- for me the best chapter of the book. There's a lot of valuable material here on both the Cube and the Magic Domino. The only weak point of the chapter is the presentation of a method called the 'throw and catch'. This method solves upper corners and middle edges simultaneously, but uses a set of more than forty routines! The reader is left to work out the details of 24 sequences, and there are errors both in diagrams (two are swapped) and notation (in one sequence an undefined subscript is used). What is more, reasonable results can be achieved using a small set of short sequences, described elsewhere in this booklet. The following two chapters cover the mathematical aspects of the cube, and the final chapter from the Hungarians is a short essay on the psychology of the cube. Singmaster's Afterword is an expert summary of the recent history of the cube (since the original Hungarian book appeared), concentrating on the cube's numerous offspring.

Alas, the bibliography is inadequate. It comes from the original Hungarian edition, and should probably have been scrapped. Many of the sources are in Hungarian, and there are relatively few in English. In several places in the book, methods too complex to be fully described are mentioned but no information is given as to where to find out more. Many of the articles in the bibliography appear to be simple newspaper accounts, and it would have been better to eliminate these and include serious articles. The illustrations are excellent, with many diagrams in full color. There is also a good index. Despite the few flaws I have listed, the book is well done, and a valuable addition to the literature on the cube.

Seven Towns Ltd. --

Full-color booklet with a rather poor solution to the cube, relatively easy to memorize but very inefficient. Stage 3 (Up corners) is particularly bad, sometimes taking 20 moves to put in a corner which can be done in three ((rdRD)5 instead of FDf). I bought an inexpensive printed copy on eBay, but the contents are available free online.

Shah, Neil --

Another book by a teenager, this is a well-illustrated beginner's solution (mostly pictures with little notation), though neither very efficient or very clear (e.g. the author fails to give an explicit method for bringing an upside-down corner to the Up layer in the first phase of solving, doesn't even suggest that Up edges should be done before Up corners, and uses slice moves where they aren't necessary). In Chapter 3 he tells beginning solvers to do the top face without regard to position (i.e. get all of the yellow facets on the same face), and then swap as necessary. This is a wrongheaded approach: although he later suggests placing them correctly to begin with, I think it is better to teach solvers the correct way from the start. He uses conventional middle-edge insertions, then solves the third layer by orienting edges, swapping them a pair at a time (inefficient if all of them are out of place), placing the corners, then using monotwists to finish. The book is written for children, teenagers, and absolute beginners, but even on that basis I cannot recommend it, especially at its substantial price.

Singmaster, David --

The first reasonably comprehensive book in English on the cube was

Singmaster, David --

Five issues of this small magazine were published by Singmaster after the publication of

Slocum, Jerry, David Singmaster, Wei-Hwa Huang, Dieter Gebhardt, and Geert Hellings --

This new book, profusely illustrated in full color, is divided into four sections, each with a different author. The first section is a short history, by Jerry Slocum, of 19th-century puzzle crazes, including tangrams and the Fifteen Puzzle, both popularized by Sam Loyd. The second section is a good history of Rubik's Cube, by David Singmaster. The next section is a survey of other twisting puzzles inspired by the Cube. The last and weakest section, by Wei-Hwa Huang and Dieter Gebhardt, is a solution to the cubes from 2x2x2 through 7x7x7. The solution to 3x3x3, designed to be similar to the solutions for the other sizes, is rather poor and inefficient, with a clumsy notation. Only sketches of solutions are given for the 5x5x5 and larger cubes, and none of them use the efficient reduction methods which are now commonplace.

Taylor, Don --

The best book for the beginner is Don Taylor's

Taylor, Don and Rylands, Leanne --

A second book by Taylor,

Thai, Minh --

Thai was the champion of the first U.S. Rubik's Cube Championship, with a winning time of 26.04 seconds, on the second of two trials. He wrote a book describing his method. Thai solves all eight corners first, then the edges of the up and down layers, then the middle edges (using a set of 9 routines for orienting or swapping middle edges, in some cases simultaneously). This is a strong solution, especially on edge technique, but is quite difficult to learn (it has over 40 different move sequences to learn).

Tyler, J. --

One of a slew of new print-on-demand books on cube solving. Another oversimplified and inefficient beginner's solution, with no notation. The book is nicely printed with a glossy cover, but the colors chosen for diagrams are poor -- black arrows don't show up properly on a dark blue background. For less than what I paid Amazon for the Shah and Tyler books, you could instead buy a 6x6x6 V-Cube.

Varasano, Jeffrey --

Jeff Varasano, a 15-year old student from New York, finished in second place in the 1982 U.S. Rubik's Cube Championship, solving the cube in 28.96 seconds on his second trial. At one time he held the U.S. record for the fastest time in a tournament, 24.67 seconds. Several months before the U.S. Championship, he wrote a book,

Wray, C. G. --

Extremely long layer-by-layer solution, using monoswaps, monoflips, and monotwists. The stage where corners are exchanged is particularly tedious.

This section was originally written as a follow-up to the survey of books on Rubik's Cube, but never published. A few of these books cover multiple puzzles; the rest are sorted out by individual puzzles.

Hofstadter, Douglas R. --

The article that introduced many people to Rubik's Cube. I solved the cube for the first time after reading this article, though it took me 45 minutes and copious notetaking (writing down long conjugation sequences so I could later undo them).

Hofstadter, Douglas R. --

Douglas Hofstadter brought Scientific American readers up to date on the latest twisting puzzles in his July 1982 column. Hofstadter described the many new puzzles which had appeared or were soon to appear, several of them from the catalog of Uwe Mèffert, the inventor of Pyraminx (one of the puzzles described). Hofstadter himself suggested the name Skewb, which was adopted to replace the puzzle's original name Pyraminx Cube. Among the other puzzles described are magic octahedrons, dodecahedrons, icosahedrons, a sphere called IncrediBall (later produced as Impossi*Ball), and a 5x5x5 cube with shaved corners and edges, called Pyraminx Ultimate. No solutions to any puzzles are given, but theoretical results by several puzzlists are reported.

Hofstadter, Douglas R. --

A hardback collection of Hofstadter's articles for

Kiltinen, John O. -- Oval Track and other permutation Puzzles (and just enough group theory to solve them), 2003, The Mathematical Association of America, 305 pp., ISBN 0-88385-725-1, $43.00

A thick paperback textbook on the mathematics of group theory, covering the Fifteen Puzzle, Top-Spin, and Hungarian Rings. Profusely illustrated in black and white, and includes a CD-ROM with simulation software for Windows and Macintosh. The math is somewhat heavy going, but even non-mathematicians can find useful information here.

Kosniowski, Czes and Ewing, John --

Kosniowski, like Nourse, wrote a follow-up to his book on the cube, covering other puzzles of a similar sort. Kosniowski and Ewing emphasize the use of group theory to solve the cube, Pyraminx, the Fifteen puzzle, and others. For the mathematically oriented reader, this will be a useful book in explaining the structure behind the cube and a number of related puzzles. Included is another 2-page catalog of moves for the cube (most of which can be found in Jackson or Singmaster). There is a small section on the pyramid, describing a few basic moves which can be put together to produce a workable (but inefficient) solution. No coverage is given to the Missing Link or the Magic Snake, but other puzzles (including two new ones created by Kosniowski) are discussed.

Nourse, James G. --

James G. Nourse, who topped the best seller list with

Ideal Toy Corporation --

Although its solution booklet for the cube was only of average quality, Ideal Toy Corporation produced a fine booklet on solving The Missing Link. This is basically a catalog of move sequences showing how to exchange two tiles in almost any position relative to each other, without, disturbing any other tiles. This gives the solver great flexibility in working out his own solution, but enough instruction is given so that he will not feel lost. The illustrations, in color, are clearly drawn so that the effect of each sequence can be clearly seen.

Stat, Bob --

For collectors only. The full title is The Altogether Fun and Absolutely Understandable Solution to Rubik's Pocket Cube. It actually contains an overlong and overcomplicated solution to the 2x2x2 cube. Reading it even now, after almost 27 years of experience, I find it hard to follow, even though Stat goes to a great deal of effort to help the solver understand why each sequence has the effect it does. It has only five sequences to learn, all of them easy and fairly short, but is a very inefficient solution, taking an average of about 40 moves. Instead of placing Up corners directly, Stat's solution puts them in position and the orients them if necessary. Later he explains how to twist pairs of Up corners using what he calls The Adjacent Cell Twist (a meson similar to the one we describe in our solution), which twists URF anticlockwise and UFL clockwise. Right after doing it once (in the course of untwisting four corners), he tells the solver to do it twice in a row on a pair of corners when ULF needs to be twisted anticlockwise and URF clockwise, failing to point out that it's not necessary to do it twice, or even to have a mirror image version, since the whole cube can simply be reoriented so that the cubes are in the correct position (the actual reorientation in our notation is [RU2]). The most astonishing mistake, however, is in the section entitled The Final Dilemma (pages 71-74). Stat states quite clearly that a diagonally opposed pair of twisted cells "can

Adams, Jeffrey --

A very slow and inefficient solution, doing corners first, then edges, and centers last. Perhaps the first book to have a reasonably short Single Edge Flip, though Adams' 15-move routine disrupts some centers (the non-disrupting one he gives later is 29 moves). Good color illustrations. Adams was at the time a 25-year-old mathematics professor at MIT. I bought a used copy.

Jean-Charles, Jérôme --

Translated from the original French, this is a corners first solution, using part of the Varasano 3x3x3 method (the second set of corners are oriented first, then positioned). Jerome introduces an even more complicated version of the MES slice notation (ES are the U/D slices, XY are the L/R slices, and αβ are the F/B slices), then largely abandons it for a pictorial notation, using a confusing diamond shape to indicate turns of the F/B slices). Better than the Adams and Mason solutions, but there is a lot to remember. There are also some editing errors (e.g. the diagrams on pp. 80-83). Illustrated in black and white. Introduction by Ernő Rubik. I bought a copy from an independent dealer through Amazon UK.

Lenard, Frank -- There is no magic to Rubik's revenge : the four by four cube, 18 pp., 1982

A rare booklet, which I have never seen. As far as I know, one of only seven English-language books devoted entirely to the 4x4x4 cube.

Mason, William L. --

This book describes a layer-by-layer solution to the 4x4x4 cube, using a consistent set of 8-move tricycles to handle centers, edges, and corners. Although the author takes pains to help the solver remember what each routine does, there is a lot to remember here and the general method is not efficient. Solutions to the 3x3x3 and 2x2x2 cubes are outlined in appendices. There is also a collection of 50 pretty patterns. Like most books from this era, it is illustrated in black and white. There are a few copies floating around the used book market, but at ludicrous prices (over $1000). I was extremely lucky to get a paperback copy from Paperback Swap.

Reid, Michael -- Mastering Rubik's Revenge, 1982, Simon & Schuster, 63 pp.

Smith, Wendy (edited by Robert Kraus) -- Rubik's Revenge, 1983, Simon & Schuster, 128 pp., paperback, ISBN 978-0671457495, out of print

Included here for completeness' sake, this is one of two English-language guides from the 1980's which I have never seen. It also doesn't seem to be available anywhere, at any price. I've been looking everywhere for the last several years: Bookfinder, eBay, Amazon, Paperback Swap, WorldCat, etc.

Thai, Minh et al. --

U.S. Rubik's Cube Champion Minh Thai also wrote a book on Rubik's Revenge, with M. Razid Black and Herbert Taylor. The solution is based on Thai's solution to the cube, with some additional steps necessary to solve the multiple centers and the double set of edges. Like his solution to the cube, this is powerful but rather difficult. The solution is marred by a consistent diagram error in Stage VI, starting on page 47: the diagrams for center-swapping routines show the center at Ldf being swapped to Bdl, but the 9-gram and 8-gram routines both move the Luf center rather than the Ldf center.

Alford, Bill and Iobst, Ken --

A reasonably efficient solution to Pyraminx; the sequences given here handle each possible situation, so that the solver need not backtrack as is necessary with Nourse's solution. The mathematics of the pyramid is discussed briefly, and five pretty patterns are shown (with solutions). The book is somewhat longer than necessary (partially due to the pictorial notation), but it is nevertheless a worthwhile book for anyone interested in Pyraminx. Inexpensive used copies are plentiful.

Nourse, James G. --

See above.

Werneck, Tom --

Although written at the request of Uwe Mèffert, the inventor of Pyraminx, this book is a disappointment. The book consists of some history and background on the pyramid, a solution, and a number of pretty patterns. The background is the best part of the book, giving some details about Mèffert and his invention of the pyramid (Mèffert invented the pyramid in the early 1970's, but did not see its possibilities as a puzzle until Rubik invented the cube several years later). The solution given, however, is by far the weakest of all of those discussed here. The basic method is poorly conceived and long-winded. It uses the small corners as signposts to which color will end up on each layer, requiring the solver to turn one or more small corners after most sequences. This adds considerably to the amount of time (and the number of moves) required. Individual sequences are also unnecessarily long (e.g. 13 moves for a sequence which twists a large corner, essentially equivalent to a tricycle). The worst section of the book, however, is the section on pretty patterns. There are 12 patterns presented here, but many of them are chaotic and unattractive, and Werneck has saddled them with rather pretentious names. The move sequences given to solve most of them are, again, generally longer than necessary. A few of them should be classified as basic sequences rather then pretty patterns. An example is the pattern called The Fire Red Cat's Paw (nothing more than a double edge flip), with a solution given that runs 16 moves (bUBrLR(UL)5). A double edge flip should only take 8 moves. Three of the patterns also appear in the section on alternative sequences with solutions which are shorter than those in the pattern section, but still much longer than the best known solutions! Werneck tries to downplay the relationship between the cube and the pyramid (instead of helping the reader by emphasizing their similarities, he advises forgetting about the cube while trying to solve the pyramid). The Pyraminx enthusiast would do well to choose another book instead. For collectors only.

Helm, Georges --

Neat 6-page booklet in German, giving a full solution to the 2x3x3 Domino, including extra routines for speeding up edge swaps. Easy to follow even if you can't read any German; the diagrams are clear and the notation is similar to Singmaster's, except that German OULRVH (oben, unten, links, rechts, vorne, hinten) = English UDLRFB.

If you run across any books or magazines in Dutch, BOLRVA (boven, onder, links, rechts, voor, achter) = English UDLRFB.

Balfour, Michael --

This is a collection of about 130 figures, and does not include solutions for the shapes. There is little background information, but Balfour poses a few interesting problems in the last section (also without solutions). There are a larger number of figures here than in Fiore's book, and many of them are grouped together thematically, including some good collections of birds, dogs, and snakes. Surprisingly few of the figures in Balfour are found in Fiore, and the real Snake enthusiast will want to buy both books. Inexpensive used copies are plentiful.

Fiore, Albie --

Albie Fiore (a former editor of the superb British magazine

Nourse, James G. --

Nourse packs 41 figures, plus information on forming the rhombicuboctahedron (the spherical shape in which the snake is usually sold, pictured on the cover of Balfour) and some spirals, into only 7 pages of his book on post-cube puzzles.

van de Craats, Jan --

101 figures with black-and-white photographs.

Sly, Susanna --

A book as silly as its title. It has less than 100 cartoons, showing snakes scrambled into various figures (not very well drawn, and titled only in the Index), with no solutions or other details. Recommended only for the most die-hard collector. I bought this for $2 in February 2010 from a used book dealer.

Endl, Prof. Dr. Kurt --

Singmaster, David -

Issue 2 mentions the Orb (called Orb-It in the U.K.); issue 3/4 mentions the Skewb (called the Pyraminx Cube in the U.K.).

Endl, Prof. Dr. Kurt --

Endl, Prof. Dr. Kurt --

Hofstadter, Douglas -- Metamagical Themas, July 1982,

Singmaster, David --

Delbourgo, Daniel and Tino --

Schlagbauer, Keith H. --

Dewdney, A. K. -- Computer Recreations, "Bills baffling burrs, Coffin's cornucopia, Engel's enigma",

Engel, Douglas A. --

Marley, Scott -- Group Theory, Rubik's Cube, and the Avenger,

Singmaster, David --

Hordern, L. Edward -- "Square-1 -- The Solution",

Hanegraaf, Anton -- "The Many Faces of Square-1",

Hewlett, Clarence --

Hewlett, Clarence --

Kopský, Vojtech -- "The Story Behind Square-1",

Snyder, Richard B. --

Binary Arts -- "Top-Spin Solutions", 7pp., 1989

Hordern, L. Edward -- "The Top-Spin Puzzle",

Lammertink, Ferdinand -- "About the Design of Top-Spin",

Wiezorke, Bernhard and Anton Hanegraaf, "Top-Spin Processes",

Fletterman, Ronald -- "Smart Alex",

Lavery, Angus -- Rubik's Clock: A Quick Solution (Authorized Edition), 1988, Pan, 48 pp, paperback, £1.99, ISBN 0-330-30866-1

A straightforward solution to the Clock (which came to the author in a dream!), with large black-and-white diagrams. Also shows all 16 possible positions of the buttons and which clocks turn when each wheel is turned (there are only two cases for each position).

[The following three articles are from

The Editors of CFF -- "Rubik's Clock", p.10

Hordern, Edward -- "Solution for the Clock", p.11

Schultz, Guus Razoux -- "Mathematics on the Clock", p.12-15

Adam Alexander -- inventor of Alexander's Star

Christoph Bandelow -- author of many books on twisting puzzles

Tony Durham -- inventor of Skewb

Tony Fisher -- inventor of Siamese Cubes, the first transformed twisting puzzle design, which also led to analysis of bandaged cubes where certain layers cannot be turned

Solomon Golomb -- mathematician working in many puzzle fields (particularly polyominoes, a term he coined); also coined the terms baryon and meson by analogy with particle physics

Gaétan Guimond -- developer of a complex corners-first method for solving Rubik's Cube, widely used for the Pocket Cube

Georges Helm -- author and preeminent collector of twisting puzzles and books

Douglas Hofstadter -- wrote the first mainstream article on Rubik's Cube, for

Edward Hordern -- eminent collector and analyzer of puzzles (he wrote the definitive book

Herbert Kociemba -- developer of the powerful Cube Explorer solving program

Tom Kremer -- game inventor and agent who licensed the Cube from Ernő Rubik; founder of Seven Towns Ltd. and co-founder of Winning Moves, which manufactures Rubik's Cube and other puzzles today

Uwe Mèffert -- inventor of Pyraminx and manufacturer of many puzzles.

James G. Nourse -- author of a 1981 best-selling book on Rubik's Cube, and subsequent books on other twisting puzzles as well as Rubik's Magic

Cyril Östrop -- author of an early solution book, perhaps the first to suggest placing Up corners and middle edges simultaneously. His ideas helped inspire the 3x3x3 method described in this booklet.

Ernő Rubik -- inventor of the Rubik's Cube, perhaps the first puzzle inventor in history to become a brand name

Jaap Scherphuis -- author of a very comprehensive website on twisting puzzles and other mathematical puzzles, including not only solutions but many simulator programs written in Javascript

David Singmaster -- author of an early newsletter and eventually several books on Rubik's Cube and other puzzles; devised a notation which is widely used

Christopher Taylor -- co-inventor (with Chris Wiggs) of the Orb and Rubik's Clock

Don Taylor -- author of one of the first commercially published solutions, a best-seller in 1981

Minh Thai -- winner of the inaugural Rubik's Cube World Championship in Budapest, and best-selling author. On June 5, 1982, at only 16 years old, he solved a cube in 22.95 seconds in the finals (the current world record is 5.66 seconds by Feliks Zemdegs of Australia in 2011; top solvers now average under 10 seconds!). Single times can be a fluke; nowadays the average time of five trials (discarding the fastest and slowest) is used; that record is also held by Zemdegs with 7.87 seconds in 2010. Thai went on to write two popular books explaining his methods of solving both the Rubik's Cube and Rubik's Revenge.

Morwen B. Thistlethwaite -- devised the first extremely short algorithm, initially able to solve the cube in 52 moves (later improved)

Jeffrey Varasano -- one of the competitors in the first Rubik's cube tournaments, holding a world record of 24.67 at the end of 1981; author of a book describing a corners-first method for solving Rubik's Cube which is widely used for the Pocket Cube

Panagiotis Verdes -- inventor of a new cube mechanism which allows for cubes larger than 5x5x5; so far 5x5x5 through 7x7x7 have been manufactured by his company.

Chris Wiggs -- co-inventor (with Christopher Taylor) of the Orb and Rubik's Clock

Feliks Zemdegs -- a 15-year old Australian, current world record holder (for both single solve and average of five solves) for 3x3x3 (also one-handed 3x3x3) and 5x5x5 cubes; also average record for 2x2x2 and 4x4x4. He has 9 of the 10 fastest times ever for the 3x3x3, and 13 of the 20 fastest for the 5x5x5. In an early version of this booklet, I predicted he would be first to break the 1 minute barrier on the 5x5x5, which he did at the 2011 World Championships -- he averaged 59.94 in the finals. He was also the first to break the 2 minute barrier on the 6x6x6.

Appendix 1 -- Varasano method for speedsolving 2x2x2

In the first stage, we want to put all four blue facets on the Up face, regardless of what order the four blue corners are in (that is, the upper halves of the four side faces can be any combination of colors). Turn the whole cube so that (at least) one of the Up facets is blue, and that a blue facet is at ULF (diagram above far left). [If there are two adjacent blue facets, you can skip one step and hold the cube as in the second diagram above.] Find another blue facet, preferably in one of the pink shaded locations (where a single turn puts a blue facet in place) in the first diagram. Make the turns indicated to put the second blue facet at either ULB or URF, and then hold the cube so that the two blue facets are on the left half of the Up face. Now we want to put a third blue facet at either URF or URB. The ideal positions are again shown in pink: anywhere on the right half of the cube, except the Right face. If both remaining blue facets are on the left half of the cube (or the lower half of the right face), a single Down turn puts one of them in one of the positions where the correct Right turn puts it in place.

In the rare circumstance where both blue facets are on the upper half of the Right face, we can use a special maneuver, shown above far right, to put both in simultaneously. Turn the Right face anticlockwise to put the third and fourth blue facets on the front half of the Right face, then do

If you have placed the third blue facet by itself, the fourth facet can be placed using sequences shown in the third diagram. Many of these are the same as we are familiar with in the standard 2x2x2 solution, but the shortcut sequences highlighted in yellow are worth knowing; they take advantage of the fact that we can put the fourth facet where the third one was and the third back in the remaining spot. [We will meet the sequence

Now all four blue facets are on the Up face. Turn the whole cube upside down, and remember what color is opposite blue on your 2x2x2 (white on the original Pocket Cube, and in our diagrams, but green on most newer cubes. Of course, if you chose a different color than blue, you need to know, or figure out, what color is opposite it). Use one of our six standard twisting sequences to put the four facets of the correct color on the new Up face. [The standard method uses seven sequences: the baryons (Sunes) and second double meson are identical to the ones we already know. The other four routines are shorter, but harder to remember (they also reposition some of the Up corners); there are two mesons with mirror-image orientations. You can learn these from page 28 of Varasano's book.]

Now all eight corners are correctly separated into two layers and oriented correctly with respect to their layer (so that Up and Down faces each have four facets of one color), but may be out of position. In each of the two layers, either the corners are correctly positioned (all of the side facets match in that half of the cube), two adjacent corners need to be swapped (one of the four sides has two facets of the same color), or two diagonally opposite corners need to be swapped (none of the sides match: in this case it doesn't matter which diagonally opposite pair is swapped). There are six possible cases. In one case (8 matches) both layers are fully correct and they can simply be turned so they match, and the cube is solved. Each of the other five cases requires using a different sequence which simultaneously fixes both layers. Note that the whole cube needs to be held in a particular position for each case. The two layers only need to be aligned with each other beforehand in the 2 matches case; turn the whole cube so that the correct pair on the Down layer is on the Front face, and turn the top layer so that its correct pair also goes to the Front face. After performing the indicated routine, turn the layers so they match to finish the solution (the last move of the 4 Matches routine is shown as U2, but it might be any turn (or none) of the Up face).

[Note: The 2-match routine can be done a little faster if you reorient after (or even while) you're doing the middle turn, so it actually comes out as

Mathematicians analyzing the 3x3x3 and larger cubes introduced two additional sets of notations (horrible and unnecessary in my view) for slice turns and turns of the whole cube. Since these are used elsewhere (including the official notation of the World Cube Association), I have added a quick explanation.

MES denote turns of the three slices of the 3x3x3 cube. They also denote turns of the central slice of the 5x5x5 and larger odd-order cubes, or the two center slices of 4x4 and larger even-order cubes.

M (meridian) turns the middle slice in between L and R, in the direction of L (L* in our notation). M' is R* in our notation. [Some sources reverse this]

E (equator) turns the middle slice in between U and D, in the direction of D (D* in our notation). E' is U* in our notation.

S (slice) turns the middle slice in between F and B, in the direction of F (F* in our notation). S' is B* in our notation.

Some books use H/V/C (horizontal, vertical, center) to indicate U*/R*/F*.

xyz denote turns of the entire cube.

x turns the entire cube in the same direction as an R turn, [R] in our notation. x' is [L] in our notation.

y turns the entire cube in the same direction as an U turn, [U] in our notation. y' is [D] in our notation.

z turns the entire cube in the same direction as an F turn, [F] in our notation. z' is [B] in our notation.

Note that not only is M ambiguous depending on the source, E and y are opposite turns, while S and z are turns in the same direction. The whole thing makes no sense.

Some authors also use a w suffix for deep turns: Fw means to turn the Front face and its corresponding middle slice together (FF* in our notation for 3x3x3, 12F in our 4x4x4 notation). Other authors use a lower-case letter (f instead of Fw).

Not only does this system require three different notations just for 3x3x3 cubes, it breaks down badly when you try to apply it to cubes of 6x6x6 and larger.

Copyright ©2020 by Michael Keller. All rights reserved. This booklet was edited most recently on July 2, 2020.