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Ben Reiniger wrote on Tue, Dec 12, 2023 02:59 PM UTC:

Two of the games here are best seen as playing in full 3d space, just restricted to the outermost cells of a cube (Gilman's Empty Cube Chess calls it as such, while Judith's Cube-Surface Board Chesses doesn't seem to say it, but the merging of adjacent squares I think makes it so). The rest are actually on the surface (though several are incompletely described, and at least one is really just a circular chess board).

In the latter category, bishops aren't colorbound, and their passage through corners needs a decision. Playing on the surface also means all the pieces are more long-range. A rook can keep an opposing king confined to one face, patrolling the bordering cells from four other faces all at once from any such square (and sliding around to the other side if the king gets too close). Two rooks then force mate, guarding each other along that perimeter and performing the usual net on the king's face, just from its boundary.