The loser is the one who can't make a move because all the beans are gone. Play the game with your partner. Develop a strategy. Develop a general strategy for the game and prove that it works. It has been known since the time of Euclid that there is a regular polyhedron called the icosahedron with twenty triangular sides.
Most students have only a vague idea about what a regular polyhedron is.
By thinking about the icosahedron, we will invent for ourselves a suitable definition of regular polyhedron, decide whether such polyhedra exist, and discover ways to construct them. Given a regular polygon with 400 sides, show that it can be tiled with parallelograms. Prove that, if you are presented with any such tiling, at least 100 of the parallelograms are rectangles. Consider an island in the middle of the ocean, and assume that every point on the island is on a hillside, is an isolated mountain peak, is an isolated valley bottom, or is a simple saddle point. There are no lakes on the island.