Question 65
Tom, Dick, and Harry play a game in which they each pick an integer between $1$ and $2011$. Tom picks a number first and informs Dick and Harry of his choice. Then Dick picks a different number and informs Harry of his choice. Finally, Harry picks a number different from both Tom's and Dick's. After all the picks are complete, an integer is randomly selected between $1$ and $2011$. The player whose number is closest wins $2$ dollars, unless there is a tie, in which case each of the tied players wins $1$ dollar. If Tom knows that Dick and Harry will each play optimally and select randomly among equally optimal choices, there are two numbers Tom can pick to maximize his expected profit; what are they?