Finding Cryptarithms via Computer Search



  (Two words, 8153274 096)

     The ideal cryptarithm above may be the first ever published with two 12-letter addends and a 12-letter 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 a 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              

(Two words, 29617 54038)            (Two words, 0683 742159)


    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

(Two words, 2061 795483, give an appropriate key for this discussion)