Impossible Puzzle
The Impossible Puzzle, also named Sum and Product Puzzle is a puzzle called "impossible" because it seems to lack sufficient information for a solution. It was first published in 1969,[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.
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[edit] Puzzle
X and Y are two different integers, greater than 1, with sum less than 100. S and P are two mathematicians; S knows the sum X+Y, P knows the product X*Y, and both know the information in these two sentences. The following conversation occurs.
- P says "I cannot find these numbers."
- S says "I was sure that you could not find them."
- P says "Then, I found these numbers."
- S says "If you could find them, then I also found them."
What are these numbers?
[edit] Solution
The solution has X and Y as 4 and 13 (or vice versa), 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.
[edit] References
- ^ Hans Freudenthal, Nieuw Archief Voor Wiskunde, Series 3, Volume 17, 1969, page 152
- ^ Scientific American Volume 241, December 1979
[edit] External links
- Puzzles by John Burkardt
- The Impossible Problem by Torsten Sillke
- Two Mathematicians Problem on mathforum
- Model Checking Sum and Product
- Survey: The Freudenthal problem and its ramifications
