I haven’t posted a puzzle in a while. The following problem has the usual nice characteristics; it works on a cocktail napkin or as a programming problem, via exact solution or simulation, etc.
I am thinking of a randomly selected integer between 1 and (inclusive). You are the first of players who will each, in turn, get a single guess at the selected number. The player whose guess is closest wins $300, with ties splitting the winnings evenly.
Here is the catch: each player may not guess a number that has already been guessed by a previous player. As the first player, what should your strategy be? Which player, if any, has the advantage? And what happens if we vary and ?