Pentahedron

Pentahedron, solved

Pentahedron, turned on dihedral axis

A pentahedron is a five-sided polyhedron. The term is not used much because it is ambiguous: there are two pentahedra, the triangular prism and the square pyramid. The Pentahedron puzzle is a triangular prism.

Unlike most other dihedral puzzles, the Pentahedron is dihedral to the core: it is built around a core that supports three-fold radial rotation plus dihedral flips. It is therefore symmetry-preserving: it scrambles without jumbling.

To my way of thinking, this is what you want in a dihedral puzzle. Puzzles like the Threefold Hexagonal Prism and the Dipyramid are superficially dihedral, but really they are just Rubik’s Cube variants with a dihedral shell. There is no correspondence between the core and the shell, and the puzzle jumbles on every move. It might as well be a Batman head. But the Pentahedron — ah, this is different.

A face-turning puzzle, the Pentahedron has two triangular faces and three square faces. Square face pieces can be exchanged by rotating around the radial axis. Triangular face pieces can be exchanged by flipping around any of the three dihedral axes. Pieces cannot be moved between the two different face types. As you would expect of a shape with only six isometries, this puzzle is not all that difficult. On the other hand, at least the four axes of rotation used here are a generating set for those isometries, so we’re not coming up short.