Finding Cryptarithms via Computer Search
Additions
IRRELIEVABLE
+ IRREVERTIBLE
AFFINITATIVE
(Two words, 8153274096)
The ideal cryptarithm (a puzzle in which every digit is present, and the solution is unique) above may be the first
ever published with two 12-letter coherent addends and a 12-letter coherent sum (all
three words may be found in Webster's Second New International
Dictionary (1948) and the Compact Oxford English Dictionary
(1991)). This was found by a custom program, part of the
Puzzle Virtuoso suite of puzzle tools. It works from
a candidate list of words, and checks every pair of words (including
duplicates) as addends, against every other word in the list as a
possible sum. It checks each possible puzzle (using the
cryptarithm addition solver in Puzzle Virtuoso) to find those with
exactly one solution, and in which every digit occurs (the above
example is in ordinary base 10, but the program can potentially find
solutions in other bases such as 11 or 12 (popular ones in
cryptarithmetic puzzles). The puzzle was the only
valid one turned up by a search of 651 highly-patterned 12-letter words
(extracted from a raw word list from Webster's Second, herafter
abbreviated as NI2). The extraction was done with a
different PV module which generates pattern word
dictionaries, in this case with an option set to filter out any
patterns shorter than 8 digits (the three words have patterns
12-22413743, 12-224325173, and 12-224351537). Similar searches
have been performed on shorter words (a similar list of 355 11-letter
words generated 177,092 potential puzzles, but only 10 valid
ideals). One was published in The
Cryptogram (C-13 in SO16, p.26); two others appear below:
BARBARITIES
ECLECTICISM
+ PEPPERINESS + VELVETINESS
RECRESCENCE CONCOCTIONS
(2 words, 2961754038) (2 words, 0683742159)
Smallest Addition
A different kind of search looks for an
ideal addition with the smallest possible sum. The
smallest I have found is the following:
PLACE
+ DOOM
REDID
(2 words, 1-0)
Smallest single-digit multiplications
I wanted to find an ideal multiplication
cryptarithm (with a single-digit multiplier), with the smallest
possible multiplicand. The equation ABCD x E = FGHIJ has 13
solutions, so I made a list of every possible multiplication with a
five-digit mutliplicand and five-digit product, with one repeated
digit. There are 55 possible multiplications; I copied each
one into a multiplication solver (using Puzzle Virtuoso and two
others). Six of the multiplications have no solution, 42
have multiple solutions, and there are seven ideals. From
smallest to largest multiplicand:
LLANO
SMITE THING ATLAS
x D x P x R x O
STICK
ALOOF OOZES GUIDE
(2 words,
1-0) (2 words, 1-0) (2 words, (3
words, 0-1)
6245097318)
CANTS GRUEL
LASSO
x P x O x G
BIBLE STAFF
PRICE
(2 words, 1-0) (2 words,
0-1) (2 words, 9-0)
Nine odd words
When I first tried composing
coherent additions, I used pattern word dictionaries to find pairs of
words which looked suitable (and had letters in common), then tried to
find values which produced a sum to which I could match a third
word. I was successful on occasion, and produced three
cryptarithms which were published in The Cryptogram (C-14 JA97, C-8
JF98, and C-14 ND98). When I wrote a search tool to look
for ideal additions, I put the nine words (average, decoder, organdy,
coolest, terrace, deports, octopus, scallop, results) into it to see if
it would reproduce the three additions. To my astonishment, it
turned up all three, plus six additional cryptarithms using
combinations of words from different puzzles (two of the six include a
repeated addend). In addition, there are 3 more ideals in base 11
and 2 in base 12. I do not know if this is a result of
selecting pattern words, or if many groups of 9 seven-letter words
would produce similar results. But it seems remarkable to
get 14 valid puzzles from a list of only nine words.
Roots
Another search tool in Puzzle Virtuoso looks
for a square root by testing a range of values (for example, every
five-digit number with a nine-digit square), and compares the pattern
of the desired root to the pattern of the first (five) letters of its
square. This program, at present, needs to be customized
for each run. Here is a live example I composed while
writing this. I wanted to find a square root cryptarithm with a
solitaire theme, where the square root of STOREHOUSE is equal to
DEMON. I customized the program to check every possible value of
DEMON with a ten-digit square (from 31624 to 98765) and check whether
its square starts with a five-letter pattern (STORE) in which the E's
and O's match, but no other digits are repeated. This takes a
minute or two to set up and a second or two to run. It gave
me a list of 9 possible solutions; I scanned by hand to see whether H
could be assigned a value equal to or larger than the sixth digit of
the square (three of the cases were unusable by this check). I
usually like to have roots without zeroes, and that gave me the
following cryptarithm:
D E M O N
V STOREHOUSE
E
NOR
NRS
NSEH
TTUE
NOTOU
NDTOH
HEENSE
HMMDEN
RRRRD
(Two words, 2061795483, give an appropriate key for this discussion)