Five-dimensional Rubik's Cube puzzle

MagicCube5D is a five-dimensional analog of the Rubik's Cube puzzle. It comes as a Windows executable, and requires that you grok a little n-dimensional topology before you can try to solve it — so far, three brave souls have!

Each of the d-dimensional cubies could be considered to have its faces covered by stickers of one smaller (d-1) dimension. But each cubie also only exposes a subset of its stickers to the "outside", meaning these are the stickers you could see if you lived and operated in d dimensions. We can use the number of exposed stickers as a classification of cubie types. For the 3D case, the 27 cubies are broken into 4 types, those that expose 0 stickers, 1 sticker ("centers"), 2 stickers ("edges"), or 3 stickers ("corners"). Each sticker on a given cubie has its own color, so we could also call these 1-colored, 2-colored, etc. pieces.

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(Thanks, Ekaeka!)