The Impossible Puzzle

Dutch mathematician Hans Freudenthal proposed this puzzle in 1969 — at first it appears impossible because so little information is given.

X and Y are two different whole numbers greater than 1. Y is greater than X, and their sum is no greater than 100. S and P are two logicians; S knows the sum X + Y, and P knows the product X × Y. S and P both reason perfectly, and both know everything I’ve just told you.

  • S says, “P does not know X and Y.”
  • P says, “Now I know X and Y.”
  • S says, “Now I also know X and Y.”

What are X and Y?

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