# A Surprising Number Guessing Game

Alice and Bob each chooses a positive integer and, without revealing it to the other, tells their choice to Charlie. Suppose that Alice chooses 2004 and Bob chooses 1019. Then Charlie writes the sum of the two chosen numbers on a blackboard, together with another integer he chooses as he likes. Suppose the two numbers on the board are 3023 and 3203.

Alice looks at the two numbers and announces to Bob that she does not know what Bob's number is. Bob looks at the two numbers and announces to Alice that he does not know what Alice's number is.

Alice then announces to Bob that she still does not know his number. Bob then announces to Alice that he still does not know her number. And so on.

At some point, one of Alice and Bob will be able to state what the other had chosen. Who will be the first to do so?

Of course, we make the usual assumptions that Alice and Bob are truthful and smart.

Source: Mathematical Miniatures by Savchev and Andreescu, a problem book by the MAA (vol. 43 in the New Mathematical Library).

© Copyright 2004 Stan Wagon. Reproduced with permission.

9 November 2004