Castawords -- some thematic variant logic puzzles by Michael Keller
Castawords is a logic
puzzle which I first saw in the
September 1987 issue of the now-defunct British puzzle magazine Tough Puzzles. It is also seen (under
various names) in U.S. puzzle magazines (e.g. Dell Logic Puzzles,
where it appears regularly under the name Dicey Words
or Building Blocks). I composed the first two examples rated four stars in Dell Logic Puzzles; the typical example is three stars. Unfortunately I have no idea who
invented or named the original idea. I have found
very little on the Internet regarding this puzzle (you can find a couple of examples by searching for Dicey
so I am collecting here some general information on the puzzle, some
methods of solution, and a number of problems (most of these were
published in WGR or The Games Cafe; some of them were newly created for
this article). Solutions to all of the puzzles
(including detailed solutions to many of the harder ones) are on the solution page.
Imagine that the sides of four dice or wooden blocks (cubes)
labeled with 24 different letters of the alphabet, and the dice are
rolled a number of times to produce a dozen or so four-letter words
(reading the topmost letter from each cube and arranging them in an
appropriate order). For example, if the four cubes are labeled ABCDEF /
GHIJKL / MNOPQR / STUVWX, some possible words are LURE, SOLE, and VAIN.
The object of the puzzle, given a list of words, is to deduce the
distribution of the 24 letters among the four cubes, based on the fact
that each word contains one letter from each cube (and hence letters
appearing in the same word cannot appear on the same cube).
a puzzle is properly constructed, all 24 letters are present, and there
is a unique solution (as in cryptarithms, we call such puzzles ideal).
All of the puzzles presented here are ideal, except for number
28. Non-ideal puzzles are sometimes found in puzzle magazines:
the composer specifies which two letters are missing, and other letters
not found in the word list must be placed in empty slots so that each
block has six different letters.
(1) BACK BIRD CUTE DOWN FIGS FLUX JERK MANE MYTH PLOY SODA ZIPS
Let's try solving problem 1 above. As we assign letters to the various blocks, we will call each block by the letters which we know are on it, in alphabetical order. We start by comparing DOWN with SODA, and see that S and A must combine with W and N in some order. But N cannot go with A because of MANE, so we have D / O / AW / NS. Next, M and E (from MANE) must combine with D and O in some order: we'll write this as Dme / Ome / AW / NS, the lower-case letters indicating possibilities. Consider the words BACK and BIRD; B cannot go with A or D. If B combines with NS, then C and K must go with D and O in some order -- but this is impossible, since either the D or O group must include E, which can combine with neither C (CUTE) nor K (JERK). So B must combine with O, and since C or K must combine with D, the D group must include M (not E, which we already know cannot combine with C or K). So far we have: DMck / BEO / AW / NSck. Now add I and R from BIRD: I cannot go with S (FIGS), so I goes with A and R with N. This forces C to go with N and K with D (JERK), and we can add the J to A. Next we add U and T from CUTE (T cannot go with M), H and Y from MYTH (Y cannot go with O), and P and L from PLOY (L cannot go with U). At this point we have: DKMPU / BEHO / AIJLTW / CNRSY. Only four letters remain. Look at FIGS and FLUX; F goes with B, X with C, and G with D. Finally Z from ZIPS goes with B and we have the solution.
(2) BALE BASK COVE CURD EXIT FARM MOWS PAIN PITH TEMP UGLY ZANY
A more complex, but very powerful, method is to determine the six letters of each cube one by one, starting with one of the letters of highest frequency. In problem 2, we start with the letter A, and try to find the five letters which go with it. List the letters of each word not containing A as possibilities, each group in parentheses: A(COVE)(CURD)(EXIT)(MOWS)(PITH)(TEMP)(UGLY). Now remove each letter found in words containing A (i.e., LESRMPIY). We get: A(COV)(CUD)(XT)(OW)(TH)(T)(UG). Since T is the only possibility from TEMP, we add it to A, outside the parentheses, and eliminate H and X. Now we have only four words left (not containing A or T), and each must provide a different letter. Thus we can eliminate the repeated letters O, C, and U. This produces ADGTVW as the only possibility for the cube containing A. Now follow the same procedure with another high-frequency letter not already placed -- let's try E, eliminating all other letters in words containing E, as well as the established letters ADGTVW. We get E(SK)(UR)(FR)(S)(N)(H)(UY)(ZNY), and can immediately add HNS to E, eliminating K, Y, and Z. This also leaves us U, eliminates R, and adds F, yielding EFHNSU. Neither of the first two dice contain M, so we can reconstruct a third cube starting with M. We get M(BL)(BK)(C)(XI)(I)(LY)(ZY), leading to CIM(BL)(BK)(LY)(ZY). If Y is right, L is wrong and B is right, giving us only five letters. So Y must be wrong and L right, giving us CIKLMZ. The fourth cube can be found by listing the unused letters from each word (BOPRXY), completing the solution.
Here are some more standard problems to try. All of the puzzles here have twelve words or fewer; problems 13-15 (which appeared in 1992) may have been the first published constructions with 11 words. To my knowledge a 10 word example has never been composed, nor proven impossible. Note that (5) contains all colors -- the first thematic I ever constructed; (7) omits the two most common English letters E and T; (16) is a later thematic example using birds (a bit on the obscure side -- these are brutally hard to construct):
(3) BOAT CHAP DUNE FLIP
GRAD NOSY PEAK QUIZ TRIM VEIL WHIP ZEST
(4) AXIS CRAG HALF JAMB JUTE KEYS MOLD PATH PING SHOT VENT WIRY
(5) BLUE FAWN FLAX GOLD GRAY JADE NAVY PINK PLUM RUST TALC ZINC
(6) CURB DAMP HARD JEST LONG LYNX MOCK QUAY TEAR VANS WHEY ZEBU
(7) BORN CALM DOCK FILM GRIM HAZY JAWS JINX PUSH QUID VARY YANK
(8) BECK BOND CHEF EXAM MUTE PURL QUIP THEY TOGS VISA WAIL WREN
(9) BRAY DUST FLOP GLUE HINT JIBE MAZE NAIL QOPH VIOL WARM XYST
(10) ARMY BUNT CORE DOZE FLOW GIFT JAIL MANX QUAD RISK SIGH WIDE
(11) BUSY CHEW DING FIVE FOUR JEST PIGS SWAN TANK WHOM XRAY ZERO
(12) AXLE BONY DAYS DUCT DUNK FARM JUMP POSH RAZE SCOW THUG VOLE
(13) BOWL CLIP DARK FEUD GNAT HUMP JUNK JURY MIST OXEN WAVY
(14) DIKE FROG GLIB HUNT JOKE LADY PRIM QUIT SUCH SWAM VINE
(15) BAND CENT CLOG EXPO HIVE HONK MELD QUAG SIZE SPRY TURF
(16) APUS CHAT COLY DOVE FINK GUAN JYNX LARK STIB TODY WHIM XEMA
A Red Herring
Twelve of the words below are normal, but one word is a red herring, consisting of four letters from the same group of six. Find which word it is and solve the puzzle. (Hint: If a pair of words have two letters in common, neither can be the red herring. For each possible red herring, try to find the two companion letters).
(17) BURY FACT GAZE GILD HAND HATE
HYMN JOIN MARK NECK PLUS VOTE WEST
More complex versions of Castawords
The idea of extending to five letter words was suggested by Eduard Riekstins, who at the time edited a puzzle column in the Latvian newspaper CM Cevodnya. He published several Castawords puzzles (in Lettish and Russian) in his column; he sent the following English one to me, which I published in WGR12; it obviously inspired my puzzles Game Logic and Animal Logic below:
A puzzle-lover has found in his attic five toy letter-blocks from his childhood. Each block contains six letters. With these cubes he can produce the following words (mostly related to games):
(18) BLACK BOARD CHESS CRAZE FACET JOKER LOTTO NORTH PAWNS POINT POKER QUEEN SEVEN SHOGI SIXTY TRUMP VIXEN WHIST WHITE
Here are three
more of my thematic puzzles, which appeared in 2000 in the now-defunct
website The Games Cafe:
(This is similar to a musical-instrument themed Building Blocks
I composed, which was published in the August 1996 issue of Dell Logic
Puzzles, p. 37). You have five blocks with letters of the
alphabet on (most of)
their six sides. Four of the five blocks contain five different letters
and one blank side; the fifth block has six different letters. No
letter is repeated, so each of the 26 letters appears exactly
once. By arranging and turning the blocks, you can spell each of
the 14 words below, which are all names of games. Words
shorter than five letters must use blank sides, so that HEX requires
that H, E, and X be on three different blocks, one of which contains
six letters. Can you deduce what letters are on each block?
Some of the games above may be unfamiliar: Clue (a detective game), Junta (a political game), Probe (a word game), Qubic (4x4x4 tic-tac-toe), Realm (an abstract board game), and Risk (a multiplayer war game) are commercial board games. Hex is another abstract board game, as is Oware, a form of mancala. Ghost is another word game. Durak is a Russian card game, while Vira is a card game from Sweden. Frog, Giza, and Yukon are types of card solitaire. This puzzle is not as hard as it looks.
(This is a harder variation of an animal-themed Building Blocks I composed earlier, which was published in the December 1995 edition of Dell Logic Puzzles (p. 19).) This time you have five blocks with letters of the alphabet on all of their six sides. Four of the 26 letters are found on two different blocks each, so that there are 30 letters in all -- unlike the previous puzzle, there are no blank sides. By arranging and turning the blocks, you can spell each of the 17 words below, which are all names of animals. Can you deduce what letters are on each block?
(20) APHID BISON CHIRO CIVET COYPU
FINCH HYRAX JUREL LLAMA MOOSE OKAPI QUAIL SHEEP SHREW SQUID TIGER ZEBRA
(21) Musical Logic
Dmitri Plekharanov's Lyric Suite is famous for its variety of dances, including a polka and a waltz. The major sixth chord in the opening movement is followed by a clarinet motif that emphasizes the break between registers. The second movement consists of a round, each voice playing the same melody twice.
Put 25 letters onto five blocks (each having one side vacant), so that each five-letter word in this puzzle can be spelled with one letter from each.
Castawords Goes to the Movies
(22) ANTZ BLOW CARS DAVE FIST HELP HULK JAWS MASH RUDY TAXI TRON
One or more blanks (which do not appear in the movie titles) can be on any of the five blocks.
(23) ALIEN BUGSY EVITA FARGO GILDA JANIS KLUTE MARCO MELBA PANIC PIQUE SHANE TOXIC WATER ZELIG
a two-player deductive game
I introduced into NOST (the Knights of the
Square Table, a now-defunct postal club for chess, chess variants, and
other games) a two-player version of Castawords. Each player selects a
Castawords grid (four rows of six letters). Players then alternate
guesses (four-letter words with no repeated letters). Each word is
scored according to the distribution of its letters in the opponent's
grid : a score of 1 1 1 1 indicates that each row contains one letter,
2 1 1 (the most common score) indicates that some row contains two
letters and two others contain one each, etc. (Words containing one or
both missing letters can receive scores 1 1 1, 2 1, 3, 2, or 1 1). A
sample grid and examples of each score :
AHJLUV HAUL 4 WIPE 3
BCIPWY CLIP 3 1 BATH 2 1
DGKNOX LOAN 2 2 STOP 1 1 1
FMQRSZ FLAP 2 1 1 TONE 2
(-et) BARK 1 1 1 1 DATE 1 1
Here is a problem based on the two-player version.
The following 28 words have been guessed for a certain grid; all 28
give the result 2 1
1 -- what is the grid?
CANE CHIP DOGS FARE FLOP GNAT GRIM
JAMB JOKE LOCK MAIN MAST MINK NOSE
OKAY PICK QUIZ SAME SAND SITE STAY
STEM SOUR SWAM TAKE TAME WENT WHOM
The Puzzle Laboratory program includes a module to allow you to make guesses against a random set of 4x6 letters.
A Castawords solver and some experiments
On September 18, 2012, I wrote a computer program to solve
standard 4x6 Castawords puzzles. This was used to check all of
the four-letter examples in this article. It is included in the suite
of tools in the Puzzle Laboratory program. I hope to expand
it to handle other variations. Some experimentation with various
combinations of letters (not even worrying yet about creating real
words) suggests that an ideal 10-word standard Castawords puzzle may be
impossible. The best combination I've found so far (with
ten unique letters, one quadruple, and 13 doubles), yields 26 solutions
(e.g. OAFD PAHI QAJK RALM SBHG TBCL UEFN VCEJ WGIM XDKN).
An idea I had (mentioned in WGR11) was to try and create a Castawords puzzle in which half of the letters occur once and half occur three times each. With the help of the new program I managed this, with a puzzle I worked out on September 20, 2012. I started out with nonsense words with the letter distribution I wanted, switched them around until I found an arrangement that produced a unique solution, then used a cryptanalysis program, Edwin Olson's Decrypto, to find real words that fit the letter arrangement. Remarkably, the program found a set of words, of which 11 were real words and the 12th could be anagrammed into a real word. The puzzle below is substantially more difficult than a typical 12-word Castawords:
(25) BASK COST DIRE FELT GORY HUNT JULY MUSK PAIL VARY WINK ZONE
Another idea was to try and create a Castawords puzzle in
which each of 24 letters was used exactly twice. So far I am
having little success; the best arrangement of nonsense words I've
found so far (ACLQ
ABRU BCMN DHRS DENO EFST FGOP GHTU IKVX IJPQ JKVW LMWX) has 16 solutions.
Here's another new puzzle with an interesting feature. Can you solve it and figure out what's unusual about it?
(26) BRED CLAY COIL DOPE FISH GOSH JOLT MONK OVAL SIZE TURN WALK
Another puzzle with only 11 words:
(27) BEVY CHEZ DRUG FOCI HOSE JULY KILO MIND PAIR TABU WHEN
If we drop the requirement that a puzzle be ideal (specifying which two letters are not present), we can get down to ten words. In the puzzle below, we specify that Y and Z are missing. Other letters which are missing from the word list must be placed in empty slots:
(28) AXIL BALE CAST JADE MAKE NAVE PAID RAMS SAGE TUNASolutions
Return to Puzzle Laboratory
Much of this article, including puzzles 1-15 and 17, originally
appeared in WGR11 (June 1992, pp. 23-24). Puzzles 19-21 previously
appeared on The Games Cafe (www.thegamescafe.com) in 2000; that site is no
longer in operation. Puzzles 18 and 24 originally appeared in WGR12 (January 1994,
page 4). Puzzle 16 appeared in WGR13 (February 1998, page 14). Puzzles
22 and 23 were newly composed (September 18 and 23, 2011) for this page, as were puzzles 25-28.