Charles Trigg proposed this festive cryptarithm in the American Mathematical Monthly in 1956:
MERRY XMAS TO ALL
If each letter is a unique representation of a digit, and each word is a square integer, what are these four numbers?
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L.A. Ringenberg of Eastern Illinois State College showed that there are two solutions:
A table of squares shows that ALL must be 100, 144, 400, or 900. If it’s either of the first two, then XMAS has to be 2916 or 9216, and TO can’t be a square. If ALL is 400, then XMAS has to be 1849, 3249, 6241, or 8649. Taking these four possibilities in turn:
- If ALL = 400 and XMAS = 1849, then M = 8, MERRY = 81225, and E and X now both represent 1, so that can’t be right.
- If ALL = 400 and XMAS = 3249, then M = 2, MERRY = 27556, and TO = 81, so produces a valid solution: 27556 3249 81 400.
- If ALL = 400 and XMAS = 6241, then TO isn’t a square.
- If ALL = 400 and XMAS = 8649, then M = 6 and L = 0 and there’s no solution for MERRY.
Finally, going back to the possibility that ALL = 900, in this case XMAS = 1296 or 7396:
- If ALL = 900 and XMAS = 1296, then TO isn’t a square.
- If ALL = 900 and XMAS = 7396, then M = 3, MERRY = 34225, and TO = 81, giving a second solution: 34225 7396 81 900.
(“Problems for Solution,” American Mathematical Monthly 63:10 [December 1956], 723.)
12/08/2017 UPDATE: A variant with a unique solution, from reader Jean-Noël Monette:
MERRY XMAS TO YA ALL
27556 3249 81 64 400
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