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.

Crosslinked Forum/UWR List

No theory in this article, it's just a proposal I've been thinking of but don't have the skills to program. I've never seen this in practice, but it was an idea I had a while ago and which I'm sure a lot of other people have had independently. It ought to be written up, though...

The thing that inspired this was the realization that people look to the unofficial world record (UWR) pages on speedcubing.com as a reference, but that many fast people don't post their times at all, so they are clearly not representative. I've often heard the excuse that the official times are the only times that matter, but the fact is that most events and puzzles do not have any official counterpart and the only way to compare times on them is through an unofficial list of this sort.

It seems to me that this would solve this problem, plus many additional ones which I'll talk about later. The basic idea is to have a forum where users enter their unofficial personal best times. The times would then be organized in a list in their profile, and there would also be a UWR section which would automatically sort and display the times that users have submitted. Ideally it would be very easy to submit times, so that people would grow used to submitting new PBs as often as they post in Accomplishment threads. A timer could also be added, and then users' single and rolling average-of-12 times could even be automatically updated when they finish a session. From a database point of view it would be a relatively simple addition to the forum system. The way I would do it would be to take a working forum and then add the databases of records and categories; then we would just have to make a way to submit records, display them in people's profiles, and display them in the UWR format. Other additions such as timers or a compare-users function would just involve different ways of interacting with this system.

This doesn't sound like that much of an improvement over a forum like speedsolving.com, but it would solve a lot of problems. There wouldn't be many problems with fake times (like there apparently are on the UWR list) because the mods could simply delete or ban the offending account. Signatures wouldn't need to contain people's best times because they would be easily found in the user's profile. Nothing would have to be manually updated on the unofficial record list, so whenever someone updated their time or e-mail the list would update, and you'd never have duplicate entries. Similarly it would be easy to add new events: you could allow people to create their own events, with certain restrictions, and perhaps show all events with at least two people in them. Since the UWR list and forums would be on the same website, it would be a lot easier to compare people, and you could even have a function which takes two users and compares all of their personal records automatically. You'd also end up with an account for everyone who wants to be on the unofficial list, which means that the community could be more organized than ever; it wouldn't be much of a problem to make different forum sections for people from different countries or people who speak different languages, and perhaps eventually everyone could be using the same website as their speedcubing community (rather than having speedsolving.com for most English-speaking people and then a bunch of other separate websites for various other languages or countries). This would also be a very convenient centralized place for people to talk about their records, since it would all be on one site.


The question is, though, should everyone move from speedsolving or wherever to use this? I kind of think that we should agree on some kind of website that would satisfy all of our needs as a community for the next few years. Cubing is definitely not dying down; in fact it is picking up faster and faster, and I think it's about time we become very organized with the community. Speedsolving is a pretty good forum but it has a big problem with posting times: people can't really display much of their accomplishments because of small signatures, so you have to already know who people are in order to figure out who's amazingly fast. In fact some of the faster people I know don't even know what their best times are, which seems kind of odd given that so many other people want to know that. This tells me that more organization is needed. There's also a problem of lots of smaller, regional cubing forums; this means that the community isn't all in one place, so it can be tricky to find someone who isn't on speedsolving, and if something is posted in one place it won't necessarily reach everyone. So I figure we could use this kind of certainty.

The last thing I think I should say is that, once we are sure we have a forum that will satisfy everything we want, moving to it won't be that hard. Initially you will have people using the old and new system concurrently, but eventually the old system will just become an archive of past cubing discussions. This seems to be the eventual fate of the speedcubing Yahoo group, which was replaced by speedsolving because the Yahoo group message system is much less convenient and organized than the forum system. I am pretty sure that, if a community website can be made which is clearly superior to the existing community forums, we will eventually have everyone in the new website, and it will be much easier to share records and ideas. It will definitely take a while to program and perfect this, but I think that if we use lookahead we can see that it will be well worth it in the long run.

Number of Positions of Generalized Twisty Polyhedra

I wrote this on 3/30/08.

The goal of this article is to determine the number of positions of various twisty polyhedra. I will first show an algorithmic theory where we can find the number of positions in well-behaved twisty polyhedra (that is, puzzles which are symmetrical and have no move constraints based on the position) without difficulty, and then use it to find the number of positions in a few infinite families of theoretical twisty polyhedra, and a few examples of how to modify this theory for trickier puzzles.

Theory of Positions

Before we start determining the number of positions of a puzzle we must fix the orientation somehow. This is necessary since simply turning a puzzle around without doing moves on it will not change its position. The normal way to do this is to fix one specific piece, or a set of pieces which are fixed relative to one another. If this is impossible we can divide the number of positions we calculate later by the number of possible positions of a solved puzzle (most usually 12 for a puzzle with tetrahedral symmetry, 24 for a puzzle with cubical or octahedral symmetry, or 60 for a puzzle with icosahedral or dodecahedral symmetry) but it is not certain that this will give us the correct result. Note, however, that if we have a few choices for what piece to fix, you will get the same answer no matter which piece you choose.

After this the first step to finding the number of positions of a twisty puzzle is to determine the orbits. An orbit is the set of positions that a given non-fixed piece can be in, and orbits containing the same positions are considered the same. In a 3x3x3 cube we will typically fix the orientation by forcing the centers to be in specific places; then there are two orbits, one for the 12 edges and one for the 8 centers. Note that there can be more than one orbit containing similar-looking pieces; for example on the 3x3x3 square group (where only half turns can be used) there are in fact three edge orbits and two corner orbits.

In general we will proceed by finding the number of positions in each orbit and multiplying them together. Because of parity concerns, the best way to do this is to order the orbits in some way and consider them one by one, assuming that while considering one orbit all pieces in the previously considered orbits are solved. This way, when two orbits have the same parity, we see that one does not obey a parity constraint and the other does, so we correctly divide by 2 only once.

So how many positions are in a single orbit? This is generally an easy problem, as it can be directly calculated with a little casework. First we consider the permutations. If there are n pieces, we start with n!; if there are n pieces each of k types, as in the centers of a higher-order Rubik's Cube, we start with (nk)!/n!^k. If the pieces of the orbit have to obey a (permutation) parity constraint then we divide by 2; note that this is only possible in the first type because if we have indistinguishable pieces we can always make a 2-cycle of same-colored pieces to cancel out a wrong parity. Next we consider the orientation of the pieces. If the pieces cannot be oriented differently when they are in a given position we do not have to consider orientation and we can just multiply by 1. Otherwise, if we let k be the number of possible orientations one piece can have if it is a given position and n be the number of pieces then we should multiply by k^n. If the pieces of the orbit obey an orientation parity, we divide by its order, which is generally 2 or 3 but can be higher.

Let's work through an example to illustrate the technique: the 3x3x3 cube. As I already mentioned, if we fix the orientation by fixing the centers' positions, we have two orbits: edges and corners. First we'll consider edges: there are 12 pieces (12!) with no parity (since corners are unknown), and each has an orientation of order 2 (2^12) with an orientation parity (/2), so the total number of possibilities here is 12! * 2^11. Then we consider corners: there are 8 pieces (8!) with parity (/2, since edges are fixed), and each has an orientation of order 3 (3^8) with an orientation parity (/3), so the total number of possibilties is 8!/2 * 3^7. Thus the total number of positions is (12! * 2^11)(8!/2 * 3^7) = 43252003274489856000, as I'm sure you already know.

Number of Positions of Infinite Families of Twisty Polyhedra

The most important concept here is that of layers. It is difficult to put into words, but the number of layers of a puzzle is basically the number of parallel, twistable slices (not counting slices on the opposite side of the puzzle's center). Puzzles with one layer include the 2x2x2, 3x3x3, Megaminx, and Dino Cube; puzzles with two layers include the 4x4x4, 5x5x5, Gigaminx, and Lattice Cube. A family of puzzles is basically a set of puzzles that function similarly (have the same axes, the same fixed pieces, the same shape, and so on), but which differ in the number of layers, a constant which I will call n throughout the rest of the article. I don't count 'trivial tips' (twistable elements of a puzzle which do not affect any other elements, are not affected by any other elements, and can always be solved in one move) as layers, because they do not share any pieces with the rest of the puzzle; thus, although most families only have twistable puzzles for n >= 1, there are some families where the case for n = 0 has more than 1 possibility.

There are many families of puzzles; however, most families have only had one puzzle constructed or even considered, and there are other families (such as stacked pucks and 2x2x(2n+1) cuboids) where adding one layer simply multiplies the number of combinations by a fixed amount. In this article I will only consider the number of combinations for families of which at least two puzzles have been constructed or designed. Of course, these are the most popular families to consider. All numbers I give have been calculated using the formula given here.

Odd NxNxN Cubes: Just like the 3x3x3, we will hold the centers constant. Then our orbits are: 12 edges with 2 orientations each (12! * 2^11), 8 corners with 3 orientations each (8!/2 * 3^7), n-1 orbits of 24 edge wings ((24!)^(n-1)), and n^2-n different orbits of 24 centers, all of which are made up of 4 pieces each of 6 types ((24!/4!^6)^(n^2-n)). Thus the total number of positions is 12! * 8! * 24!^(n^2-1) * 4!^(-6n^2+6n) * 2^10 * 3^7. For this family the existing puzzles are n=1 (3x3x3, with 4.325 * 10^19 positions), n=2 (5x5x5, with 2.829 * 10^74 positions), and n=3 (7x7x7, with 1.950 * 10^160 positions).

Odd NxNxN Supercubes: This is like the odd NxNxN cubes, except that there are much more parities since the center pieces have no duplicates, and also the centers have orientation. Again keep the centers fixed, then the orbits are 6 centers with 4 orientations each (1 * 4^6), 12 edges with 2 orientations each (12!/2 * 2^11), 8 corners with 3 orientations each (8!/2 * 3^7), n-1 orbits of 24 edge wings ((24!)^(n-1)), and n^2-n orbits of 24 centers ((24!/2)^(n^2-n)). So the total number of positions is 12! * 8! * 2^(-n^2+n+21) * 3^7 * 24!^(n^2-1). The puzzles that have been made so far are n=1 (3x3x3 Supercube, with 8.858 * 10^22 positions) and n=2 (5x5x5 Supercube, with 5.289 * 10^93 positions).

Even NxNxN Cubes: We can't hold centers constant, because there are no centers, but if we fix one specific corner we will always be guaranteed a fixed orientation. So our orbits are: 7 corners with 3 orientations each (7! * 3^6), n-1 orbits of 24 edge wings ((24!)^(n-1)), and n^2-2n+1 orbits of 24 centers, each of 4 pieces in 6 colors ((24!/4!^6)^(n^2-2n+1)). The total number of positions here is 7! * 24!^(n^2-n) * 4!^(-6n^2+12n-6) * 3^6. The puzzles in this family that have been constructed are n=1 (2x2x2, with 3.674 * 10^6 positions), n=2 (4x4x4, with 7.401 * 10^45 positions), and n=3 (6x6x6, with 1.572 * 10^116 positions).

Even NxNxN Supercubes: This is very much like the odd NxNxN supercubes. Keep a corner fixed, and our orbits become 7 corners with 3 orientations each (7! * 3^6), n-1 orbits of 24 edge wings ((24!)^(n-1)), and n^2-2n+1 orbits of 24 centers, all with parity ((24!/2)^(n^2-2n+1)). So the total number of positions is 7! * 24!^(n^2-n) * 3^6 * 2^(-n^2+2n-1). The existing puzzles are n=1 (2x2x2, with 3.674 * 10^6 positions) and n=2 (4x4x4 Supercube, with 7.072 * 10^53 positions).

Megaminx Family: These puzzles work just like the odd NxNxN cubes, except with larger numbers of pieces in each orbit. However, in these puzzles all turns create 5-cycles, so every orbit has a permutation parity. There are 30 edges with 2 orientations each (30!/2 * 2^29), 20 corners with 3 orientations each (20!/2 * 3^19), n-1 orbits of 60 edge wings ((60!/2)^(n-1)), and n^2-n orbits of 60 centers, in 12 colors of 5 pieces each ((60!/5!^12)^(n^2-n)). So the total number of positions is 30! * 20! * 60!^(n^2-1) * 5!^(-12n^2+12n) * 2^(28-n) * 3^19. The existing puzzles are n=1 (Megaminx, with 1.007 * 10^68 positions) and n=2 (Gigaminx, with 3.648 * 10^263 positions).

Pyraminx Family: These puzzles are slightly more tricky to calculate since some of them have middle centers (where there are only 4 pieces instead of 12) and some do not. I am not including the trivial tips in the calculation because they are not an inherent part of the puzzle, but of course to put them back in we can just multiply by 3^4. In this puzzle we are keeping the corners in the same position, although their orientation can change. Note that, just as in the Megaminx family, all turns create even permutations, so every orbit has permutation parity. In this family our orbits are 4 corners with 3 orientations each (1 * 3^4), 6 middle edges with 2 orientations each (6!/2 * 2^5), n-1 orbits of 12 edge wings ((12!/2)^(n-1)), a total of floor((n-1)^2/3) orbits of 12 centers, in 3 centers for each of 4 colors ((12!/3!^4)^(floor((n-1)^2/3))), and ((n-1)^2 mod 3) orbits of 4 centers ((4!/2)^((n-1)^2 mod 3)). Thus the number of positions is 6! * 2^(5-n-((n-1)^2 mod 3)) * 3^4 * 12!^(n-1+floor((n-1)^2/3)) * 3!^(-4floor((n-1)^2/3)) * 4!^((n-1)^2 mod 3). The existing puzzles are n=1 (Tetraminx, with 9.331 * 10^5 positions), n=1 with tips (Pyraminx, with 7.558 * 10^7 positions), and n=2 with tips (Master Pyraminx, with 2.172 * 10^17 positions).

Magic Octahedron Family:This family works just like the Pyraminx family, but now there are parities involved with some of the edges since a turn creates edge 4-cycles. Again, we want to fix the corners, and we're not going to fix the tips. The orbits are 6 corners with 4 orientations each (1 * 4^6), n-1 sets of 24 edge wings ((24!)^(n-1)), one orbit of middle edges (12!/2 * 2^11), a total of floor((n-1)^2/3) orbits of 24 centers, 3 each in 8 colors ((24!/3!^8)^(floor((n-1)^2/3)), and ((n-1)^2 mod 3) orbits of 8 centers ((8!/2)^((n-1)^2 mod 3)). So the number of positions is 2^(22-((n-1)^2 mod 3)) * 24!^(n-1+floor((n-1)^2/3)) * 12! * 3!^(-8floor((n-1)^2/3)) * 8!^((n-1)^2 mod 3). The existing puzzles are n=1 (Magic Jewel, with 2.009 * 10^15 positions), n=1 with tips (Magic Octahedron, with 8.229 * 10^18 positions), and n=2 with tips (Master Octahedron, with 1.029 * 10^47 positions).

Trajber's Octahedron Family: Although the Trajber's Octahedron functions in an analogous way to an odd NxNxN cube, the pieces are colored differently, which means that a different method is required when solving it, and it has a different number of positions. Thus it makes sense to consider it a separate puzzle. Note that octahedra made from even NxNxN cubes are entirely different, in that they have many solutions and no piece has multiple stickers. We will hold the corners constant, although allowing them to rotate. Thus our orbits are 6 corners with 4 orientations each (1 * 4^6), 12 middle edges with two orientations each (12!/2 * 2^11), n-1 orbits of 24 edge wings ((24!)^(n-1)), n^2-n outer center orbits, with 3 pieces each of 8 colors ((24!/3!^8)^(n^2-n)), and 8 inner centers (8!/2). Thus the total number of positions is 12! * 8! * 2^21 * 24!^(n^2-1) * 3!^(-8n^2+8n). The existing puzzles are n=1 (Trajber's Octahedron, with 4.050 * 10^19 positions) and n=2 (5x5x5 Trajber's Octahedron, with 3.429 * 10^78 positions).

Dino Cube Family: This is an interesting set of puzzles to think about, although it is easy to solve, since you basically just have a large number of orbits that are all solved like a dino cube. Note that a true n-layer Dino Cube will have 2n(n+1) stickers on each face, and also has two solutions - if you solve it to a mirrored color scheme you will not encounter any problems. This doesn't change the number of combinations, but it does make the puzzle slightly 'easier'. There are n(n+1)/2 orbits on this puzzle, each of which must have an even permutation because all turns create 3-cycles, and no pieces have orientation since they are in a given orientation whenever they are in a given position. We will fix one piece in the centermost orbit (11!/2) and then keep the other ones free ((12!/2)^((n^2+n)/2-1)), so the total number of positions is 11! * 12!^((n^2+n-2)/2) * 2^((-n^2-n)/2). The existing puzzles are n=1 (Dino Cube, with 1.996 * 10^7 positions) and n=2 (Lattice Cube, with 1.145 * 10^24 positions).

Number of Positions of Some Other Interesting Twisty Polyhedra

We can use these techniques for basically any twisty polyhedron, but in some specific cases we need to use a trick to find the correct answer, because if we are not careful we will miss an important detail about the puzzle. I'll illustrate each one with a puzzle. I will make a large table with the number of positions for each of a number of existing puzzles in another article, so if you are interested in the number of positions of a specific puzzle that isn't mentioned here it should be on that list.

Siamese Cube: The Siamese Cube is composed of two separate puzzles which do not share pieces. Since they are the same, you can find the number of positions by finding the number of positions for one of those puzzles and squaring it. Fix the block and the number of positions is ((24)(120 * 3^5)(11!/2 * 2^10))^2 or 2.046 * 10^32.

Helicopter Cube: The tricky thing about this cube is that (in its non-jumbleable form) it actually has four orbits of centers, which each contains one center of each color, and at least one orbit can have an odd permutation as long as the permutation of all of the orbits combined is even (since each turn flips parity on two orbits but doesn't change the overall parity). So if we fix a corner the number of positions ends up as (7! * 3^6)(6!)(6!)(6!)(6!/2) or 4.937 * 10^17.

Alexander's Star: In the standard coloring scheme each piece has a duplicate. So we want to choose one pair of pieces and fix one. There are two identical pieces, though, so we can fix each position exactly two ways, so we have to divide the result by 2. But note that there still may be positions which look the same in more than one of these states (which we assume to be different), which would mean that the result we calculate is actually too small. Generally there are a very small number of these, not enough to influence the calculated result if we write it scientific notation, but it is important to realize that if we wrote out the number in full it would almost certainly be incorrect. The number of positions is approximately (29*(28!/2!^14) * 2^28)/2 or 7.243 * 10^34.

Here is another way to look at this phenomenon: consider a "puzzle" with six tiles arranged like the faces of a cube, where you can switch any two tiles at any time. Suppose we color these tiles so there are three red ones and three blue ones. Now intuition tells us there are exactly 2 positions; our calculation, on the other hand, says that we can fix one of them in a specific orientation and get (5!/(2!*3!))/3*4 = 120/144 or 0.833... positions. Of course this is a contrived result, with many more identical positions than a real puzzle would have, but it does show how the calculated number is too small.