Cryptarithms

Theory

Cryptarithms, also referred to as alphametics, present puzzles in which an arithmetic equation is provided, but the digits are substituted with letters, each letter representing a distinct digit. Your task is to decipher the arrangement, so to speak, to uncover the numerical value that each letter represents.


Just for fun, many of the times, the letters actually spell words. One of the most famous alphametics, spelling out ‘SEND MORE MONEY’ was first published by Henry Dudeney, a British puzzlist, in 1924.

Five rules govern alphametics:

  1. Identical digits are replaced by the same letter. Different digits are replaced by different letters.
  2. After replacing all the letters with digits, the resulting arithmetic expression must be mathematically correct.
  3. Numbers cannot start with 0. For example, the number 0900 is illegal.
  4. Each problem must have exactly one solution, unless stated otherwise.
  5. The problems will be in base 10 unless otherwise specified. This means that the letters replace some or all of the 10 digits – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Problems

1. Solve these cryptarithms:

  • THIS + IS = EASY
  • FORTY + TEN + TEN = SIXTY
  • EINS + EINS + EINS + EINS = VIER
  • EINS + EINS + EINS + EINS + EINS = FUNF

2. Decipher this cryptarithm: replace the asterisks with numbers so that equalities are satisfied in all rows and each number in the last row is equal to the sum of the numbers in the column under which it is located.

3. In the cryptarithm shown below, the operations in each line are performed in a row from left to right, although the brackets are not placed. Each number in the last row is equal to the sum of the numbers in the column under which it is located. The result of each row is equal to the sum of the numbers in the column with the same number. Not a single number in the rebus is equal to zero or starts with zero, but numbers can end with zero. Decipher it!

4. Solve this equation: T – W – O = T ÷ W ÷ O = 2 (different letters are different numbers).

5. Bearing in mind that none of the numbers below have 0 as a leading digit, can you find a solution to all of these cryptharithms?

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