Question 18
Two friends $A$ and $B$ must solve the following puzzle. Each of them receives a number from the set $\{1, 2, ..., 250\}$ but does not see the number received by the other. The goal is for each friend to discover the other's number. The procedure they must follow is to announce, in turns, positive integers, not necessarily distinct: first $A$ says a number, then $B$ says a number, then $A$ again, etc., in such a way that the sum of all the announced numbers is 20. Prove that there is a strategy by which, through a prior agreement, $A$ and $B$ can achieve the goal, no matter which number each one receives at the beginning of the puzzle.