This section is devoted to puzzles having similar pieces which must be permuted, often as groups, in order to progress from a random state to a solved state. This group forms a sub-class of the general class of "Sequential Movement" puzzles. Achieving any given state, or arrangement, depends on previously achieved states and sequences of movements, often known as "operators."

Many of these puzzles are mass-produced (or hand-crafted modifications to a mass-produced puzzle), colorful and made of plastic. There are a few sub-categories, the first of which is the "Twisty Polyhedra." Every puzzler knows about Rubik's Cube, the quintessential representative of this group. These puzzles are in the form of a Platonic or an Archimedean solid, "sliced" along various planes to permit certain axes of rotation of pieces or groups of pieces. They contain clever internal mechanisms which keep the moving pieces coherent. (You can see many patents showing the mechanisms at Joshua Bell's site.) A lot of group-theory mathematics applies to this category.

There is also a sub-category where the moving pieces are essentially grouped in two halves which can move relative to each other and exchange pieces - the Dihedral group.

Another sub-category includes the 3-Dimensional Sliding Block/Ball/Hole puzzles.

• Twisty Polyhedra
• Dihedral Twisty Puzzles
• Unusual Permutation Puzzles
• 3-Dimensional Sliding Block Puzzles

These are the websites I consider to be the reference for these types of puzzles:

• Jaap's Puzzle Page, maintained by Jaap Scherphuis, is the best for analysis and solutions.
• TwistyPuzzles.com, maintained by Sandy Thompson, has the most comprehensive catalogue/database, and a vibrant message board community.
• Hendrik Haak's Puzzle-Museum is an impressive on-line collection, and includes many custom-designed puzzles hand-crafted by Tony Fisher, Anthony Greenhill, and others.
• Georges Helm's collection

The illustration above shows many of the twisty polyhedra puzzles that are now or have in the past been mass-produced and commercially available. The illustration below is my attempt to provide a fun "map" of the Twisty Polyhedra puzzle landscape, including most of the commercially produced puzzles as well as several of the interesting hand-made custom modifications. I have exercised personal judgement as to what to include or exclude, and though I have tried to be comprehensive there is no way I could be complete. Sadly, I own copies of only a small portion of these puzzles. Photos are from several sources, including Sandy's TwistyPuzzles.com, Jaap's Puzzle Page, and Hendrik Haak's PuzzleMuseum.

The basis of the map is a central pentagon, having the five regular Platonic solids at its vertices (the yellow circles). At the center of each vertex circle is the key commercially produced puzzle having that shape. Those and other key commercially produced puzzles are outlined in red. Spherical puzzles radiate outward from the center of the pentagon. For the most part, derivatives of the key puzzles are shown near their relations, though some placements may be problematic. Some interesting cube sticker variations and bandaged cubes are shown in the upper left, and cube derivatives in the upper right. The families of derivatives of the Skewb and the Square-1 are shown in bubbles on the left. A group of rhombic octahedra appear on the right, and a group of dihedral puzzles in the lower right. Radiating "arms" show the different sizes of Rubik's Cube, and puzzles related to the Dino Cube.

The next section contains several definitions. You can click the "_" symbol to hide it, if you'd like.

Definitions:

The basic form of a "twisty polyhedron puzzle" is a polyhedron (MathWorld, Wikipedia), although rounded shapes are traditionally included in the class, for example spheres, and cylinders or pucks (or "UFO"s). Polyhedra have been studied extensively since antiquity and formally classified, based on their features, and whether they are convex or concave (non-convex). For a comprehesive list, see George Hart's Encyclopedia of Polyhedra, or the Wikipedia page for Polyhedron. Only a few shapes have been used again and again as the basis for twisty polyhedra puzzles. Some have only appeared in custom puzzles. Polyhedra that are too spiky, too rounded, or too irregular might be of interest to a collector but probably won't make popular puzzles as they are difficult to handle and/or too difficult to visualize when solving. The five Platonic Solids (regular convex polyhedra) - tetrahedron, cube, octahedron, dodecahedron, and icosahedron - have proven to be the most popular shapes, though the icosahedron is infrequently used. There are four Kepler-Poinsot (regular non-convex) polyhedra - These shapes are also the result of stellation of some Platonic Solids. The tetrahderon and cube have no stellations. The octahedron has one stellation, called a stella octangula - that shape has appeared in the Starburst and the Dino Star. The dodecahedron has three stellations: the small stellated dodecahedron, the great dodecahedron, and the great stellated dodecahedron. Only the great dodecahedron has been used in a commercial puzzle - the Alexander's Star. There are 59 stellations of the icosahedron, one of which is the great icosahedron. Some of the Prisms and Anti-Prisms have been made. The larger category here is Prismatoid polyhedra, which also includes pyramids, cupolas, frustums, and wedges. There are also several varieties of Bi- or Di-pyramids. Of the 13 Archimedean (semi-regular convex) polyhedra, the truncated tetrahedron, cuboctahedron, truncated cube, truncated octahedron, rhombicuboctahedron, and truncated icosahedron (soccer ball) have appeared. The 13 Archimedean Duals, or Catalan Solids have not been much used, except for the rhombic dodecahedron and rhombic triacontahedron. There are three stellations of the rhombic dodecahedron, the first, second, and third. These shapes have been used more for interlocking puzzles than twisty puzzles. The 92 Johnson Solids have not been used much at all. A polyhedron becomes a twisty puzzle when it is cut into distinct pieces, and an internal mechanism preserves the coherence of all pieces while allowing groups of one or more pieces to be moved (a twist). After a move or twist, pieces have exchanged positions with other pieces and potentially changed their orientation. The objective of the puzzle is to mix up the pieces, then restore them to a specific target configuration. There are a variety of cutting schemes and internal mechanisms ("cores") that have been developed. You can see many of the relevant patents at Joshua Bell's website. Most mods are based on commercially available puzzle cores, though with the broader availability of improved design tools such as CAD and 3D printing, new and more complex mechanisms are easier to realize (such as those used in the Pentultimate and the 24 Cube). It may be better to classify puzzles by their use of internal mechanism, but I'm not sure I like that method. I think it's more interesting to challenge designers by specifying a shape, slicing format, turning mechanics, and other features, then let them figure out the necessary internals. Of course, this does make it hard to chart the various mods made from non-standard mechanisms such as the Square-1, or adaptations of the mechanism for one shape to make a different shape, resulting in weird asymmetric cuttings (Tony Fisher's Hexaminx - a cubic Megaminx - comes to mind). Chris Lohe has a nice, though not exhaustive, chart at his website, organized by mechanism versus shape. Here are several mechanisms and/or cutting formats: Cuboids AxAxA - A=2,3,4,5,6,7 AxBxB - 1x2x2 (Morph), 1x3x3 (Okamoto's Floppy Cube), 2x3x3 Domino, 3x2x2 Slim Tower and Franken Tower, 3x4x4 Specter and Chaos, 4x2x2 (Fisher 2003), 4x3x3 Phantom Cube, 5x3x3 Grown Tower and TF 2004, 5x4x4 TF 2003 AxBxC - 1x2x3, 2x3x4 Step Up Tower and TF 2003, 2x4x5 Chaos These can be made in fully functional, extended, or "chaos" (multiple core + extended) forms. The EastSheen A4 cube core allows interesting overlapping effects Pyramorphix Pyraminx/Skewb Dino/Stella Octangula Helicopter/Bevel Hybrid Chop (e.g. the 24 Cube) Megaminx Dogic Key features of a polyhedron include faces, vertices, and edges. Most twisty polyhedra puzzles will have different types of pieces corresponding to each of these features. A given face may have one or more center pieces, edge pieces (shared between faces), and corner pieces (again, shared between faces). Almost always, an edge joins two and only two faces, and the corner pieces (if present) are at vertices of the polyhedron. Faces and edges meet at a vertex, and define a particular vertex figure. A design or variant might be contrived to disguise or eliminate centers, corners, or edges or combinations thereof. This may necessitate allowing pieces to overlap as they turn past one another as the trajectory of the cut on the surface of the puzzle no longer follows the linear path of a great circle equivalent. This will differentiate one puzzle from another otherwise of the same order, and dictate different solution methods. I use a 3-letter code to represent the design scheme, as follows: CEC = corners edges centers all present CEN = no centers CNC = no edges CNN = no edges and no centers NEC = no corners (this is what people usually mean when they say "edges only" though technically centers are present, too) NEN = no corners and no centers NNC = no corners and no edges - faces only

Members of this puzzle category are distinguished from their Sliding Piece brethren in the Sequential Movement class in several ways. For twisty puzzles, pieces move in distinct groups. There is no frame and there are no levers or plungers. Also, the space of possible states is typically very large and it is unreasonable for a person to navigate directly from a mixed state toward a solved state (i.e. using "God's algorithm") - instead, a variety of operators or algorithms (i.e. specific sequences of particular twists) must be used in series, each of which accomplishes a subgoal leading towards better order by moving subsets of pieces in determined ways while leaving other subsets undisturbed. Expert solvers have memorized many useful operators, can quickly recognize patterns or states when a given operator will be useful, can keep track mentally of where they are in the midst of a sequence of moves, and can apply operators (sequences of twists) very rapidly with great manual dexterity. Sometimes operators learned for one puzzle can be useful on another type, but often a distinct set of operators must be learned for each different puzzle type. A Deep Cut puzzle's cuts divide it into halves. Otherwise, the puzzle is Shallow Cut, though there are gradations of shallowness, and some puzzles might have a few deep cuts mixed with shallow cuts. Alternatively, circumscribe a sphere around the puzzle, and project the cutting planes to intersect the sphere. If the projections make great circles on the sphere, the cut is deep. This disqualifies the Pyraminx ( Pyraminx setting on Jaap's Sphere) but includes the Skewb ( Skewb setting on Jaap's Sphere). Read a debate about the definition of "deep cut" in the TwistyPuzzles forums. If the cutting scheme results in sets of regular pieces, essentially interchangeable within their groups, then it is customary to distinguish them by coloring their individual faces. The overall pattern of colors will define the goal state of the puzzle. In other cases when the pieces are distinctly proportioned, the puzzle might be monochromatic and its goal state defined by shape alone. I've defined the order based on the number of parallel cuts (either deep or shallow) between opposite pairs of faces, vertices, or edges, or for tetrahedrons, opposing face-vertex pairs. Note that for some shapes, the count of parallel cuts does not increase linearly. Also note that the term slice is used ambiguously to mean either a cut itself, or a layer between two cuts or between a cut and the surface of the puzzle. My definition of order might be at odds with they way some people think about these puzzles - for example I define the 2x2x2 Pocket or Mini Cube as order 1 but some might define it as order 2 based on it having two "slices" in each dimension. Note that it is possible to construct puzzles with asymmetric slicing schemes - i.e. different numbers of parallel cuts between different pairs of opposite features. These would be characterized using compound orders. It is also possible to arrange the cuts so that they are either equally or unequally spaced between the features that bound them - the resulting slices are correspondingly either equal or unequal. Another irregularity results when not all opposite features have the same set of cuts between them, though those that do, have the same number of cuts between them - i.e. subsets of opposite features are treated differently. I call this a Partially Cut puzzle and distinguish it from a Fully Cut puzzle. This can be a consequence of creating a shape modification with a particular underlying core mechanism originally clothed in a different shape. An example is the dodecahedral Skewb Ultimate which has a Skewb core, originally clothed in a cube. The cube has eight vertices, and in the Skewb every pair of vertices has one (deep) cut between them, so four cuts suffice. The dodecahedron has 20 vertices or 10 pairs of opposing vertices. Since there are only four (deep) cuts, there is a cut between each of only four out of the ten pairs of opposing vertices. If all the pieces in a face move during a twist, the puzzle is Face-turning. If instead all the pieces around a vertex move, the puzzle is Vertex-turning. Similarly, if all the pieces along an edge move, the puzzle is Edge-turning. Hybrids are possible, but the internal mechanisms become complex, and the puzzle becomes very "floppy" since it is difficult to hold without inadvertently twisting something. In tetrahedra, the distinction between face and vertex turning is blurred, since a vertex is opposite a face. A more useful distinction is whether cuts divide edges into equal parts or not. Only a few of the many possible species of twisty polyhedra puzzles have been realized; fewer still have made it into mass production, and even fewer remain readily commercially available. Many of the more esoteric shapes and cutting schemes, when they exist, are hand-made modifications or "mods." The art of "twisty modding" has advanced considerably only recently. A sphere has no faces, vertices, or edges. A more useful distinction is how the cuts divide the surface, and how they intersect. The cuts can make great circles or small circles. By convention, two points at diametrically opposite locations on an arbitray great circle are chosen as poles. All great circles passing through the poles are longitudinal. One great circle can perpendicularly bisect all longitudes - this is the equator. Small circles parallel to the equator are latitudinal. Symmetrically arranged small circles are typically centered on regular polyhedra inscribed within the sphere. A cylinder or puck has two parallel circular faces with a curved surface between them. Cuts can be radial and divide the circular faces, or sectional and lie between the faces. So to classify the various species of twisty polyhedra puzzles by their physical characteristics, one could use the following criteria in some priority: Shape Order Turning type - Face, Vertex, (Face/Vertex for tetrahedra), Edge, Hybrid Cutting style - Deep vs. Shallow, Equal vs. Unequal, Full vs. Partial, Symmetric vs. Asymmetric; for spheres, layout of longitudinal, latitudinal, and small-circle cuts; for cylinders/pucks/ufos, layout of radial and sectional cuts Features - Corners/Edges/Centers present? Internal core mechanism Decoration/coloring scheme One could additionally or alternatively use the total number of unique possible permutations or states, or some (subjective) rating of difficulty, or of rarity.

A while back I snagged a Usenet post of a list compiled by Mark Longridge (March 22, 1996) in which rearrangement puzzles were ranked by number of combinations. That list gave me the idea for the organization scheme of the table below. Jaap's Puzzle Page was invaluable in teaching me what puzzles were out there, and for providing combinations data for several puzzles, allowing me to add them to the table at the apropos rank. I include a few puzzles I do not own, for reference - they will be noted as such.

The number of permutations, positions, or states a puzzle can achieve is not always a good indicator of difficulty. Many cubers rank the Square-1 as more difficult to solve than many puzzles with larger numbers of permutations. I have not attempted to rank the puzzles by difficulty.

My favorites include the Pyraminx (I worked out solution procedures myself), the Square 1 (I wrote a program to explore moves), the Impossiball (I've had it for a long time though I've never solved it, and I love its organic motion), and the Skewb (its motion is so precise). I also like the Orb[-it]. I got a Masterball in Japan and the Tonne in Germany.

Puzzle	Name and Notes	Positions	Mechanism
	Gigaminx, Teraminx, Petaminx (Custom puzzles - included for reference to show number of positions - I don't have these.)	Gigaminx: 3.65*10263 Teraminx: 1.16*10525 Petaminx: 3.16*10996	Gigaminx: Tyler Fox; Teraminx, Petaminx: Drew Cormier
	V-Cube 7 from Verdes Innovations.	1.95*10160	Panagiotis Verdes
	V-Cube 6 from Verdes Innovations.	1.57*10116	Panagiotis Verdes
Get new ones from Meffert's.	Dogic Jaap's page	2.199*1082	Zoltan and Robert Vecsei
	5x5x5 Rubik's Wahn Professor Cube also Eastsheen A5 (a different mechanism)	2.8*1074	Udo Krell 6-armed spider
	Megaminx (Tomy version, Meffert's tiled version, Hungarian Supernova re-issue, tiled Chinese version, stickered black Hong Kong version, stickered white Hong Kong version)	1.0*1068 (12 color version) 6.144*1063 (6 color version)	Kersten Meier Ben Halpern 12-armed spider
	Meffert's Pyraminx Crystal First patented in 1987 by Uwe Meffert: DE8707783 (U1). Katsuhiko Okamoto had created a version he called the Mega Crystal. Aleh Hladzilin created a version he eventually named the Brilic - he made around a dozen, some of which sold for over $1000. At first he used a Dogic core, then later a Megaminx core. Noah Hevey has written a nice history of this puzzle - see topic 85537 in the TwistyPuzzles forums. Also see thread 7711 for a discussion of solution methods. While a twist on the Megaminx moves 5 corners and 5 edges, a twist on the Crystal moves 5 corners and 10 edges.	1.68*1066	A build-up of the Megaminx, without centers.
	Mozaika	6.27*1049	Rudolf Destics
	4x4x4 Rubik's Revenge also Eastsheen A4 (a different mechanism)	7.4*1045	Peter Sebesteny grooved sphere
	The Ball.B From Poland - website here. This shape - a spherical Megaminx (aka Ballminx) - was first explored by Jürgen Brandt.	The version with dots is like an edges-only Megaminx, since for this version the corner orientations don't matter. 7.12*1040 The version with flags is equal to the Megaminx.	?
	Alexander's Star (Equal to a Megaminx with no corners and no centers.)	7.2*1034	Adam Alexander
	Face-Turning Octahedron (compared to the Magic Octahedron)	8.49*1028	?
	Logi-VIP	2.7*1025	1982 Logitoy AG, Austria Hubert Petutsching Patent WO8101638
The Thomasball is essentially the same but with a different mechanism	Impossiball (Equal to a Megaminx with no edges and no centers.)	2.4*1025	William O. Gustafson Wolfgang Kuppers
	3x3x3 Picture Cube 4 distinct orientations for all centers e.g. Hoey's Tartan Cube I don't have this.	1.772*1023 ? 8.8*1022	Erno Rubik Dan Hoey
play with a virtual calendar cube here	Rubik's Perpetual Calendar (Kalender Kubus) The "O" character on one center has only 2 distinct orientations	4.4*1022	Marvin Silbermintz
	Super Square 1 (4-layer) Produced by cube4you. See this thread at the cube4you forums. Watch a video.	1.19*1022	?
	Rubik's Cube 4th Dimension distinct center orientations	1.1*1022	Erno Rubik
	Rubik's World	2.7*1021	Erno Rubik
	Masterball (Geomaster, aka Rainbow version) See other versions at Les Casse-Tete de Chantal Masterball Homepage Mullen & Robinson solution	4.1*1020	Dr. Geza Gyovai patent 4856786
	3x3x3 Rubik's Cube Solutions here. Rubik's Ball and the Cuboctahedron are essentially the same. Algs for moving edge from bottom to FL: Start in FD matching F: DL - D'L' - D'F' - DF Start in LD matching L: D'F' - DF - DL - D'L' For bottom corners, see 2x2x2 entry - those algs mess up bottom edges, so do them next. Algs for fixing bottom edges: 3-cycle (if one is in position, hold it in front): L'R - F - LR' - D2 - L'R - F - LR' or, where ( means L'R, and ) means LR': (F)D2(F) flip DL & DB: L'R - F - LR' - D' - L'R - F' - LR' - D' - L'R - F2 - LR' or (F)D'(F')D'(F2) You may have to repeat these 2 algs in sequence.	4.325*1019	Erno Rubik 6-armed spider
	Rubik's Mirror Blocks (aka Bump Cube) designed by Hidetoshi Takeji. The Bump Cube was entered in the IPP 2006 Design Competition. The hand-crafted version had been for sale at $320. I got one boxed copy signed by Hidetoshi-san.	(Same as 3x3x3 cube.)
	Trajber's Octahedron Purchased from David Calzone. David made a batch by casting pieces molded from 3D-printed masters. The Trajber's Octahedron is a vertex-turning puzzle and has a 3x3x3 cube core. The group shot shows various kinds of octahedral twisty puzzles - the vertex-turning Magic Octahedron, the Trajber's, Meffert's Skewb Diamond (face-turning), and a new face-turning octahedron from Taiwan (the next higher order from the Skewb Diamond):	4.05*1019	?
	Dodeca Nona A faces-only dodecahedron. This type of 3D edgematching puzzle is included here because I consider them "faces-only" versions of twisty polyhedra - no corners and no edges. 12 magnetic pentagonal 2-sided tiles fit to the faces. Each face has the numbers 1 through 5 arranged around its corners - all 24 possible arrangements are included. Place the tiles so that at every vertex of the dodecahedron, the numbers add up to nine.	3.99*1018 1122 solutions.	?
	The Void Cube, designed by Katsuhiko Okamoto. Manufactured by Gentosha Toys. Purchased from Torito. The Void Cube won the Jury Grand Prize in the IPP 2007 Design Competition. When solving the Void Cube, you might run across a parity problem. To see the internals, see this thread on TwistyPuzzles.	1/12 of a normal 33 3.60*1018	Katsuhiko Okamoto
	Tantrix The Rock A truncated octahedron. This type of 3D edgematching puzzle is included here because I consider them "faces-only" versions of twisty polyhedra - no corners and no edges. Tantrix Home Page Jaap's page	1*1018 "over a billion billion"	?
	Mobius This type of 3D edgematching puzzle is included here because I consider them "faces-only" versions of twisty polyhedra - no corners and no edges.	?	?
	Octagonal Prism Jaap's page	4.5*1017	?
	Magic Octahedron	2.0*1015 (not including the trivial tips) 8.23*1018	?
	Christoph's Magic Jewel (a Pyraminx Octahedron minus the tips) I don't have this.		Christoph Bandelow 6-armed spider
	Square 1	1.2*1013	Dr. Vojtech Kopsky
I have three knock-offs one is broken :-(	3x3x2 Rubik's Domino	4.0*108	Erno Rubik
	Rainbow Cube Comes in 7-color and 14-color versions. Jaap's page	2.4*108 239,500,800	Bethel Japan
	Skewb Ultimate Jaap's page The Skewb Ultimate is the "most difficult" of the Skewb family - every piece has a proper orientation, unlike, for example, the face centers on the Skewb. The hierarchy is: Positions Moves to Antipode Puzzle 100,776,960 14 Skewb Ultimate 3,732,480 12 Halpern-Meier Tetrahedron 3,149,280 11 Skewb 933,120 11 Pyraminx 138,240 10 Skewb Diamond 2,160 6 Meffert's (4 color) Beachball	1.0*108	Uwe Meffert
	Rubik's UFO	4*107	Erno Rubik
Jaap says these are analogous:	Dino Cube #1 6-color each side different I don't have this Dino version.	1.9*107	?
	Halpern's (Halpern-Meier) Tetrahedron (Also comparison with Skewb.) This was produced commercially - but I have a custom-built example made by Matt Davis from cast pieces and a Skewb keychain core. Solve as a Pyraminx, then fix centers. My operator to flip two edges in place: L T' R T R' T L' T' When 3 edges around the top would be solved if only you could circulate them clockwise: (R T' R') T' (R T' R') Fix centers - swap F-L and R-D: (T' R' T R)*3	3.7*106 3,732,480	Ben Halpern Kersten Meier
	2x2x2 Pocket Cube Jaap's page Algs to finish bottom corners (begin by positioning 2 or 4): Swap adjacent front bottom corners: R'D' - R F - D - F'R' - D R - D2 Swap front left bottom diagonally: R'D' - R F - D2 - F'R' - D R - D Leave front left, turn other 3 CCW: R'D' - R - D'R' - D2 R D2 CW (inverse of above): D2 R' D2 - R D - R' - D R	3.7*106 3,674,160	Erno Rubik 6-armed spider. Quarks, from Fourier Idea, Inc.
	Meffert's Fisher's Golden Cube Perhaps the most famous of the Skewb mods, now produced commercially by Meffert.	?	Tony Fisher
	Skewb Jaap's page Meffert's solution. L means twist around down-front-left corner R means twist around down-front-right corner B means twist around down-back-right corner 0) Pick a top face, get 4 corners around it irrespective of their rotation. 1) Swap top-front-left with top-back-right if necessary: LBL 2a) Rotate TFL CW: LR L'R' 2b) Rotate TFR CCW: R'L' RL 3) Rotate bottom corners - depends on position of bottom-color facelets: 3a) None done: hold so 2 bottom-color facelets face right: LR'LR' L'R L'R 3b) Two done: hold so one not done is at DFR: do alg above, then do 3a. 4) Fix faces: swap F w/ D and L w/ R (fiddle combos until done) if 3 faces in a row are done, hold them L-B-R if 3 faces around a corner are done, hold them L-T-B [LR' L'R]x3	3.1*106 3,149,280	Tony Durham
	Einstein Cube A faces-only cube (granted, the faces are rounded). This type of 3D edgematching puzzle is included here because I consider them "faces-only" versions of twisty polyhedra - no corners and no edges. 12 positions per face. A similar puzzle called "Turn Twelve" has 24 positions per face.	12^6 = 2,985,984 (Einstein w/ 12 pos./face) 3*106 24^6 = 191,102,976 (Turn 12) 1.9*108	?
	Diamond Cube Jaap's page	2.0*106 2,425,500	?
	Bandaged Cube Jaap's page Andreas Nortmann has investigated bandaged cube variations - read his article (thread 3217) in the TwistyPuzzles forum. He says there are 7336 different bandaged cubes.	1.0*106 1,108,800	?
	Pyraminx My operator to flip two edges in place: L T' R T R' T L' T' When 3 edges around the top would be solved if only you could circulate them clockwise: (R T' R') T' (R T' R')	9.3*105 933,120 (not including trivial tips) 7.6*107 75,582,720 (with tips)	Uwe Meffert 4-armed spider
	Tetraminx (Snub Pyraminx - same as Pyraminx with trivial tips removed.)		Uwe Meffert
	Brainbow Jaap's page	623,760	?
	Skewb Diamond Jaap's page	138,240	Uwe Meffert
The Starburst is a custom mod.	Pyramorphix Jaap's page I find I can solve the Pyramorphix using only four operators (beyond fiddling to get it into a tetrahedron shape and properly position the corners): Move the bottom face down: D2 R2 L2 R2 Exchange the left and right faces: R2 L D2 R L2 D' (this twists corners,too) Twist Up corner clockwise: (R L' R' L)x2 Twist Up corner counter-clockwise: (L' R L R')x2	136,080	Rubik, Barry Lockwood
There are four versions of the Dino Cube: 2 colors 312.55/1 4 colors 528.99/1;535.55/1 6 colors, no dinos 337/1 6 colors, with dinos 667.96/1	Dino Cube 4 color version I find this very easy to solve even without operators. The four Dino versions shown for reference. I've only got the 4-color, but the others are just sticker variations.	42,000	?
	Tonne	?	?
	Morph (1x2x2)	Only 6 positions possible! I find this harder to mix up than to solve.	?

This section contains dihedral puzzles - puzzles whose halves can move relative to each other and permit the exchange of pieces between them. Their shapes aren't convex polyhedra, nor simple spheres, cylinders, or pucks.

Puzzle	Name and Notes	Combinations
	Smart Alex Dumitru A. Pop, patent on 26 May 1992 Jaap's page	6*1020
	Octo Bracelet	3.7*1018 (?)
	Sando Ring (aka King Ring)	4.97*1014
	Tricky Disky Jaap's page	2.1*1013
	Hungarian UFO	2.1*1013
	Brainball Andreas Unsicker Jaap's page	2*1012
	Netblock UFO Wai K. Chan	2.0*108 200,121,075
	Gerdig UFO Gerhard Huncaga Jaap's page	130,040
Group shot:	Roundy (4-leaf/4-color version) Purchased at IPP28 in Prague.	40320
	Roundy (3-leaf/6-color version) Fritz Gruber 12/7/93 patent 5267731	23040
	Saturn Jaap's page	5040
	Roundy (3-leaf/3-color version) Fritz Gruber 12/7/93 Jaap's page	2880
	Clever Disk	?

This section contains variations of puzzles, and custom-built puzzles.

You can hide the following notes by clicking the "_" button.

Notes About Sticker Variations

A 3³ has 26 cubic units visible on its surface - 8 corners, 6 face centers, and 12 edges. Each corner exposes three "facelets" - each face center one, and each edge two, for a grand total of 8x3+6x1+12x2= 24+6+24 = 54 facelets, each of which is usually covered with one sticker. In the standard configuration there are nine stickers of each of six different colors, and each of the six faces of the cube is stickered with a solid color. The basic sticker pattern has varied with respect to the six colors used and how they are arranged relative to each other. In addition, one or more stickers might bear a logo of some sort.

The cube is usually made from black plastic, but has appeared in other colors. A collector will have to decide if color variations of the body material are worth collecting for their own sake.

The corners and the edges permute in separate groups. Furthermore, the six face centers of the basic 3³ are fixed relative to each other and do not permute.

a) Corners can be "twisted" among 3 orientations
b) Face centers can be twisted among four orientations
c) edges can be flipped between 2 orientations

As the cube is twisted, the units permute within their groups, and the color arrangement becomes scrambled. The objective of the puzzle is to unscramble a scrambled cube, restoring the canonical color pattern on all faces simultaneously.

Sticker variations can serve four basic purposes:
1) pure decoration
2) advertising promotion
3) modify the difficulty of the basic coloring, without changing the basic objective of restoring a canonical pattern
4) alter the goal of the puzzle (e.g. calendar cube, sudo-kube)

Personally, I am uninterested in sticker variations of the 1st or 2nd varieties, except insofar as they simultaneously accomplish the 3rd or 4th purpose.

The "standard" stickering actually reduces the potential difficulty by making (b) unimportant - all face center orientations are indistinguishable. However, (a) and (c) are important. Some sticker variations can make (b) important, by making it possible for from one to all six of the face centers to have two or four distinct orientations. Some sticker patterns go further and add confusion by making the individual units and/or their proper orientations very difficult to visually distinguish.

Notes About Twisty Mods

To "mod" or modify a twisty puzzle is to create a customized version of a puzzle, usually by starting with one of the commercially available puzzles and making various modifications, or sometimes by building a new variety from scratch.

New custom puzzles will continue to be made and there will always be some new design-of-the-hour not covered here. To keep abreast of the latest developments, you should monitor the Twisty Puzzles forums.

Here is where to go to see additional custom-design puzzles:

• Ton Dennenbroek's site
• 3D Puzzle Reconstruction Studio - Katsuhiro Okamoto (Japanese only).
• Funny-Looking Twisty Puzzles - Carter Tarrer
• chrono.org - Jin Kim (An outdated FAQ is also here.)

Some designs will make it into production while others will be forgotten or remembered only as impractical curiosities. Artists will come and go, and pass away.

There are several people who have become fairly well-known in the twisty puzzle community for their custom creations, and many creations which have become recognized as "classics."

• Tony Fisher
• Anthony Greenhill
  Youtube video of Greenhill's Tetrahedron (Square 1 mod)
• Katsuhiko Okamoto
  Read an article by Carter Tarrer on Okamoto's Puzzles at TwistyPuzzles.com.
  Here is a video of Tony Fisher demonstrating Okamoto's Master Pyraminx.
• Jurgen Brandt
  Read an article by Geert Hellings about Jurgen Brandt's Puzzle Designs, on TwistyPuzzles.com.
• Hidetoshi Takeji
• Aleh Hladzilin
• Jin H. Kim
• Ton Dennenbroek
• Carter Tarrer
• Tyler Fox - ceated the Gigaminx.
  Here is a video of Tony Fisher demonstrating Fox's Gigaminx.
• Adam Cowan
• Drew Cormier
• Matt Shepit

There has been a recent flowering of new custom-crafted designs, brought on by broader knowledge of and availability of CAD design tools that can output STL (e.g. Solidview, Alibre, CoCreate by PTC, 3D Studio Max by Autodesk, Kompas-3D from Ascon, ViaCAD from Punch, VariCAD, BRL-CAD, others listed on Peter Eland's site ) (see Tyler Fox's tutorial on YouTube - part 1 - turn up the volume all the way), 3D printing services (e.g. 3dpartz.com, fdmonly.com, Shapeways, printo3d.com) that can take STL input and make master parts, and materials and techniques for casting parts from polyurethane plastic resin (e.g. Conap, Alumilite - get some at Hobby Engineering) using silicone rubber (e.g. Oomoo 25 or 30 - longer pot life, but longer cure time) two part molds poured in a Lego box. Don't forget a mold release agent such as Mann Ease Release 800 or 200. See the articles at TwistyPuzzles. Also this thread.

The traditional methods include cut-downs using a Dremel or hacksaw, and build-ups using Apoxie Sculpt (also here), Milliput, 1/8" ABS plastic sheet, or .040" polystyrene sheet. You can find materials at McMaster.

Many mods will use a black DIY core.

Use a Stika or some other vinyl sheet cutter (e.g. US Cutter) to create the stickers from Oracal 651 vinyl adhesive sheet.

Some of the amazing custom creations:

• Gigaminx (Tyler Fox aka TBT Tyler)
• Pentultimate by Jason Smith - see Jason's website Puzzleforge.com for a detailed look into the genesis of this puzzle. Read more about its history on TwistyPuzzles.com in this thread.
• By Drew Cormier (aka Drewseph0011)
• Gigaminx V1.5
• Teraminx
• Master Tetrahedron
• Kilominx (by David Litwin, also by Thomas)
• By Adam G. Cowan (aka Puzzlemaster42)
• Helicopter Cube
• Super-X Cube
• Trigonal, Pentagonal and Hexagonal Prisms (Lee Tutt)
• Several creations by Matt Shepit of New Zealand (aka the.drizzle aka kicktothegroin)
• Cheeseblock
• Toru
• Rua (face turning rhombic dodecahedron)
• Danger Cube (pillowed tetrahedral square-1)
• The 24 Cube (edge-turning Dino) - see thread and video

Scott Bedard has created a cooperative site named bedardpuzzles.com where some of these puzzles will be available for sale.

Which custom creations should a collector strive to obtain? All are hard to find, produced in very limited quantities (sometimes only one exists, or only instructions and piece examples or molds have been made available), and can be very expensive. You might consider building your own.

Puzzle	Name and Notes
	Tony Fisher's Mental Block 3x3x1 aka Rubik's Layer Custom-made by Tony, from a full-sized Skewb
	Floppy Cube 3x3x1 Custom-made by Katsuhiko Okamoto This won First Prize at the IPP26 Design Competition.
	Fisher's Cube / Diagonal Cube 8-color sticker variant
	A fully functional 2x2x3 - this mod is known as the "Slim Tower."
	Double and Triple Cubes Available commercially from various sources, made from keychain 2x2x2 cubes. Two cores share a corner in "Siamese" configuration.
	King Pillow Cube A commercially produced shape variation.
	Ultimate Cube A commercially produced sticker variation. I have an original in its packaging.
	Truncated Rhombic Dodecahedron This cube-variant is almost a rhombic dodecahedron, except the four-color centers are flat, not pyramidal. The three-color corners are pyramidal.
	3x3x5 Extended Cube This simple extended cube-variant has an extra piece glued to each of the nine facelets of two opposite faces.
	3x3x4 Extended Cube This simple extended cube-variant has an extra piece glued to each of the nine facelets of one face.
	Mini Evil Twin Designed by Mike Grimsley.
	Axis Cube Designed by Adam Cowan, made by Frank Schwartz.

Here are other unusual and interesting takes on the permutation puzzle...

The Planets puzzle consists of four spheres arranged in a tetrahedron within a frame. The spheres have various craters in them and are contrived to interlock so as to only permit certain rotations depending on where the craters are at any moment. Rotate the spheres so that each side of the tetrahedron is a uniform color.	Cmetrick is from eLogIQ. There are 6.9*109 possible positions. Jaap's page eLogIQ has also released the Cmetrick Mini.	I got an Enigma from Norman Sandfield at the 2005 NYPP. He said the reason they've been so hard to find is that the firm that makes them only sells them in bulk for advertising promos. However, recently I've seen a color version for sale at the Puzzle-Shop. [Jaap's Enigma page]
This is a variant of the Enigma, a French puzzle called "Combinescion."	This is the Spectra, by Eng's I.Q. Co. Ltd. 1987. 3072 positions. Jaap's page	Hoppa Gula
Rubik's Clock	Rubik's Rabbits	Rubik's Pen by Ideal from 1982.
This is a Boomdas puzzle from Asia. It is an interesting take on the 2-dimensional sliding puzzle, but using a linking mechanism similar to that of the Muto Cube, and with no frame. One side is numbered 1 to 9, the other has a stylized drawing of a figure.
The Virus	Kinato Hex Pro (Warning: website requires Chinese character set) and Kinato Hex 7	Orbik
Kabalabda Ball	Magellan This turns out to be based on the Four-Color Map Theorem. The objective is to ensure that all adjacent areas contain different colors on their wheels, on both sides of the puzzle at once.	Labyrint
Gear Up - designed by Oskar van Deventer made by George Miller	Eggcentric - designed by Oskar van Deventer

Slider Position	Cylinders which move
Down	1, 4, 5
Middle	3, 5
Up	2, 4, 5

To scramble the puzzle, when all 5 red segments are showing through the window, turn the egg small end down so that an internal weight moves to the small end. Push the slide one or two positions and turn the base several times. To solve, get the egg to stand upright on its large end. You have to line up all red segments in the window again. Each of the 5 cylinders has 10 segments one of which is red. So there are 10⁵=100,000 possible states. The 3-position slider controls which cylinders turn when you turn the egg's base. The instructions say all turns should be clockwise. From my experience, sometimes the cylinders "misfire" or skip. Here is a movement chart from the instructions - the slider position is either up (towards the small end), in the middle, or down (towards the large end). Number the cylinders 1-5 starting from the small end.

One of these went for $150 on eBay.

There are many 3-dimensional sliding piece (or sliding block) puzzles. Some consist of a framework or container inside of which are movable colored cubes. In some, there are moving marbles or beads instead of blocks, and in some cases the frame itself can be re-configured. Usually there is a single "hole" which can be thought of as moving around. Sometimes, however, the moving frame accomplishes the permutations of the piece positions and no hole is needed. In yet another sub-category, there are flat plates which can overlay each other. Still another sub-category accomplishes permutation using pieces as segments of interlocking rotating disks.

• Movable Gap, Rigid Frame
• Movable Gap, Movable Frame
• No Gap, Rigid Frame (Interlocking Orbits)
• No Gap, Movable Frame
• Overlapping Plates

Movable Gap, Rigid Frame
Pepsi Can Start with an idea as simple as mapping a 15-like puzzle onto a cylinder. This puzzle has advertised several popular drinks.	Billiards 9-Ball created by Joshua Frankel 3,628,800 positions Jaap's page	Massage Ball Otto Wu patent on 14 Feb 1995 6.1*1019 positions Jaap's page
Vadasz Cube (2x2x2 and 3x3x3 versions) (Also 4^3, which I don't have.)	Minus Cube (Russian)	Varikon black
Peter's Black Hole 5.4*1027 positions Jaap's page Twistypuzzles.com has an article by Ad van der Schagt titled "The History of Sliding Block Puzzles Before Peter's Black Hole" (PDF).	Clark's Cube	I-Qube
This is called the "Switch" or the "Knox Transposition Puzzle." It was issued by Mag-Nif in 1970 and also appears in their "Game Chest" set. The pegs slide in channels in the base. The object of the game is to exchange the sets of colored pegs in 24 moves or less. This actually borders on a non-jumping (exchange-only) type of Peg Solitaire.		Crossteaser 2.7*1011 positions [Crossteaser home page]
Inversion	Mad Marbles	Magic Jack
Tumbler - van Deventer	Pionir Cube	Panex Panex Puzzle resources page at Baxterweb Play a level-4 version online at cheesygames.com.
? (Hungarian barrel)	Diamond Bob's Billiards Eight Ball, and Diamond Bob's Diamond 8	Rubik's Brain Racker
	Bolaris Designed by Hannu Hjerppe of Finland - website at www.bolaris.fi. Purchased at IPP28 in Prague.
Movable Gap, Movable Frame
Mind Twister aka Wisdom Ball Yang Ju-Hsun 1 June 1993 1.7*1075 positions Jaap's page	Saturn - LD Belgium white and black versions	Tomy Great Gears 1.46*1020 positions Jaap's page
This is called Entrapment. There are also some newer "clones" available. The clear plastic on the old ones is yellowed with age.	Atomic Chaos Christoph Hausammann 2.1*1012 positions Jaap's page	Pakovalec aka Xylinder 1.3*1010 positions Jaap's page
Missing Link and the rarer Limited Edition Marvin Glass & Associates 8.2*1010 positions	Whip-it 5.7*108 positions (for the 3) Jaap's page	Bola RUVI (Whip-it Ball)
Ivory Tower and Babylon Tower both 6 rows x 6 cols 1.9*1040 positions Jaap's page	Varikon 4x4, 5x4 and 7x7 1.4*1014 positions Jaap's page	Backspin and a clone by Jaru Ferdinand Lammertink 6.4*1028 positions Jaap's page
Tomy Rack 'Em Up Mizunuma Masanori and Watanabe Hiroyuki 1984 6.3*107 positions	Tomy Row By Row Mizunuma Masanori and Watanabe Hiroyuki 13 Nov 1984 2.8*1031 positions Jaap's page	SpongeBob Puzzlepants 10,080 positions
Russian Flower	Touchdown	Calendar/Bank
Da Vinci's Mona Lisa Codebreaker Twisted Mind - another version, using numbers, and with a transparent case.	Heartache - Kohner	Double Sliding - Dario Uri
No Gap, Rigid Frame (Interlocking Orbits)
Equator, and Hungarian Globe 1.1*1025 positions Jaap's page	Hungarian Rings Endre Pap Also pictured - vintage cardboard/wood version Race War Puzzle Between Gold & Silver (I don't have). See U.S. Patent 507215 - Churchill 1893. 7.5*1019 positions Jaap's page	Magic 8
Rubik's Rings 1.9*1014 positions Jaap's page a source	Circle Puzzle 369,600 [Jaap's page]	Rotascope Raoul Henrique Raba 1982 9.1*107 positions I obtained this Rotascope which is a souvenir of the sixth IPP at Jerry Slocum's house. The front contains invitation text and Jerry's home address and phone number, which I don't want to display here. This is a picture of the back - not very puzzling without a pattern to scramble.
Tsukuda Magic Puzzle (Turnstile) Douglas Engel 6.3*109 positions Jaap's page
Lotica	Turn Push	Whirligig
Mad Triad 3.1*1045 (symbols matter) Jaap's Page	Handy Mad Triad 8.3*1023 positions Jaap's page	Rubik's Shells 4.7*1014 positions Jaap's page
Cmetrick Too There are colored disks riding in "craters" in spheres embedded in the frame. The spheres rotate and can exchange disks.	Cmetrick Too Hard In this more difficult version, the centers of the disks are colored, too.
No Gap, Movable Frame
Topspin Ferdinand Lammertink 2.4*1018 positions Jaap's page	Trillion - red, black Gunpei Yokoi 1.0*109 positions Jaap's page	Nintendo Ten Billion Barrel and Club Nintendo Star Barrel Gunpei Yokoi 2.7*1014 positions Solution site here.
I have seen this design from several places. I believe it has been called "Sortospherical."	The Orb[-it] Christopher C. Wiggs and Christopher J. Taylor 7.4*1028 positions Jaap's page	Astrolabacus John D. Harris Pat. 8 Jul 1997 3.6*1016 positions Jaap's page
Port to Port Triple Cross Ferdinand Lammertink Pat. Aug 6 1996 5.9*109 positions Jaap's page	Gripple Murray J. Gould, patented 5 April 1988 2.0*1013 positions Jaap's page	Russian Gripple
Magic Sphere	Rotos Jaap's page	Magic Cross (Zauberkreuz)
Flip Side - Thinkfun	Swissmad 369,600 [Jaap's page]	Tsukuda's Square
Rubik's Fifteen	Binary Bisect 5 - Doug Engel	Palette 7 - Doug Engel
Elemental: Neon (aka Biohazard) #051, designed and made by David Litwin		Uriblock (A custom version purchased at IPP28 in Prague.)
	Tri-Trick
Overlapping Plates
Mind Lock	3-Level Puzzle Dollar Tree	Jushbox

• Hessport's Rubik's Shop
• Hendrik Haak's Puzzle-shop.de
• Uwe Meffert's Site
• Cubikon.de
• Blake O'Hare's Nerd Paradise
• Rubik's Cube Museum
• Christian Eggermont's Square 1 Analysis
• Christoph Lohe's site
• George Helm's List of Puzzle Solutions
• Four-Dimensional Rubik's Cube