The Pyraminx series of puzzles are face/vertex-turning tetrahedron puzzles built around a tetrahedron core. (Recall that the tetrahedron is its own dual, so it is meaningless to distinguish between face-turning and vertex-turning tetrahedron puzzles.)
The core of the Pyraminx puzzles has four turning axes, with each allowing for 120° turns. The outer tetrahedron shell is built around these turning axes so that they align with four tetrahedron axes of symmetry: the ones that pass through a vertex and the middle of the opposite face. These axes of symmetry also have 120° rotation, so turns always move pieces from isometry to isometry, and thus preserve the shell’s tetrahedron shape. These are puzzles that can be scrambled but not jumbled.
A negative of the Pyraminx series is that the corner cubies rotate independently of the rest of the puzzle, and can be trivially aligned with their neighbouring centres. Kinda pointless really.
The pyraminx puzzles are:
- Pyraminx (3 layers)
- Master Pyraminx (4 layers)
- Professor Pyraminx (5 layers)
- Royal Pyraminx (6 layers)
- Emperor Pyraminx (7 layers)
There is no 2-layer Pyraminx because that would contain only those pointless corners, for a total of 12 possible positions.
There are several other tetrahedron puzzles, including Jings Pyraminx and the Pyramorphix series. Jings Pyraminx is similar to the Pyraminx but has a different cut. The Pyramorphix series are edge-turning puzzles and therefore completely different.
The trivially oriented corners on the Pyraminx are just silly. The Tetraminx is a Pyraminx in which these corners have been removed, resulting in a puzzle in the shape of a truncated tetrahedron. It is otherwise identical to the Pyraminx.
The vertex-turning Octahedron Puzzle, on the other hand, despite having the same internal mechanism as the face-turning cube puzzles, is extremely similar to the Pyraminx. Like the Pyraminx, it has corners and centres that are trivial to solve, and the algorithms for solving the edges are the same.
Don’t let the name confuse you: the Pyraminx Crystal is an unrelated dodecahedron-shaped puzzle.