Impossible Puzzle
Encyclopedia
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, and the name Impossible Puzzle was coined by Martin Gardner
. The puzzle is solvable, though not easily. There exist many similar versions of puzzles.
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.
What are these numbers?
Initially P does not know the solution, since
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.
Puzzle
A puzzle is a problem or enigma that tests the ingenuity of the solver. In a basic puzzle, one is intended to put together pieces in a logical way in order to come up with the desired solution...
called "impossible" because it seems to lack sufficient information
Information
Information in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...
for a solution. It was first published in 1969, and the name Impossible Puzzle was coined by Martin Gardner
Martin Gardner
Martin Gardner was an American mathematics and science writer specializing in recreational mathematics, but with interests encompassing micromagic, stage magic, literature , philosophy, scientific skepticism, and religion...
. The puzzle is solvable, though not easily. There exist many similar versions of puzzles.
Puzzle
X and Y are two different integers, greater than 1, with sumSUM
SUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...
less than 100. S and P are two mathematicians; S knows the sum X+Y, P knows the product
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
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?
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.
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