From 360: Shinju, a geometrical game. You're given an arrangement of shells in some of the squares of a square grid. One of the shells hides a pearl. Your goal is to find it, opening at most four shells. When you open a shell, it either contains the pearl, or a number. That number is the distance to the pearl, in the (T)chebyshev distance.
There's a nice little result hiding here:
Problem 1: Given any arrangement of shells, prove that it is possible to find the shell in four clicks.
Since you always get four clicks (at least as long as I played), the game becomes trivial if you can find a constructive proof of that fact. (If you're the sort of person that likes to rack up points in video games, though, I think you get more points if you don't use all your clicks -- so how do you set things up so that you're likely to solve the puzzle in less than four clicks?)
Problem 2: Generalize (to higher dimensions, different metrics, etc.)