TPCP: Type Cf

The Cf puzzles are very simple to describe: they are just the very well-known 2x2x2 and 3x3x3 cube. There are, of course, much more efficient methods than those described here.

Puzzle Cf1: Gelatinbrain Number: 3.1.2 Common Name: 3x3x3 Rubik's Cube Types of Pieces: 6 fixed centers, 12 edges, 8 corners. Solution: Solve all edges using U-layer turns and the 3-cycle U2 R' L F2 R L'. Then solve the permutation of corners with commutators of the type (L) (U' R U) (L)' (U' R U)'. Finally orient all corners with commutators of the type (R U2 R' F' U2 F) (D) (R U2 R' F' U2 F)' (D)'.

Puzzle Cf2: Gelatinbrain Number: 3.1.1 Common Name: 2x2x2 Rubik's Cube Types of Pieces: 8 corners. Solution: Solve the D face intuitively. Then solve the U face using R U R' U R U2 R' U2 to twist three U-layer corners clockwise. Finally solve the puzzle using R U2 R' U' R U2 R' F R' F' R, a corner transposition.

TPCP: Type Te

These puzzles are tetrahedral with edge turns. Puzzles Te3 and Te4 in this series have extra moves if you remove the restriction that they must be tetrahedral, since in fact they can be considered as shape modifications of the 3x3x3 and 2x2x2 Rubik's Cube.

	Puzzle Te1: Gelatinbrain Number: n/a Common Name: n/a Types of Pieces: 4 fixed centers, 6 trivial edges, 12 centers in 3 orbits, 4 corners. Solution: Solve the edges by twisting them in place. The centers can then be solved with four-move commutators of adjacent edges. Finally the corners can be solved with commutators of the form (DL) (DU DR DU DR) (DL)' (DU DR DU DR)'.
	Puzzle Te2: Gelatinbrain Number: 5.2.1 Common Name: Senior Pyraminx (?) Types of Pieces: 6 trivial edges, 12 centers in 3 orbits, 4 corners. Solution: Solve the edges by twisting them in place. The centers can then be solved with four-move commutators of adjacent edges. Finally the corners can be solved with commutators of the form (DL) (DU DR DU DR) (DL)' (DU DR DU DR)'.
	Puzzle Te3: Gelatinbrain Number: 5.2.3 Common Name: Mastermorphix (half turns only) Types of Pieces: 6 trivial edges, 4 corners, 4 triangular centers, 12 trapezoidal centers in 3 orbits. Solution: Solve the edges by turning them in place. Solve one triangular center and one adjacent corner using four-move commutators of adjacent edge turns, then solve the rest of the triangular centers and corners (and the edges) by alternating the two turns which do not disturb the two that were solved before. Finally the trapezoidal centers can all be solved with commutators of the form (DL) (DU DR DU) (DL)' (DU DR DU)'.
	Puzzle Te4: Gelatinbrain Number: n/a Common Name: Pyramorphix (half turns only) Types of Pieces: 4 corners, 4 centers. Solution: First pair up one corner with one center. Then there are only two possible moves which do not break up this pair; alternate between them until the puzzle is solved.

TPCP: Type Tv

These puzzles are tetrahedral with vertex turns. Because the tetrahedron is the unique regular polyhedron to have a face opposite a vertex, all face-turning tetrahedral puzzles can also be considered to be vertex-turning, and for simplicity's sake they will thus fall under the Tv classification.

	Puzzle Tv1: Gelatinbrain Number: n/a Common Name: n/a Types of Pieces: 1 fixed center, 4 trivial tips. Solution: Turn each trivial tip, in any order, to match the fixed center.
	Puzzle Tv2: Gelatinbrain Number: 5.1.1 Common Name: n/a Types of Pieces: 1 fixed center, 4 corners, 6 edges. Solution: Turn each corner, in any order, to match the fixed center. Then all edges can be solved with simple 4-move commutators of adjacent vertex turns.
	Puzzle Tv3: Gelatinbrain Number: 5.1.2 Common Name: Pyraminx Types of Pieces: 4 corners, 6 edges. Solution: Turn each corner so that they are solved relative to each other. Then all edges can be solved with simple 4-move commutators of adjacent vertex turns.
	Puzzle Tv4: Gelatinbrain Number: 5.1.3 Common Name: Halpern-Meier Tetrahedron Types of Pieces: 4 corners, 6 edges, 4 centers. Solution: Turn each corner so that they are solved relative to each other. Then all edges can be solved with simple 4-move commutators of adjacent vertex turns. If centers are unsolved they must be in a double transposition, which can be solved with (UDR UDL UDR' UDL')3.

Twisty Polyhedron Categorization Project

There are a lot of twisty puzzles out there, and they can be categorized in a multitude of ways. This project, the TPCP, is an attempt to categorize all of the most basic twisty puzzles into a list, ideally giving each one an identification code, an image, the gelatinbrain number or common name if applicable, a list of the types of pieces and their orbits, and a basic solution. I do not expect to complete all of it but I hope to get a good chunk of it done.

The most basic twisty puzzles, as I define them, satisfy the following characteristics:
- They are based around a regular polyhedron.
- There is exactly one type of axis around which turns can be made.
- All possible turn axes are represented, and they all have the same depth. Thus the puzzle is symmetrical and remains mathematically unchanged under a rotation.
- Each face of the solved puzzle has one unique solid color and the goal is to bring the puzzle into a state where each face has one solid color.
- After any turn, the puzzle is still in a regular-polyhedral form.

The foundation for the TPCP is very mathematical. Each puzzle belongs to one type, which is described by a capital letter denoting the polyhedron (T, C, O, D, and I for tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively) and a lowercase letter denoting the type of axis (f, e, and v for face, edge, and vertex turns, respectively). Thus for example the 3x3x3 cube belongs to type Cf. Within the types, each puzzle gets a natural number, starting with puzzle 1 for the shallowest cut and increasing for deeper cuts. Puzzles with different-depth cuts which function in the same way are considered to be the same. Even trivial puzzles will be considered, except for puzzles with a cut depth of 0 where turns do not move any pieces and thus no scrambling at all is possible. In this case the 3x3x3 cube would be Cf1.

For standardization of images I have chosen to use an image size given by the downloaded Gelatinbrain applet: 312x312 pixels. Of course I have used the lossless .png format so the images should not take too much time to load. Images for puzzles which are not represented in the downloadable applet have been created by me.

Note that in this project if a set of pieces is described as "trivial" the pieces cannot be permuted relative to each other but do have an orientation.

Although it is not addressed in this project, it is possible to use this categorization system to create types for higher-order puzzles (which I call compound types), and I may investigate some of these series in the future. The type of, for example, a dodecahedral puzzle with two types of edge turn and one type of vertex turns would be Deev, and the Super-X would belong to type Cfv. Although the numbers within the types cannot be as unambiguous as in the TPCP puzzles, we can still assign each puzzle in a compound type a number, with the expectation that someone wishing to do research in this topic would look up a puzzle's number in an online list.

With that said, let's start to classify some of the types.