GeoGebra Files for Twisty Puzzle Classification

In 2009 and 2010, I actually ended up making several more GeoGebra files for exploring the possible twisty puzzles. I don't think I ever actually published them, and they may not be 100% finished, but they definitely ought to be out in the wild. So here's a zip file with what I have: http://www.mzrg.com/filepost/GeoGebra_TPCP.zip

First, there are some 2D state diagram views. The files here are Tvv, Tee, Tev (that's all the T ones, since vertex and face turns are equivalent); Cee, Cvv, Cff, Cfv; Dff; Oee; and a partial Iee (there are so many Ie puzzles by themselves!). The display has one face of the polyhedron, and a 2D state diagram.

Second, and perhaps more interesting, are the files marked "Cview" etc. These display colored polyhedra with 6 sliders at once: two v, two f, and two e. I haven't actually done the work of classifying the puzzles, of course, but you can set up and view some pretty complicated things, and it's neat to play with the sliders and see how the puzzles change. The icosahedron one is the most complicated and does not completely work, but I haven't touched it in years and I don't remember what points and lines do what...

Speedsolving Posts: 2D State Diagrams

The posts here are from January 2009 and can be found in this topic: http://www.speedsolving.com/forum/showthread.php?8628

This is a topic on puzzle theory - specifically, it deals with the theory of symmetrical twisty puzzles. There's no real "theory" section in this forum but it doesn't belong in Off-Topic, so I've put it here so more people will see it. (This does have applications to speedcubing, in terms of solving gelatinbrain puzzles.)

So, as some of you may know, symmetrical twisty puzzles can be described by talking about the shape and the number of separate types of turns in each of the three basic categories (face turns, edge turns, and vertex turns) which are named around where the axis (which the pieces rotate around) points. A 3x3x3, for instance, is a cube with one type of turn (in the face turn category); in my categorization scheme that would make it type Cf (C for cube, f for face turn). The only other type-Cf puzzle is the 2x2x2, which has a deeper turn that just happens to line up with the turn on the opposite face. (The 4x4x4 and 5x5x5 have two face turn types, so they are type Cff. Similarly the 6x6x6 and 7x7x7 are type Cfff.) Note that I'm considering a 3x3 with slightly shallower or deeper cuts to be the same, because it has the same number and types of pieces and the exact same solution, so the only difference is in the shape of the pieces.

If we want to talk about symmetrical puzzles with only one type of turn, it's relatively simple, because we can describe the depth of the turn as a single number. But if we have two types of turns, we need two numbers: for instance, on a 5x5x5, if we imagine a turn of depth 0 to turn nothing and a turn of depth 1 to turn the entire cube, the two turns have depths 1/5 and 2/5. But it's hard to imagine exactly what numbers will give a certain puzzle, and which numbers will give a different one. So the question is this: how many possible puzzles are there of a given type, and, more importantly, how can we understand where one puzzle begins and another ends?

My answer to this is the 2D State Diagram. This particular one describes the Tvv-type polyhedra (that is, tetrahedra with two vertex turn types - note that a vertex turn and a face turn are the same thing on a tetrahedron). The two axes correspond to the two depths of turn, and each one goes from 0 (no part of the puzzle is turned) to 1 (the entire puzzle is turned).

So what are the lines in the diagram? Each one represents some kind of puzzle which only exists for a very specific set of turn depths. The diagonal line from the top-left to bottom-right represents the degenerate case where the two depths are the same (so, type Tv puzzles with only one type of cut). Each of those big spaces between the lines, however, is a specific puzzle, which remains the same anywhere in that space. There are also puzzles on the lines themselves, and each place where lines intersect is another puzzle, albeit a very specific one because even a small change in either of the two depths will make it different. The Pyraminx, for example, is located at the intersection of a horizontal or vertical 2/3 line, and the diagonal line from bottom left to top right.

Finally, I have prepared an interactive simulation (!) of this state diagram, so that you can play with it to see how it works out. It uses the GeoGebra geometry program, which I'm really fond of. You can download it from my website. To use it, just move the red point around inside the square (use the white pointer symbol to move a point around), and watch how the puzzle changes when you cross or move along lines.

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I wrote up the state diagram for type Dff puzzles (dodecahedra with two different depths of face turns). The Gigaminx is from this category. Note that this category is MUCH more complicated than the Tvv type I showed in the first post - I'm not completely sure of the count, but I think there are a whopping 98 puzzles of this type!!! I'm really glad I made an interactive simulation, because there is no way I'd ever want to draw all of them out by hand. You can also see that there are 7 puzzles along the diagonal (these are the Df type puzzles).

In the image, I have marked a few lines as green. Those lines are distinctive because, although they separate different puzzles, the puzzles on the lines themselves are equivalent to one of the puzzles on one side of the line. For instance the Megaminx and Supernova, although they look like they might different puzzles, function exactly the same way.The full way to count the number of puzzles is:
- Any open space is a separate puzzle, although you have to make sure to only count puzzles on one side of the main diagonal (for everything here).
- Any line segment (i.e. a line between two intersections) is a separate puzzle, except (i) if the line segment is green; (ii) if the line segment is on the main diagonal; (iii) if the line segment is on the top or left. The second and third cases remove puzzles with essentially only one depth of turn, and the first removes a puzzle that looks like a different puzzle but isn't.
- Any intersection of lines is a separate puzzle, as long as (i) it isn't on the the top or left or the main diagonal, and (ii) it has at least two black lines crossing it.
The Tvv case gives 11+10+2 = 23 puzzles like this (I think). Note that, although I haven't added it in yet, the halfway lines and the bottom-left-to-top-right diagonal in the Tvv diagram should also be green. It's only important to know what lines are green and what are not when you are counting puzzles, though.

Finally, there is a GeoGebra simulation of this, although I have made the two-dimensional slider much larger to compensate for the complexity. That way you can really see what lines you are crossing.

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For each polyhedron there are actually six types of second-order puzzles: the familiar vv, ee, and ff, but also fv, ev, and ef. The first three use only one type of cut, so they're symmetrical, but the other three use two different types of cuts and have much more complicated drawings. As an example I've made one for the Cfv group. There are around 50 puzzles in this group, most of which have probably never been seen before. I hope to eventually make an interactive simulation of all 5*6 categories of second-order regular twisty puzzles. I imagine there will be over 1000 puzzles in total!

TPCP: Type Ce

(Old post from 2008)

These puzzles are rather complicated and there tend to be a lot of them, which tends to be the case with edge-turning types. They are also quite difficult to produce in real life, although one or two of them have actually been created. These are the first really complex puzzles we've seen. Note that these puzzles are related to type Oe.

	Puzzle Ce1: Gelatinbrain Number: n/a Common Name: n/a Types of Pieces: 6 fixed centers, 12 trivial edges, 24 triangular centers in 4 orbits, 8 corners. Solution: n/a
	Puzzle Ce2: Gelatinbrain Number: 3.3.1 Common Name: Helicopter Cube, Bevel Cube Types of Pieces: 24 centers in 4 orbits, 8 corners. Solution: n/a
	Puzzle Ce3:Gelatinbrain Number: 3.3.2 Common Name: n/a Types of Pieces: 6 inner centers, 24 pentagonal centers, 24 inner triangular centers in 4 orbits, 48 outer triangular centers in 2 orbits, 12 edges, 8 corners. Solution: n/a
	Puzzle Ce4:Gelatinbrain Number: 3.3.3 Common Name: n/a Types of Pieces: 6 inner centers, 24 inner triangular centers, 48 outer triangular centers in 2 orbits, 12 edges, 8 corners. Solution: n/a
	Puzzle Ce5:Gelatinbrain Number: 3.3.4 Common Name: n/a Types of Pieces: 6 inner centers, 24 adjacent triangular centers, 24 diagonal triangular centers, 48 tetragonal centers in 2 orbits, 12 edges, 8 corners. Solution: n/a
	Puzzle Ce6:Gelatinbrain Number: 3.3.5 Common Name: n/a Types of Pieces: 6 inner centers, 24 inner triangular centers, 48 outer triangular centers in 2 orbits, 12 edges, 8 corners. Solution: n/a
	Puzzle Ce7: Gelatinbrain Number: 3.3.6 Common Name: n/a Types of Pieces: 6 inner centers, 24 pentagonal centers, 24 inner triangular centers, 48 outer triangular centers in 2 orbits, 12 edges, 8 corners. Solution: n/a
	Puzzle Ce8:Gelatinbrain Number: 3.3.7 Common Name: Little Chop Types of Pieces: 24 centers. Solution: n/a

TPCP: Type Cv

The Cv puzzles were some of the first simple twisty puzzles to be developed. One in particular, the Skewb, is probably the puzzle with the most distinct shape modifications. These puzzles are related to the Tv and Of puzzles.

	Puzzle Cv1: Gelatinbrain Number: n/a Common Name: n/a Types of Pieces: 6 fixed centers, 8 trivial tips. Solution: Turn the trivial tips to solve them.
	Puzzle Cv2: Gelatinbrain Number: n/a Common Name: n/a Types of Pieces: 6 fixed centers, 8 trivial corners, 12 edges. Solution: First position all trivial corners. Then the edges can all be solved intuitively by 4-move 3-cycle commutators of adjacent corners.
	Puzzle Cv3: Gelatinbrain Number: 3.2.4 Common Name: Dino Cube Types of Pieces: 12 edges. Solution: Use basic block building to solve all edges.
	Puzzle Cv4: Gelatinbrain Number: 3.2.2 Common Name: Master Skewb Types of Pieces: 6 inner centers, 24 outer centers in 2 orbits, 12 edges, 8 corners. Solution: Solve inner centers and corners like Cv5. Then use commutators of the type (2DLF) (URF URB' URF') (2DLF)' (URF URB' URF')' to solve the edges. Finally use commutators of the type (2DLF) (ULF DRF' ULF' DRF) (2DLF)' (ULF DRF' ULF' DRF)' to solve the outer centers.
	Puzzle Cv5: Gelatinbrain Number: 3.2.1 Common Name: Skewb Types of Pieces: 6 centers, 8 corners. Solution: Pair up the top center with its 4 corners intuitively. Then solve all centers using DLF' DRF DLF DRF'. Finally use (DLF' DRF DLF DRF')2 to orient all remaining corners.

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.