Download Cryptarithms and more Study notes Calculus in PDF only on Docsity! Cryptarithms Fall 2016 ARML Power Contest A cryptarithm 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 M cannot be zero, it must be a one. Then the number MORE is less than 2000, so MONEY is less than 12000. Thus, the letter O must represent a 0 or a 1. But it cannot represent 1, since M already does, so it must represent 0. Next, look at the hundreds column where it tells us that E + 0 = N. Since E and N must 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 N is one more than E (or that E is 9 and N is zero, but that is ruled out because we already know that O is 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 S is 9. The table below summarizes what we know so far: 0 1 2 3 4 5 6 7 8 9 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. 1
