# Rainbow Cube Rainbow Cube, solved.

The Rainbow Cube is a cuboctahedron-shaped puzzle. Only the triangular faces turn, so this puzzle has eight turning faces arranged on four axes of rotation.

The Rainbow Cube is a surprisingly easy puzzle to solve. You start solving faces, expecting it to be fairly easy until you get to the last face. Having solved the penultimate face, you set yourself for the challenge of solving that last face, only to find it already solved!

Why is this puzzle so easy to solve? Since the turning faces are triangles, the axes have three-fold symmetry… and there are four such axes. Sound familiar? This is precisely how vertex turning cube puzzles turn. Indeed, a cuboctahedron is just a cube with the vertices cut off, so these face axes correspond exactly with cube vertex axes.

That means we would expect the Rainbow Cube puzzle to have many of the characteristics of vertex turning cube puzzles such as the Dino Cube. In fact, the Rainbow Cube corresponds exactly to the Dino Cube.

All of these vertex-turning cube puzzles share the characteristic of being much easier than they look. Why? Because the vertex-turning axes of a cube are not a generating set for all of the cube’s isometries. For each isometry, there are isometries that cannot be reached using vertex turns alone. What this means for vertex-turning puzzles is that some pieces cannot be manipulated into some positions and orientations. Specifically, there is only one possible orientation for each piece in each position. That means that it is impossible to put a cubie into its correct position without also orienting it correctly. And this makes puzzles like the Rainbow Cube much easier to solve than they look.