15 Puzzle - Futility Closet

15 Puzzle

A problem from the 1999 Russian mathematical olympiad:

Show that the numbers from 1 to 15 can’t be divided into a group A of 13 numbers and a group B of 2 numbers so that the sum of the numbers in A equals the product of the numbers in B.

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Suppose this were possible, and let x and y be the numbers in B. Then: (1 + 2 + … + 15) – x – y = xy 120 = xy + x + y 121 = (x + 1)(y + 1) We know that x and y fall between 1 and 15, so the only possible solution is (x, y) = (10, 10). But x and y must be two different numbers. So the task is impossible.

July 2, 2013July 12, 2013 | Puzzles