Sum and Product Puzzle

The Sum and Product Puzzle, also known as the Impossible Puzzle because it seems to lack sufficient information for a solution, is a logic puzzle. It was first published in 1969 by Hans Freudenthal,[1] and the name Impossible Puzzle was coined by Martin Gardner.[2] The puzzle is solvable, though not easily. There exist many similar versions of puzzles.

Puzzle

X and Y are two different whole numbers greater than 1. Their sum is no greater than 100, and Y is greater than X. S and P are two mathematicians (and consequently perfect logicians); S knows the sum X + Y and P knows the product X * Y. Both S and P know all the information in this paragraph.

The following conversation occurs:

What are X and Y?

Solution

The solution has X and Y as 4 and 13, with P initially knowing the product is 52 and S knowing the sum is 17.

Initially P does not know the solution, since

52 = 4 × 13 = 2 × 26

and S knows that P does not know the solution since all the possible sums to 17 within the constraints produce similarly ambiguous products. However, each can work out the solution by eliminating other possibilities following the other's statements and that is enough for the reader to find the solution given the constraints.

See also

References

  1. ↑ Hans Freudenthal, Nieuw Archief Voor Wiskunde, Series 3, Volume 17, 1969, page 152
  2. ↑ Gardner, Martin (December 1979), "Mathematical Games: A Pride of Problems, Including One That Is Virtually Impossible", Scientific American, 241: 22–30.

External links

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