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rbpl

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**Game Theory a problem which is a bit similar to the "Impossible Puzzle"**

From numbers 1 to 10, two integers X, and Y (not necessarily distinct) are chosen by a referee . The referee informs secretly to Joe the integer U where U = X + Y . The referee informs secretly to Bob the integer V where V = X^2 + Y^2. Before the referee makes his choice, he explains this rule to both players in their common presence. The players take turns to guess the numbers X; Y selected. In case a player is not sure of the numbers he can say "I don't know". Then it is the turn for the opponent. Suppose the game starts with Bob. Suppose he says, "I don't know" and immediately Joe says, I know. When will this happen?

How do I get started?

I thought about the problem it is a bit similar to the "Impossible Puzzle" but the main idea of the "Impossible Puzzle" is that you get the sum and the product here I have X^2 + Y^2 instead of X*Y. However to point of such an exercise is to find the actual values.

I know that when you add X and Y where they are between 1 and 10 you will get 20 different solutions, but when you add X^2 and Y^2 you get more than 20 solution thus it should be easier to find the solution since you have the answer for V.

The part that throws me off the most is the question "When will this happen?"