Cryptarithms
Fall 2016 ARML Power Contest
Acryptarithm is an arithmetic puzzle where digits are replaced by letters and the puzzle
is to figure out which digit each letter stands for. One of the best known examples is a classic
puzzle by one of the greatest puzzle creators of all time, Henry Dudeney.
S E N D
+ M O R E
---------
M O N E Y
Dudeney’s puzzle is a special kind of cryptarithm called an alphametic, where the letters
actually spell words that fit together in a sentence. The rules of cryptarithms are: 1) each
letter stands for exactly one digit, and it is the same digit each time the letter is used, 2)
no digit is represented by two or more different letters, and 3) the leading digit of a number
cannot be zero.
Dudeney’s puzzle can be solved as follows. (It is highly recommended that you get out
some scratch paper and work along here!) First, a four-digit number is less than 10000, so
MONEY, being the sum of two four-digit numbers, must be less than 20000. But since the
leading digit Mcannot be zero, it must be a one. Then the number MORE is less than 2000,
so MONEY is less than 12000. Thus, the letter Omust represent a 0 or a 1. But it cannot
represent 1, since Malready does, so it must represent 0.
Next, look at the hundreds column where it tells us that E+ 0 = N. Since Eand Nmust
represent different digits, the only way that could happen is if there had been a carry from
the addition in the tens column. So we know that Nis one more than E(or that Eis 9 and N
is zero, but that is ruled out because we already know that Ois 0). It also means that there
is no carry from the hundreds place to the thousands place. Then we know that S+ 1 = 10
so Sis 9.
The table below summarizes what we know so far:
0123456789
O M S
Additionally, we know that the pair EN represents 23, 34, 45, 56, 67, or 78. You will later
be asked to finish this puzzle.
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