Create a game based on the mental challenge of visualizing compositions of rotations in 3 dimensions. Rotations in 3D do not commute. Such visualization is something people can get better at with practice.
Two people, one cube with distinctly marked faces. They take turns rotating the cube according to some rules about what rotations are permitted for a given player on a given turn. Whoever achieves the goal orientation state wins. Or dodecahedron.
Rotations of a sphere (a Lie group). Perhaps some mechanical device that limits the rotations permitted from the current state.
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