I found the following prompt online:
Consider the unit square, $S=[0,1]^2$, and imagine that you have to place a point in S so as to minimize the largest circle that fits inside S while avoiding your point. It is intuitively clear that you should place your point in the center of S, i.e. at $\left ( \frac12 , \frac12 \right )$. Can you prove it? Consider now a game with two players, each of which can move one of two points. At the start, the points are placed randomly in $S$. The players take turns making a move. On her turn, a player chooses which point to move, and whether to move it horizontally or vertically. After her move, the player’s score is the radius of the largest circle her opponent can draw in S that avoids both points. The goal is to have the minimum score possible. Design a strategy. What if there were $N$ (e.g. $3$) points, and players? What if N approaches infinity?
But I wasn't able to find a solution, and there was no specification on the number of moves, so I assumed it went forever. Any ideas?