Consider a game mechanic consisting of rotating a virtual non-shapeshifting twisty puzzle, which brings certain things, represented by the facets or cubies, next to each other. Then, start (or continue) a simulation or adventure based on those adjacency conditions.
Maybe the sticker colors represent different biome types, inspired by Minecraft, on a cubical planet.
Or, the colors represent chemicals which react, so on each face is a reaction of the 9 chemicals on that face.
We consider better colorings than traditional (all stickers on each face the same color) because solving the cube is no longer the goal. But keep it symmetric when solved to distribute colors evenly. By tradition, consider 3x3x3. Colored by cubie rather than face, i.e., all stickers on a cubie the same color, to avoid stickers of different colors being eternally adjacent across a cube edge. Possibilities for the 8 corner cubies: all 1 color, 2 colors for the 2 tetrahedra, 4 antipodes, 8 different colors. Possibilities for the 12 edges: 1 color, 12 colors, 6 antipodes, 3 orthogonal bands, and:
4 clusters of 3: pick one of the two possible regular tetrahedra among cube vertices. Each cluster is adjacent to a tetrahedron vertex. This one feels weirder, less symmetric, than the previous ones because one has to first choose one of the two tetrahedra. It's not quite chiral but feels that way because of the two possibilities. It also feels a little like a skewb.
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