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][2] and the name Impossible Puzzle was coined by Martin Gardner.[3] The puzzle is solvable, though not easily. There exist many similar puzzles.
X and Y are two whole numbers greater than 1, and Y > X. Their sum is not greater than 100. 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 problem is rather easily solved once the concepts and perspectives are made clear. There are three parties involved, S, P, and O. S knows the sum X+Y, P knows the product X·Y, and the observer O knows nothing more than the original problem statement. All three parties keep the same information but interpret it differently. Then it becomes a game of information.
Let us call the split of a number A into two terms A=B+C a 2-split. There is no need for any advanced knowledge like Goldbach's conjecture or the fact that for the product B·C of such a 2-split to be unique (i.e. there are no other two numbers that also when multiplied yield the same result). But with Goldbach's conjecture, along with the fact that P would immediately know X and Y if their product were a semiprime, it can be deduced that the sum x+y cannot be even, since every even number can be written as the sum of two prime numbers. The product of those two numbers would then be a semiprime.
