Rearrangement / Permutation

This section is devoted to puzzles having similar pieces which must be re-arranged, or permuted, often as groups, in order to progress from a randomized (mixed or scrambled) state to a solved state. This group forms a sub-class of the Sequential Movement puzzles.

The "Twisty Polyhedra" comprise a large and growing sub-class. Many of these puzzles are mass-produced (or hand-crafted modifications to a mass-produced puzzle), colorful, and made of plastic. 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. Useful sequences of moves are known as "operators" or "algorithms."

 My sections: Definitions Notes on Variations and Mods Twisty Polyhedra Family Charts Catalogue (go there for a detailed menu) Quick links: Cuboids, 2x2x2, 2x2x3, Domino, 3x3x3, Fisher's Cube, Speedcubes, Skewb, Mosaic Cube, Pyraminx, Halpern-Meier Tetrahedron, Megaminx Dihedral Twisty Puzzles - a sub-category where the moving pieces are essentially grouped in two halves which can move relative to each other and exchange pieces Strange Twisties Unusual Permutation Puzzles 3-Dimensional Sliding Block/Ball/Hole Puzzles These are the websites I consider to be reference standards for twisty 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. The Speedsolving.com Wiki Georges Helm's collection

The illustration below shows many (not all!) of the twisty polyhedra (and some non-polyhedral) puzzles that are now or have in the past been mass-produced and commercially available.

Starting in 2010, a new wave of twisty puzzles hit the market, and many custom designs made it into mass production. More continue to arrive and it is impossible to keep up. Some puzzles are members of a series exploring variations on a theme - e.g. the "Planets" series of Crazy cubes and dodecahedrons, various bandaged cubes, and the "Bermuda" cubes - I have not attempted to show all the series members. Also, there are many different brands and types of the conventional face-turning cubes, including speed cubes, which often turn better than the Rubik's brand version. Usually the mechanical differences are not apparent in photos. I have also not attempted to show the many shape extensions of the FT cubes, including the "Crystal" shapes, nor all the 2x2x2 heads and objects.

Newcomers to twisty collecting have asked for a short list of essential puzzles, and many TP forum members have provided guidance.

In the illustration below, I have identified 10 fundamental puzzles most collectors feel every twisty collection should contain.

Based on collating various folks' recommendations, I have also identified a further 33 puzzles to extend a good collection. Of course, you eventually have to have them all, right?

Tetrahedra --   -- FT Cubes & Cuboids --   -- VT Cubes --   -- Skewb Vars. --   -- Dino Vars. --   -- ET Cubes --   -- Square 1 --   -- Axised Cubes --   -- Crazy/Circle/Super Cubes --   -- Other Cubes --   -- Shape & Sticker Mods --   -- 2x2x2 Vars. --   -- FT Octahedra --   -- VT Octa. --   -- ET Octa. --   -- Other Octa. --   -- FT Dodecahedra --   -- VT Dod. --   -- ET Dod. --   -- Other Dod. --   -- Rhombic Dod. --   -- Icosahedra --   -- Other Shapes --   -- Dihedral --   -- Spheres --   -- Edgematching --

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

Definitions:

Notes on Variations and Mods

Notes About Sticker and Feature Variations

A standard face-turning 33 has 26 visible moving parts (usually referred to as "cubies")- 8 corners, 6 face centers, and 12 edges, plus an internal core - a six-armed "spider."

As faces are turned, the corners and the edges permute (i.e. exchange positions with each other) in separate groups. The six face centers of the basic 33 are attached to the ends of the internal spider, so are fixed relative to each other and do not permute. Each cubie can also assume different orientations depending on type:

• Face centers can be twisted among four orientations - depending on the coloring, either all rotations are distinguishable, or 180 degree rotations are indistinguishable, while +/-90 degree rotations remain distinguishable
• Edges can be flipped between 2 orientations
• Corners can end up "twisted" among 3 orientations

We assume that neither corners nor edges can be re-oriented if they cannot first be permuted, since unlike the face centers they do not twist in place.

Each corner exposes three "facelets" - each face center one, and each edge two, for a grand total of 8x3+6x1+12x2= 54 facelets. A sticker pattern is applied to the facelets, and is used to either make apparent or hide the permutations and orientations of the cubies.

Each facelet 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.

As the cube is twisted, 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
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.

So, for a 3x3x3 with a uniform pattern, we have the following possible fundamentally different sticker variations:

3x Faces: (permutations not possible) - NO orientations visible (N), ALL orientations visible (A), or 180s obscured (O)
3x Edges: neither permutations nor orientations visible (N), permutations only visible (P), or both permutations and orientations visible (P+O)
3x Corners: N, P, or P+O

This gives 27 possibilities:

F:N
• E:N
• C:N = an unstickered cube, eq. to a 1x1x1
• C:P = eq. to the PyraDiamond - a 2x2x2 with truncated corners
• C:P+O = eq. to the standard 2x2x2
• E:P
• C:N
• C:P
• C:P+O
• E:P+O
• C:N = the traditional or void "edges-only" cube
• C:P = eq. to the truncated Trajber's Octahedron
• C:P+O = the standard 3x3x3 pattern
F:A
• E:N
• C:N = the simple babyface
• C:P = an 8-color simple overlapping cube
• C:P+O = a 12-color simple overlapping cube
• E:P
• C:N
• C:P
• C:P+O = eq. to the 3x3x3 Rhombic Dodecahedron
• E:P+O
• C:N = eq. to the Magic Octahedron or Christoph's Jewel
• C:P = an 8-color cube; eq. to the Trajber's Octahedron
• C:P+O = a "Super Cube" - commercially produced as the Ultimate Cube
F:O
• E:N
• C:N
• C:P
• C:P+O
• E:P
• C:N
• C:P
• C:P+O
• E:P+O
• C:N
• C:P
• C:P+O

Andreas Nortmann explores the first 18 members of this series in the TP forums.

We might also imagine applying the pattern non-uniformly - for example, making only a subset of the edge and corner permutations matter (and none of the orientations matter), as in the tri-color cube, or making only a subset of the face orientations matter, as in the Fisher Cube or the Rubik's Cube Fourth Dimension.

 Tri-color Cube Fisher's Cube Fourth Dimension

Note that sticker variations are different from feature variations. Feature variations arise when actual physical features are present or absent. Any features that are present can, in turn, be stickered according to the possible variations. Also, feature variations can be used in lieu of sticker variations to enforce the visibility of some specific permutations or orientations.

A physical design might be contrived to disguise or eliminate face centers, corners, 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 path of a great circle equivalent. Lately it has also become possible to eliminate the central core, forming a "Void" or "Holey" cube.

If we apply a uniform variation to all like features of a cube, we have the following possibilities:

• FEC - all features present
• FEN -  faces and edges but no corners - can be implemented in different ways:
• - this is what is typically referred to as an "edges-only" cube even though technically that's inaccurate (4x4x4 also shown)
• - by simply shaving down and blacking out the corners
• FNC -  faces and corners but no edges - can be implemented in different ways:
• - as a "simple overlapping cube" (4x4x4 version also shown)
• - by simply shaving down and blacking out the edges - the "Corners Cube"
• - taking this to its extreme, we end up with Oskar van Deventer's "Gerardo's Cube"
• FNN -  faces only - can be implemented in different ways:
• - a "babyface" cube - each face can be turned (oriented) in place, colored to make an edgematching challenge (4x4x4 Babyface also shown)
• - Oskar van Deventer has created the PantaCube, which has only five of six faces but allows those five faces to be permuted.
• NEC -  edges and corners but no faces - can be implemented in different ways:
• Eliminate the core, as in the Void or Holey Cube
• Hide the faces using overlapping, as in the "Brilicube"
• Just sticker all six faces gray or leave them unstickered - i.e. ignore them - which isn't really a feature variation after all, more a sticker variation
• NEN -  true edges-only - can be implemented in different ways:
• Using a Void Cube, cutting down and hiding the corners, then extending the edges to overlap them - this has been mass-produced.
• Extending edge plates to overlap hidden face centers and corners, so the cuts make a big X on each side (4x4x4 version also shown)
• NNC - true corners-only - since any size cube has only eight corner pieces, this boils down to the basic 2x2x2
Also consider the Nightmare Cube, in which overlapping corner pieces hide an internal 3x3x3 that has been bandaged.
• NNN - nothing moves - this is the trivial 1x1x1

Andreas Nortmann discusses these variants in the TP forums.

Edges and corners can also be truncated rather than entirely eliminated. Furthermore, we might elect to apply feature variations selectively to only a subset of cubies. Andreas Nortmann has created the series of all possible corner truncations of the 3x3x3 and shows it in the TP forums. (And earlier, here.)

Sticker and/or feature variations will differentiate one puzzle from another otherwise of the same order, and can dictate different solution methods. In the most complex case, you need algorithms to permute corners, orient corners, permute edges, orient edges, and orient faces. (The PantaCube, and the Cubedron family, require algorithms to permute faces.)

Lastly, all we've said here about sticker and feature variations regarding the face-turning 3x3x3 cube, could be applied sytematically to other orders, to other turning regimes, and to other shapes.

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.

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." Here are just a few:

• Tyler Fox - ceated the Gigaminx.
• Adam Cowan - Axis Cube, Helicopter Cube
• Jason Smith - Pentultimate - 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.
• Drew Cormier - Master (Halpern-Meier) Tetrahedron, Master Skewb, Teraminx, Petaminx - see Drew's YouTube channel
• Matt Shepit - Cheeseblock, Toru, Rua (face-turning RD), Danger Cube (pillowed tetrahedral Square-1), The 24 Cube thread and video - see Matt's YouTube channel
• "Pink"

There has been a flowering of new designs, brought on by broader knowledge of and availability of CAD design tools that can output STL, and 3-D Printing services (e.g. Shapeways, 3dpartz.com, printo3d.com ) that can take STL input and make master parts or full prints.

There is also greater awareness of 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.

This section contains a series of charts I made to organize my thinking about the relationships among various puzzles of each shape. You can click the "_" symbol to hide it, if you'd like.

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. 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.

Catalogue of Twisty Polyhedra

The following sections catalogue polyhedral twisty puzzles in my collection.

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 I originally employed here. However, I have revised the catalogue to better group related puzzles.

I include a few puzzles I do not own, for reference - they are noted as such by using this particular background color for their row.

Custom-made or 3D printed puzzles are highlighted like this.

For the best catalogue of twisty puzzles, I don't think one can beat the museum section at TwistyPuzzles.com. The curators and Andreas Nortmann in particular have done a fantastic job!

Jaap's Puzzle Page was also invaluable in teaching me what puzzles were out there, and for providing combinations data for several puzzles.

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.

Tetrahedral Twisties

Face/Vertex-turning Order 1
==>  Move cuts from tips inwards until they touch, forming inverted triangles on faces...
Space Pyramid

Space Pyramid
Just trivial tips.
"Junior Pyraminx" [T]; "Mini Pyraminx" [T]
Hoberman Brain Twist

Hoberman Brain Twist

From a company not known for twisty puzzles. A unique take on what would otherwise be a trivial vertex-turning tetrahedron - the tips invert and you can turn the whole puzzle kind of "inside-out." This idea has not been used on any other twisty AFAIK.

Continue moving the cuts from the vertices towards the opposite faces -
three different basic forms result...
(All can have tip cuts, which result in either trivial tips on vertex-turning forms,
or edge-turning O2 forms.)

GB 5.1.1 (No name.)

The first of these is GB 5.1.1. It has stationary centers. AFAIK, no-one has built this.
Pyraminx - GB 5.1.2 - Meffert

9.3*105 = 933,120 (not including trivial tips);
7.6*107 = 75,582,720 (with tips)
- Uwe Meffert - 4-armed spider

The Pyraminx
An original issued by Tomy, loose and in package, a Meffert's 25th Anniversary version, and Meffert's New Pyraminx, in black.
The Pyraminx is not deep-cut.

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')

Bandaged Pyraminx

The tetrahedral version without trivial tip cuts is the Bandaged Pyraminx, built by Thomas de Bruin [T]
I don't have this.
Tetraminx (Pyraminx sans trivial tips)

Tetraminx
(A Snub Pyraminx - same as Pyraminx with trivial tips removed.)
Mefferts version, and transparent version from Smaz.
Cubominx (Tony Fisher)/ Eye-Skewb (Eitan Cher)

A Pyraminx/Tetraminx (or Skewb) can be modded to a Cubominx [T] [T] [T] [Y] which is kind of a "half-Skewb" - curvy cuts give Eitan's Eye-Skewb [T]

Halpern-Meier Tetrahedron - GB 5.1.3 - Matt Davis

3.7*106 3,732,480 - Ben Halpern, Kersten Meier

This version of the order-1 tetrahedron is called Halpern's Tetrahedron or the Halpern-Meier Tetrahedron.
The HMT is deep-cut - all cuts go through its center.

This was produced commercially but that vintage version is very rare. I have a custom-built example made by Matt Davis from cast pieces and a Skewb keychain core. Also shown is a comparison with a Skewb.

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

Jing's Pyraminx - GB 5.1.3 - Meffert

Meffert produced a Reuleaux version of the HMT, designed by Adam Cowan. They call it Jing's Pyraminx. It is fairly large. Also shown is the Jade Club Pyraminx, which is smaller than the custom HMT.
Edgeless HMT - Rob's Pyraminx - Pyraminx Duo

I am honored that the multi-talented Dutch puzzlist Oskar van Deventer [W] [S] has named one of his puzzle designs after me - Rob's Pyraminx. [T] Oskar named this previously unrealized (AFAIK) design after noticing it in my "family tree" chart of tetrahedral twisty puzzles.

Meffert's has mass-produced Rob's Pyraminx - for commercial release it is called the Pyraminx Duo.

The Pyraminx Duo is arguably the simplest non-trivial twisty puzzle - it is easy to solve intuitively and makes a fun gift for folks who normally wouldn't consider themselves avid puzzlers. If you would never even think of attempting a Rubik's Cube, don't worry - you can solve this! With its intriguing geometric shape, it looks great, too, and would be equally at home on an executive's desk or a teen's bookshelf.

Oscar Roth Andersen has even devised a game that can be played with the Pyraminx Duo!

You can purchase black or white versions at Meffert's online shop, the Jade Club, or hknowstore.

Cornerless HMT

The HMT has been made in cornerless form, by "jesseking." Also by "kwsjack054" [T]
24 Tetrahedron (magnetic) - Tanner Frisby

24 Tetrahedron
6 Cuts - every edge is bisected - each cut is a plane that goes through the midpoint of an edge and coincides with the opposite edge. A turn moves half the puzzle.
Magnetic version by Tanner Frisby [T]

Claus Wernicke built one based on the 24 Cube by Matt Shepit. [T]

Face/Vertex-turning Order 2
Master Pyraminx - GB 5.1.4 - Evbatyrov / Cowan / Meffert

2.2*1014 positions ignoring trivial tips.

The Master Pyraminx, issued by Mefferts
Designed by Timur Evbatyrov and Adam Cowan
Another great design mass produced beautifully by Meffert!

For the longest time, Meffert could not find a way to produce the next higher order relations to his Pyraminx. Early custom puzzles made by Japanese modder Katsuhiko Okamoto [T], while clever and beautiful, were considered too fragile for the mass market. Back in late 2008 Cowan pioneered the use of the reuleaux tetrahedron to solve some of the design issues: wikipedia, [T], and Timur Evbatyrov was the first to offer a reasonably priced master pyraminx via his shapeways shop.

Shim's Master Pyraminx - Timur Evbatyrov

I dyed, assembled, and stickered my Shim's Master Pyraminx. I really like this puzzle! It was designed by Timur Evbatyrov and is available on Shapeways.

Two photos show relative size - a comparison with an original Tomy Pyraminx, and a group photo including various tetrahedral twisty puzzles.

The group photo includes, left to right, row by row from the top: the Hoberman BrainTwist, Meffert's Jing's Pyraminx (designed by Adam Cowan), Meffert's NGP (Platypus), Tomy Pyraminx, a custom Halpern-Meier Tetrahedron (keychain Skewb core) made by Matt Davis, Meffert's Pyramorphinx (a curvy Mastermorphix), Traiphum Prungtaengkit's (Traiphumi's) Mastermorphynx (a custom-made edge-turning Pyraminx), a keychain Meffert's Pyramorphi[n]x, Shim's Master Pyraminx, and a reuleaux Babymorphix custom-made by Taylor Howell.

Master Tetrahedron

The order-2 HMT is called a Master Tetrahedron. First made by Drew Cormier [T] [T].
Drew also made a pillowed version he called the RX.

I obtained a Reuleaux Master Tetrahedron made by Hung Nguyen (cublem).

Face/Vertex-turning Order 3+
Professor Pyraminx - GB 5.1.10 - Timur Evbatyrov / Meffert

The Professor Pyraminx, issued by Mefferts
Designed by Timur Evbatyrov
Yet another great design from Timur, mass produced beautifully by Meffert! I love this puzzle!
Elite Tetrahedron - Chris Hemerich

Elite Tetrahedron

The Elite Tetrahedron was first made by Drew Cormier [T] [T] [T]

This pillowed version was made for me by Chris Hemerich [T] [T] [S] [Y]

The Elite Tetrahedron is also shown in comparison to Meffert's Professor Pyraminx and Vergo's Master Pentultimate.

Royal Pyraminx - Timur Evbatyrov

The Royal Pyraminx
Designed by Timur Evbatyrov
Royal Tetrahedron - Drew Cormier

The order-4 HMT is Drew's Royal Tetrahedron. [T]
Edge-turning Order 1
Pyramorphix / Pyramorphinx / Ruby's Triangle / Figurenmatch

136,080 - Rubik, Barry Lockwood

Pyramorphix aka Pyramorphinx, with various clones
Also the East German FigurenMatch
Jaap's page

This can be solved in the same way as a 2x2x2.

Babymorphix - Taylor Howell

The Babymorphix is a Reuleaux Pyramorphix, custom-made by Taylor Howell.
Edge-turning Order 2
Mastermorphix - GB 5.2.3

Mastermorphix

This is a 3x3x3 mod. Center orientations matter. Back in 2005, these were not mass-produced, and hand-made custom versions could sell for over \$100.

Traditional version designed by Tony Fisher, produced in China, offered with official Fisher stickers by Meffert's - black and white versions. Meffert's "Master Pyramorphinx" (the "curvy" version). Also known as a "Rice Dumpling" (stickerless).

Edge-turning Order 2
Mastermorphynx - GB 5.2.1 - Traiphum Prungtaengkit

From Traiphum Prungtaengkit, of Thailand, an Edge-turning Pyraminx! He calls it a Mastermorphynx. [T]
Also shown compared to a curvy Mastermorphix and a Pyraminx.
Edge-turning Order 2
Edge-turning HMT (No name.)

An edge-turning HMT. Not made AFAIK.
Edge-turning Order 3+
Megamorphix

The Megamorphix. Based on a 4x4x4 cube.

Made by Adam Zamora [T], Apollo [T], Curvy Megamorphix by Traiphum [T]

Centerless Megamorphix - GB 5.2.4

A Centerless Megamorphix.
Ultramorphix

Ultramorphix - based on a 5x5x5 cube.

Made by Danny [T], Traiphum [T], judochiou625 [T]

Traiphum has made many orders of curvy morphix [T] - 6x6x6 "HexaPhobic" [T]; 7x7x7 [T]

Geared
Gear Pyraminx - Timur - Meffert

Gear Pyraminx - from Meffert's, design by Timur Evbatyrov
(I have black and white versions.) Also a Series II in black.
Gear Mastermorphix

A Gear Mastermorphix
Circle / Crazy
Crazy Tetrahedron - Mf8 / DaYan

Mf8 DaYan Crazy Tetrahedron (Jupiter)
also Standard version (all circles turn with opposite vertex).
There are several versions.
Related
Starburst

The Starburst has been mass produced, but was also a common custom mod of a Pyramorphix - Just add tetrahedrons to the "empty" triangular faces (they do not rotate in place).
Vulcano / Trignis - Timur / Meffert

From Meffert's, the Vulcano (aka Trignis) designed by Timur Evbatyrov.
Dinomorphix - Traiphum

Dinomorphix - designed by Traiphumi, produced by Calvin Fan [T] [T]

A hybrid face-turning and vertex-turning puzzle - each of the four faces turns like a Babyface puzzle. In addition, each of the four vertices can turn, like the trivial-tipped puzzles with this cut pattern, but since the faces also turn, the corners are tri-partite, making them non-trivial.

Fun fact: I helped Triaphum choose the name for this puzzle!

Platypus / Jackpot / NGP (New Generation Puzzle) - Meffert

Mefferts Jackpot and NGP. Designed by Yusuf Seyhan. Patented back in 2003.

Same turning regime as Triaphum's Dinomorphix - the faces turn, and the tripartite tips turn. Issued prior to the Dinomorphix. However, unlike the Dinomorphix, there are hexagonal face-centers and the tips have triangular faces - both of which show orientation that can matter. There are several variations and knock-offs. [W] [W] [W] [W]

Jaap says these are related to the Dino Cube, which is very easy to solve. Jaap's page.

Hexahedral (Cubic) Twisties

Some of these face-turning cubes are also shown in the Cuboids section. Carl Hoff has created a vertex-turning cuboid he calls the Bubbloid, based on some ideas by Anthony Villaveiran (boublez). [T] [T]

Face-turning Order 1
Rubik's 2x2x2 Pocket Cube - GB 3.1.1

3.7*106 = 3,674,160 - Erno Rubik; 6-armed spider.

Rubik's Pocket Cube

Solve using a subset of the 3x3x3 algs.

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

Various 2x2x2 Cubes

Rubik's Pocket Cube, Studio Mini Cube, Ice Cube, Yuxin 2x2x2, V-Cube V-2, New Spring clear (with transparent stickers), and a version with interior tinted pieces

Rubik's Soft Cube
A fully functional 2x2x2, about 4" on a side, with fabric-covered soft cubies.

Bump Cube Jr.

Asymmetric spacing of cuts results in the Bump Cube Jr. made by Thomas [T]
Face-turning Order 2
The 3x3x3 Cube

4.325*1019 - Erno Rubik - 6-armed spider - BE887875

3x3x3
Rubik's Cube

Solving the 3x3x3

20 moves suffice!

In August of 2010, Morley Davidson, John Dethridge, Herbert Kociemba, and Tomas Rokicki proved that every scrambled position of Rubik's Cube can be solved in 20 moves or less. (Finding those 20 moves, however, can be quite a challenge!) Read more at www.cube20.org. In January of 1995, Michael Reid proved that the "Superflip" position (corners correct, all edges placed but flipped) requires 20 moves, so 20 was known to be a minimum.

 See the Speedsolving.com Wiki. Also see Ed Karrels' page. I learned using a layer-by-layer (LBL) method: Top Edges (intuitively) Three Middle Edges (intuitively) Top Corners (intuitively, using unsolved middle edge as keyhole) 4th Middle Edge (2 algs) Permute Down Corners (2 algs) Orient Down Corners (1 alg + inverse) Permute Down Edges (3-cycle alg, optional Zperm & Hperm) Orient Down Edges (3 algs) Step 4: move an edge from D to FL: Start in FD matching F: D L D' L' - D' F' D F Start in LD matching L: D' F' D F - D L D' L' Step 5: permute D corners: (begin by positioning 2 or 4): Swap adjacent FD corners: {R'D' R F} D {F'R' D R} D2 Swap front right down diagonally: {R'D' R F} D2 {F'R' D R} D' Step 6: orient D corners: 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 Step 7: permute D edges: If an edge is already in position, hold it in DF. clockwise (viewed from bottom) 3-cycle of DL->DR->DB: L'R - F - LR' - D2 - L'R - F - LR' aka, where ( means L'R, and ) means LR': (F)D2(F) counterclockwise: (F')D2(F') You can always use the 3-cycle repeatedly, but here are a couple of shortcuts: Zperm - swap DF-DL and DR-DB: (M = middle vertical layer, same dir. as L) M'D' M2D' M2D' M'D2 M2D Hperm - swap DF-DB and DR-DL: M2D' M2D2 M2D' M2 Step 8: orient last edges: Flip FU and FR in place: R'D'L' { U' F' U F'} L D R { U F U' F } Flip FU and FD in place: { U' F' U F' } L D R { U F U' F } R'D'L' Flip all four D edges in place: (F2) D2 (F) D2 (F2) D' Here are some extra algs: Superflip (every edge flipped in place) algs - see Michael Reid's page, Walter Randelshofer. Here is one: U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2 Here is the checkerboard pattern: U2 D2 F2 B2 R2 L2 Other patterns: Michael Reid. "Six Spot" - U D' R L' F B' U D' Cube in cube - F L F U' R U F2 L2 U' L' B D' B' L2 U
Various Rubik-brand 3x3x3 Cubes

Various 3x3x3 Speedcubes

 Notes on speedcubes Types I have are highlighted like this. Types I have and prefer are highlighted like this. Rubik's Brand Regular Rubik's storebought 25th Anniv. Cube Rubik's Icon Cube Rubik's Studio Cube Rubik's Cube Deluxe (tiled) Rubik's Game Rubiks.com DIY 3x3 Assembly Cube Politoys Cube Rubik's brand Japanese speed cubing kit - JSK (There is also a JSK clone.) Chinese Cubes - Type A All Type As are designed and manufactured by the Chinese company "Guo Jia." There are two sets of Type A’s - the second set has the extra designation "quanfeng" which means “all sealed.” Type A I (or "old"): GuoJia Type A II: GuoJia 2 Type A III: GuoJia QuanFengBi Type A IV: parts come on trees - plastic washers - IV has tracks on edge piece, V does not; fragile corners, non-cubic edges Type A V Other Chinese Cubes Type B: GuoYi (made by ShengEn) Type C: GuoBing (Rubik's DIY replica) Type CII - sealed Type D: GuoYou (made by YongJun, a.k.a. YUGA) Type DII - YUGA II - sealed Type E (Diansheng no.222) Type E (Diansheng no.333) Mini Diansheng Type F: GuoYi BanFengBi (made by ShengEn) Type FII - sealed Type G (hknowstore) Other Speedcubes Cubeforyou.com brand DIY Ghost Hand Cube (aka Popbuying Fingertip Dancing) Magic Cube Ming Ho (see hknowstore) Clown Cube ("Revenge Cube" with clown picture on package insert; 8906B; painted "stickers") Slick Cube - www.slickcube.com (type D clone?) HeShu (see kcobe.com) Edison Cube (Korean) Joy Cube (Korean) DAISO (1 dollar store) - painted - fake type F Haiyan cubes - Haiyan Zhuang holds a blindfolded world record - www.cubehaiyan.com Haiyan's cube - Memory = New Type A V + Sanding Haiyan's cube - Haiyan = Haiyan's New Cube DaYan I Tai Yan (aka Big Goose) DaYan II GuHong Cube - allows reverse corner cuts - my old favorite 3x3x3! DaYan III LingYun - even better than the GuHong II. DaYan IV LunHui - after re-tensioning and lubing, my new favorite - even better than the ZhanChi V DaYan V ZhanChi - a great cube right out of the box CUNI 3x3x3 with metal internals by Alpha Cube

The Chinese twisty puzzle company DaYan has offered a series of five 3x3x3 cubes, each with a different internal design.
I took this comparison photo showing an edge piece and corner piece from each, to help keep them straight.
From left to right: 1 Tai Yan (Big Goose); 2 GuHong; 3 Ling Yun; 4 Lun Hui; 5 ZhanChi.
My current favorite is the DaYan 4 Lun Hui.

Face-turning Order 2 with Feature Modifications (not Shapemods)
Bump Cube / Rubik's Mirror Blocks - Hidetoshi Takeji

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 a mass-produced boxed copy signed by Hidetoshi-san.

A key innovation and variation on Rubik's theme - in this case, you don't solve by color. Rather, the center has been offset so that every piece's three dimensions differ from every other piece's dimensions. The cube must be solved according to shape! When scrambled, it becomes a bumpy mess that can be very confusing on first encounter. Fortunately, it can be solved using the same algorithms developed for Rubik's Cube.

Hanzoh aka Half-turn Cube - Oskar van Deventer

Hanzoh aka Half-turn Cube produced by Oskar [T], also made by Hidetoshi Takeji [T], based on an idea by Takafumi Haseda
Void Cube - Katsuhiko Okamoto

1/12 of a normal 33; 3.60*1018 - Katsuhiko Okamoto

The Void Cube, designed by Katsuhiko Okamoto.
Manufactured by Gentosha Toys. Purchased from Torito.

Also a more recent Rubik's Void.

The Void Cube won the Jury Grand Prize in the IPP 2007 Design Competition.

It is kind of an "inside joke" for cube fans who know how the original Rubik's 3x3x3 internal mechanism works by employing a six-armed spider fixed to the centers. The Void cube has no centers and no internal spider! How do the pieces stay together? A design triumph. Spawned other "holey" designs - e.g. the Holey Megaminx, the Holey Skewb.

When solving the Void Cube, you might run across a parity problem.

To see the internals, see this thread on TwistyPuzzles.

Edges-only 3x3x3 - Smaz

An Edges-Only cube from "Smaz."
Edges-only Void 3x3x3

An Edges-Only Void cube.
Simple Overlapping 3x3x3 - Kevin Phelan

I received a pleasant surprise in the post, in the form of this great Simple Overlapping Cube twisty puzzle, cleverly made by TP forum member "Zzupler" (Kevin Phelan, of Ireland) [T], and originally designed by David Calvo [T].

This "corners-only" 3x3x3 is kind of the brother to the "edges-only" 3x3x3 above.

It is nicely done and turns very smoothly. Thanks, Kevin!

Brilicube

The Brilicube. Originally designed by Aleh Hladzilin. [T]
It is a 3x3x3 cube with hidden face centers.
This version purchased from the Shapeways shop of "grigr." [T]
This is the first Shapeways puzzle I got in black strong flexible material.
It started very tight and difficult to move, but is better after much wearing of the pieces, and lubrication.
Face-turning Order 3+
4x4x4

7.4*1045 - Peter Sebesteny - grooved sphere

4x4x4
Rubik's Revenge
Eastsheen A4 (a different mechanism)
Maru 4x4x4
Shengshou 4x4x4
Ghost Hand 4x4x4
QJ Pillowed 4x4x4
Moyu Aosu 4x4x4 stickerless

There are now several 4x4x4 mechanisms available - some of which are superior to the original Revenge, which is very stiff and prone to the center stems breaking.
• Original Rubik's Revenge
• Eastsheen - turns smoothly but catches
• Meffert's
• QJ (aka "Clefferts" - a clone of the Mefferts)
• Maru (similar to V-6 mech)
• Mf8/DaYan
• Lanlan (similar to QJ)
• Shengshou 4x4
• Ghost hand 4x4 II (similar to Shengshou)
• X-Cube 4x4 (has "Pi mod" - core stays aligned)
5x5x5

2.8*1074 - Udo Krell - 6-armed spider

5x5x5
Rubik's Wahn
Professor Cube
Eastsheen A5 (a different mechanism)
QJ Pillowed 5x5x5

6x6x6

1.57*10116 - Panagiotis Verdes

6x6x6
V-Cube 6
from Verdes Innovations.

7x7x7

1.95*10160 - Panagiotis Verdes

7x7x7
V-Cube 7
from Verdes Innovations.

Verdes worked diligently over several years to finally come up with an internal mechanism that would allow cubes beyond 5x5x5 to be made. Prior to their achievement, most assumed this would be impossible due to fundamental geometrical limitations of the spherical cutting patterns employed inside the cube mechanism. Basically, as the cube grew larger to accommodate more edge pieces, some pieces would no longer have any way to connect to the core. Verdes' design ideas are the subject of a key patent. There have been several alleged infringements, coming out of Asia. Verdes has produced a 6x6x6 and 7x7x7, but even though their method allows up to an 11x11x11, only clones exist beyond 7x7x7. You can find 8x8x8, 9x9x9, and 11x11x11 online. They are controversial among collectors because of the perception of disrespect to Verdes. As an exercise in what is possible, Oskar van Deventer produced a 17x17x17 of his own design, called "Over the Top." [Y] [S] [W]

Vertex-turning Order 1
Skewb - GB 3.2.1

3.1*106 = 3,149,280 - Tony Durham

The Skewb - The order-1 vertex-turning cube
(Mefferts stickered and tiled versions)
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

Holey Skewb / Void Skewb

Holey Skewb twins from Meffert (designed by Tony Fisher)
Meffert's Pillowed Holey Skewb, in black, from PuzzleMaster
Asymmetrically cut Skewbs

Asymmetric spacing of cuts, or asymmetric cuttings, results in different families of puzzles, such as Okamoto's Offset Skewb.
Vertex-turning Order 2
Just as with the O2 tetrahedral puzzles, three basic O2 VT forms arise as the two O2 cuts (blue) are moved away from the original Skewb cut (red)...
 3.2.2 Master Skewb 3.2.4 Dino also Cubocta = Rainbow Cube Oskar's Redi Cube is a Dino with visible corners Compy Cube idea by Wayne Johnson [T], realized by Jason Smith [T], earlier paper model by Robert Webb [T]

Carl Hoff introduced a clever animation [T] he calls an "order-2 corner-turning multicube" - it shows how moving cuts define different puzzles of the same order.
Carl shows the cuts moving inwards from trivial tips, converging on the single skewb cut, which the animation stops before reaching:

Master Skewb - GB 3.2.2 - LanLan

Lanlan Master Skewb in black (also have one in white)

This is a relative of the FTO, and the Rex Cube is in turn a relative of this.

Here is a diagram by "Allagem" (Matt Galla) [T] showing how the Master Skewb and FTO are similar:

The MS corners have no equivalent on the FTO, but they do have equivalents (the face centers) on GB 4.1.4 [T].

Master Skewb - GB 3.2.2 - "Cublem"

A pillowed white Master Skewb made by TP forums member "Cublem" [T]

I bought this before the LanLan mass-produced version came out. It wasn't the first or the last time I have bought an expensive hand-made (or 3D printed) puzzle only to see an inexpensive mass-produced version appear later. Most collectors run this risk and accept it.

Rex Cube - GB 3.2.6 - Drew Cormier

Drew Cormier's Rex Cube [T]
is equivalent to a cornerless Master Skewb [T].
First designed by Drew Cormier back in early 2009 [T] [Y], then produced commercially (without his knowledge) [T].
Now offered via Meffert, with a royalty going to Drew.

Drew's Praxis Cube (I don't have) is an "axised" Rex. [T]

Dino Cube - GB 3.2.4

There are four sticker-variations of the original vintage Dino Cube:

 2 colors 312.55/1 4 colors 42,000 states 528.99/1;535.55/1 6 colors, no dinos 337/1 6 colors, with dinos 1.9*107 states 667.96/1

James R. Holloway 1995
U.S. patent 6056290

I have an original boxed four-color version. (I traded one to Kevin Uhrik for a Master Octahedron.) The Dino has been re-issued by Smaz, who provided the version with the attractive hollow stickers. I also got a version with repro Dino stickers.
I got a pillowed void dino cube from Calvin.

Rainbow Cube

2.4*108 = 239,500,800 - Bethel Japan

Rainbow Cube
Comes in 7-color and 14-color versions.
Very easy to solve intuitively.
Jaap's page

When the Dino Cube became scarce, the Rainbow Cube was a good substitute - they are basically the same puzzle - the Rainbow Cube is a cuboctahedral form of the Dino. A common mod was to transform a Rainbow back into a Dino.

Dino Cylinder - Smaz

Hong Kong puzzle designer and craftsman Smaz has mass-produced his Dino Cylinder design [T] [T]. His original "hollow" stickers make for a beautiful puzzle! It is even shipped in a nice black velour drawstring bag.

This puzzle solves like a Dino or Rainbow and is fairly easy. However, unlike the Dino, this puzzle exposes corner pieces each of which can be independently oriented in one of 3 positions. Using the same notation I used for the Mosaic Cube, here is an algorithm to rotate the df corner clockwise by 120 degrees:

ur' df' dr' -- df dr ur -- uf df uf'

In the second photo, I have applied this to all eight corners.

Compy Cube - Jason Smith

I bought one of the large versions of Jason Smith's first run of the Compy Cube. [T] [T] The Compy Cube (aka Shallow Dino, aka Sausage's Cube) is a full custom 3-D print. It is easy to solve intuitively, requiring no memorized algorithms. I dyed my Compy Cube purple, just to be different.
Dino Skewb / F-Skewb

Dino Skewb (F-Skewb) - designed by Timur Evbatyrov, produced by DaYan

An order-2 vertex-turning, asymmetric puzzle - corners with small tips do Dino turns with their neighboring 3 square face pieces and 3 edge pieces, as well as Skewb-like turns with the further layer that move more than half the cube; corners with large 3-part tips are part of Skewb-like turns only that move less than half the cube.

Vertex-turning Order 3+
The O3 VT combines the Skewb w/ each O2 VT form...
Skewb + Master Skewb
Elite Skewb - GB 3.2.3 - Eric Vergo, Drew Cormier

This is the order-3 vertex-turning cube - GB 3.2.3 - a Skewb combined with a Master Skewb. It is called the "Elite Skewb" (but could also be a "Professor Skewb").

First made by Drew Cormier [T].

Eric Vergo made this Elite Skewb for me [T]. It's instance #1!

Skewb + Dino
Dino Skewb - GB 3.2.5 - Tom van der Zanden

Dino Skewb by TomZ [T] [W] [S]
Skewb + Compy
Compy Skewb - Tom van der Zanden

TomZ's Compy Skewb [T] [Y]
(Order 4) Lattice Cube / Master Dino - GB 3.2.7 - Okamoto, Bedard, Pfennig

GB 3.2.7 is an order 4 vertex-turning cube - a Dino with tri-partite vertices. Called the Lattice Cube or Master Dino - originally designed and hand-made by Katsuhiko Okamoto.
Later made by Scott Bedard [T] [Y] and Gregoire Pfennig, now mass-produced by Calvin Fan
(Order 4) Mosaic Cube (Fadi Cube) - Oskar van Deventer - Meffert

The Mosaic (Fadi) Cube (Meffert/Oskar van Deventer) is a Lattice with visible corners.

 The mass-produced Mosaic Cube issued by Meffert was designed by Oskar van Deventer and originally available from Oskar's Shapeways Shop as the Fadi Cube. It is a vertex-turning, order-4 cube, related to Okamoto's Lattice Cube. There has been some controversy about the stability of the Mosaic Cube [T] and Oskar designed a new spherical core. I bought a black spherical core from Shapeways and modified a Mosaic Cube to swap out the standard core. To access the screws and disassemble the puzzle, one must first remove the corner caps, which are glued on and have three small legs that fit into sockets in the underlying stem piece. An Exacto knife, used with care, is helpful to pry up the corners. One or more legs might break when a cap is removed, however this is not catastrophic - you can use "Stik Tak," "Fun Tak," or a similar product to re-attach the corners so they remain easily removable. After swapping in the spherical core and adjusting the screw/spring tensions, I find the puzzle very stable, playable, and enjoyable. The Mosaic Cube is not overly difficult to solve. It is similar to a Dino or Rainbow Cube, as one might expect. The steps I employ are: Orient the corners - they are fixed in place so this step is trivial. Once the corners are oriented, the correct face colors are evident. Solve the small square centers. These centers are attached in pairs - a single piece provides a square center to two adjacent faces. There are 12 such pieces and they can be solved in the same way as Dino or Rainbow Cube edge pieces - intuitively. Solve the large edge pieces. There are 24 of them. I have three algorithms that sometimes prove useful. To describe the algorithms and their effects, we have to agree on a notation for the Mosaic Cube - here is what I use. I hold the cube at an angle and look at it edge-on - the up and down faces are held parallel to the floor, and an edge is towards you. I label the 8 vertices using two letters describing their position: UF, DF, UR, DR, UL, DL, UB, DB - as shown in the image (DB isn't visible). Any of the 24 large edge pieces can be identified by giving the labels of the two corners it lies between, with the adjacent corner first. For example, the large edge piece on the bottom in front would be DFUF while the piece above it would be UFDF. A move is a twist of a vertex by 120 degrees either clockwise or counterclockwise from the point of view of looking directly at the vertex. A move can encompass just a corner and the 3 large edge pieces surrounding it, or those pieces plus the further "layer" including 3 more large edge pieces, and 3 center dual-pieces (accounting for six small square centers). I will symbolize the latter move, clockwise, using the relevant corner name - e.g. UF means twist the UF corner and the two layers surrounding it 120 degrees clockwise. UF' symbolizes the corresponding counterclockwise move. I will use the lowercase name of the corner to symbolize the "smaller" move of just twisting the corner and the 3 large edges surrounding it - e.g. uf and uf' for counterclockwise. To 3-cycle DFUF => URUF => URDR, use: UF' ur UF ur' To twist just the single UF corner piece 120 degrees clockwise, use: (UR uf UR' uf)*2 To 3-cycle the far edges about the UF corner clockwise - i.e. DFUF => ULUF => URUF, use: ur UF' ur' UF' ur UF' ur' I figured that one out myself :-) - I do a setup (ur UF' ur'), then use the previous 3-cycle UF' ur UF ur', then undo the setup (ur UF ur'), which strung together looks like: (ur UF' ur') UF' ur UF ur' (ur UF ur') The adjacent ur' ur near the end cancels out, and the resulting adjacent UF UF simplifies to UF' giving the concise 7-step algorithm.
Edge-turning Order 1
24 Cube / Little Chop - GB 3.3.7 - Matt Shepit

24 Cube aka Little Chop - designed and made by Matt Shepit [T] [T] [Y]
also made (from Shepit's design) by Taylor [T], Karl-Heinz Diekmann [Y], Pantazis Houlis [Y]

Here we have one of the most elusive puzzles in the twisty zoo - what is known as the "24 Cube" or "Little Chop." [T] [Y]
Only a few have been produced, based on a design by Matt Shepit of New Zealand, but AFAIK none turn very smoothly. The design uses what is known as a "shells mechanism" and is very complex. To date AFAIK no-one has produced a viable alternative mechanism for this puzzle, and not for lack of trying. People have cheated by using a steel ball as a core and embedding magnets in the pieces, but such implementations don't work satisfyingly well. [S]

Edge-turning Order 2
Here is Carl Hoff's animated "order-2 edge-turning multicube" [T] which illustrates nicely how the spacing of the two cuts between pairs of opposing edges can be varied - in the Helicopter, the spacing is such that the layers touch but do not overlap. In 3.3.3 (Drew's Quad-X) the layers overlap and the spacing is symmetric. More and less overlap is possible, with different pieces [dis-]appearing.

Helicopter Cube - GB 3.3.1 - Okamoto, Cowan

I am very pleased to have finally obtained a custom-made Helicopter Cube from Adam! The Helicopter Cube was first discussed in the TwistyPuzzles forums in thread 6253 - a particularly rich thread in which several ideas, including the concept of jumbling as opposed to shape-shifting, were broached. (More discussion on jumbling: 13071, 11126 .) Katsuhiko Okamoto mentions that he had completed his equivalent Bevel Cube the previous month. Robert Webb extrapolates a rhombic dodecahedral puzzle and Matt Shepit hints of its realization - it will be Shepit's Rua. Various folks have discussed their attempts to make their own Helicopter Cubes: 13856, 13520, 12030, 12423, 11679.
Helicopter Cube - GB 3.3.1 - Meffert

The Helicopter Cube has also been produced commercially. I bought a black one and a white one.
Helicopter Cuboctahedron - Garrett Ong, Eric Vergo

Garrett Ong designed and Eric Vergo made an order-2 vertex-turning cuboctahedron, which is equivalent to a cuboctahedron Helicopter [T].
Partially Unbandaged Helicopter Cube - Eric Vergo

A Partially Unbandaged Helicopter Cube designed by Eric Vergo [S] - this is copy number 1, obtained from Eric at NYPP2011.
When a jumbling move is made, a triangular face piece can swap places with a corner - this is not possible on the regular Helicopter Cube.

Vergo also made a 2x2x2 + unbandaged heli hybrid [T]

Curvy Copter - GB 3.3.0 - Tom van der Zanden - Meffert

Meffert's is offering the mass-produced Curvy Copter created by Tom van der Zanden [T]. Tom's Curvy Copter has been very popular as a custom-produced 3D printed puzzle [S], and is now available at one tenth the price. I bought the black and white "twins" pair. The Curvy Copter functions like a Helicopter cube, but it exposes central edge pieces that must be correctly oriented, making it a more difficult challenge.
Curvy Copter Plus - Tom van der Zanden

TomZ's Curvy Copter Plus, mass-produced by Meffert [T] [S] has additional cuts that partially unbandage some turns
Curvy Copter II - GB 3.3.8 - Tom van der Zanden

Curvy Copter II by Tom van der Zanden [T] [Y]
Curvy Copter III - Mf8

Mf8 Curvy Copter III - stickerless version
Quad-X - GB 3.3.3 - Drew Cormier

Drew's Quad-X [T] [Y] (cubic Rua)
DeCETH (Deeper Cut Edge-Turning Hexahedron) - Gregoire Pfennig

DeCETH (Deeper Cut Edge-Turning Hexahedron) - Gregoire Pfennig [T] [Y]

Master Little Chop - Ben Streeter

Master Little Chop - Ben Streeter [T] [Y]

Deeper still than the DeCETH

Hexaminx - Tony Fisher

Fisher's Hexaminx (cubic Megaminx) [T]
Hexaminx - Pillowed version by Traiphum Prungtaengkit

Pillowed Hexaminx from Traiphum Prungtaengkit, of Thailand [T]
(Shown with Helicopter Cube)

I traded this to Kevin Uhrik for a Tutt's Icosaminx.

Pillowed Hexaminx (mass-produced)

A Pillowed Hexaminx - mass-produced by Calvin Fan;
this cubic shapemod of a dodecahedral Megaminx was first designed by Tony Fisher
then produced in a beautiful pillowed form by Traiphum Prungtaengkit.
Mini Hexaminx - Grégoire Pfennig

Here is a Mini-Hexaminx, designed and made by Grégoire Pfennig, printed by Shapeways. [T] [S]
Shown in comparison to a U.S. quarter, a Pillowed Hexaminx hand-made (cast) by Traiphum Prungtaengkit, and a Tomy Megaminx.
This small wonder is very stable and usable. I am impressed that something so compact works so well. Nice work, Greg!
Edge-turning Order 3+
GB 3.3.12 - 24+Heli

O3 would be 3.3.12 24+Heli
GB 3.3.16

O4 3.3.16
Eitan's Master Helicopter - Eitan Cher

O4 - Eitan's Master Helicopter - his first version was really a Master Curvy Copter [T] [T]

Eitan revised the design and also offers a "true" Master Helicopter (no exposed edge centers) [T] [Y] [T]

O1,O1 Hybrids
GB 3.4.1 - Skewb + 2x2x2 (aka Super-O, SuperZ, Skew-by-2, Skewb-by-2)

Attempted by several designers. I don't know of one that works well enough to be sold, though...
• Cowan [T]
• timselkirk [T]
• Eitan [T]
• TomZ [T]
• iaroslavski made a magnetic version [T]

BTW, Noah coined the term "Skew-by-2" but he meant something different: [T] - a Simple Overlapping 3x3x3 Cube, with the corner caps modified so the whole thing looks like a Skewb. It would twist like the SOC, not a Skewb.
"smashy" suggested the same thing and called it "Skewbik's Cube" [T]

GB 3.5.1 - 24 + 2x2x2 - 48 Cube

A 24 Cube crossed with a 2x2x2

PuzzleMaster6262 attempted one with his "Sand mech" [T] [T] generating some controversy... [T] [Y] [Y]

Skewb+24

A 24 Cube crossed with a Skewb - never attempted AFAIK.
GB 3.7.2 - Skewb + 24 + 2x2x2

A triple cross - 24 Cube crossed with a Skewb and a 2x2x2.
Not made, but explored by Carl [T]
O2,O1 Hybrids
Each O2 puzzle can be combined with an O1 puzzle to make a (2,1) hybrid. O2's can also be combined with O2's to make even more complex hybrids. And of course O1's can be combined with O3's , etc. ad infinitum (in principle anyway). And let's not forget combining more than two types together in one puzzle! Yikes! Hybrids quickly get difficult to categorize, let alone to build. Also, without the clever use of magnets or detents, the resulting puzzles can be "squishy" - hard to hold without inadvertently moving something.

In the table below, the fundamental O1 cubes - the FT 2x2x2 and the VT Skewb - are represented in the rows. I have left out the row for the 24 Cube. The fundamental O2 cubes are represented in the five columns.

NOTE that an O1 combined with an O2 of the same turning regime simply gives an O3 of that type - I have outlined those four puzzles and they are already discussed above.

 FT 33 VT Mast. Skewb VT Dino VT Compy ET Heli FT 23 FT O3 - Master / Revenge Cube 2x2x2 + Master Skewb = Hyper X GB 3.4.2 - Super-X 2x2x2 + Compy = ? Heli + 2x2x2 VT Skewb 3x3x3 + Skewb = ? VT O3 GB 3.2.3 - Elite Skewb VT O3 GB 3.2.5 - Dino Skewb VT O3 - Compy Skewb Helicopter Skewb

Master Skewb + 2x2x2 - Hyper X - Gregoire Pfennig

Master Skewb + 2x2x2 - Hyper X - Gregoire Pfennig [T] [S]
Dino + 2x2x2 - GB 3.4.2 - Super-X - Tom van der Zanden

In Berlin, I got one of Tom van der Zanden's Super-X cubes [T] - the Super-X turns like a Dino plus a 2x2x2.

An early design by Wayne Johnson [T],
Realized as a full-custom design by Adam Cowan [T] [T]
Improved by Drew Cormier, who added magnets to stabilize it [T] [T] [Y]
Also made by jesseking from a Rainbow and keychain 2x2x2 [T]

Tom's version uses printed-in detents for stability, and having played with different versions, I would venture to say that Tom's is the best to date. Visit Tom's Shapeways shop.

Helicopter + 2x2x2

Unbandaged Helicopter + 2x2x2 made by Eric Vergo in Jan 2011 [T]

Clay's Combo Copter [T]

Garrett Ong was working on one [T]

Also RubixFreakGreg [T] [S]

Helicopter Skewb - Tom van der Zanden

Helicopter Skewb - Tom van der Zanden [T] [S] also an unbandaged version [T] [S]

"namegoeswhere" also built one [T] [Y]

Helicopter + Skewb + 2x2x2

[no image]

Helicopter + Skewb + 2x2x2
Higher Order Hybrids
Redi 3x3x3 - Eric Vergo

Redi 3x3x3 - Eric Vergo
This puzzle turns at its vertices like Oskar van Deventer's Redi Cube, plus like a 3x3x3. Announced on the TwistyPuzzles forums, and available at Eric's Shapeways shop. Very clever, Eric!
Lattice-X - Olz

Lattice Cube with a 2x2x2 - a (4,1) hybrid. - a Super-X with the extra twisty tips of the Lattice Cube. Designed & made by Olz [T] [Y]
GB 3.6.1 - Dino + Helicopter - "Vestar"

A Dino + Helicopter - aka Vestar
GB 3.4.5 - Dino + 3x3x3 - Tom van der Zanden

A Dino + 3x3x3 - made by Tom van der Zanden [W]
GB 3.4.7 - Master Skewb + 3x3x3 - Minh Shenghsu

Master Skewb + 3x3x3 - Minh Shenghsu [T]
Ultra-X - Eric Vergo

Eric Vergo's Ultra-X [T]
turns like a Rainbow Cube + 23 + small vertices
Geared
Gear Cube - Oskar van Deventer - Meffert

Meffert has produced Oskar van Deventer's Caution Cube [S] and calls it the Gear Cube. The Gear Cube Extreme has four edge pieces in one layer replaced with alternatives that have less gearing. The Gear Cube Ultimate has alternative stickers requiring proper permutation of the small central gear pieces on each face.

Positions:
Gear Cube: 41,472
Gear Cube with edge base (small U piece) stickers: 165,888
Gear Cube Extreme: 2.56*1014
Gear Cube Ultimate (Extreme with edge base stickers): 3.28*1016

Gear Shift - Oskar van Deventer - Meffert

Oskar's Gear Shift - Meffert
(black and white versions)
Video solution on Bram's YouTube channel
Geared Mixup Cube - Oskar van Deventer - Meffert

Geared Mixup Cube - designed by Oskar van Deventer
produced by Meffert
A gift from Rox - thanks!
David's Gear Cube - Oskar van Deventer - Meffert

David's Gear Cube - conceived by David Tzur (Alex Polonsky),
developed by Oskar van Deventer, issued by Meffert.
Gear Skewb

Gear Skewb - designed by Timur Evbatyrov
Produced by Calvin's Puzzle
Timur originally called it "Skewb des Soleils" [T] [Y]
based on his gear HMT
Circle / Crazy
Crazy 2x3x3 designed by Daqing Bao

The Crazy 2x3x3 designed by Daqing Bao.
Genuine DaYan versions made by WitEden.
Available at Cube4you.
Crazy 3x3 Plus Cubes - DaYan

The set of eight types of DaYan Crazy 3x3 Plus Cubes - "Eight Planets"
The circle pieces either do or don't turn with the face. The eight types are different ways of arranging dos and don'ts.
I got mine from Mefferts but you can find them at several vendors.
Super 3x3x4 - WitEden

Witeden Super 3x3x4 black
Crazy 4x4 I - Mf8

The Crazy 4x4 I from Mf8.

This cube was discussed on the Twistypuzzles forums in threads 14856 and 7918. You can see how this cube moves on YouTube here.

3.23*1053 positions

Crazy 4x4 II - Mf8

The Crazy 4x4 II from Mf8.

3.1*1061 positions

Crazy 4x4 III - Mf8

The Crazy 4x4 III from Mf8, purchased via Mefferts

3.1*1061 positions
Same as version II according to Jaap.

Cube and Cuboid Shapemods and Sticker Variations
Bump Floppy

A "Bump Floppy"
QJ Heart-to-Heart

QJ Heart-to-Heart
A "dual-core" floppy.
Tonne

The "Tonne" is a barrel form of the 2x2x2. It was produced in a few different color schemes. I bought one a while ago in Germany.
2x2x2 Shape Variations

Some 2x2x2 variations - Duff Beer can, Golden Syrup, Socube Rhinocerous, Lanlan Dodecahedron, the Trick Haus

As of late 2014, there is a new series of 2x2x2 head-shaped twisty puzzles, called Mad Hedz, issued by Intex Entertainment (aka New Entertainment).
I got four of the six from Amazon and two from Lightake.

Mental Flop (2x2x3 mod) - Grégoire Pfennig

The Mental Flop by Grégoire Pfennig. [T] [S]
Visually, it's a cross between a 1x3x3 Floppy and Tony Fisher's Mental Block, hence the (great) name. Mechanically, it is isomorphic to a 2x2x3 (Slim Tower).
Shown in good company - with my original Floppy Cube hand-made by Okamoto, and my original Mental Block hand-made by Tony Fisher, along with a U.S. quarter.
Very stable and playable!
Ultimate Cube

Ultimate Cube
A commercially produced sticker variation. I have an original in its packaging.
Calendar Cube

4.4*1022 - Marvin Silbermintz

Rubik's Perpetual Calendar
(Kalender Kubus)
The "O" character on one
center has only 2
distinct orientations

Rubik's Cube 4th Dimension

1.1*1022 - Erno Rubik

Rubik's Cube 4th Dimension
Four centers must have distinct orientations

Qubami - Kelvin Stott

Qubami
Designed and produced by Kelvin Stott
The objective is to get 3 different colors and 3 different symbols on every row and column of every face.
Octagonal Prism

4.5*1017

Octagonal Prism
Jaap's page

Diamond Cube

2.0*106 = 2,425,500

Diamond Cube
Jaap's page

Truncated Corners 3x3x3 Cube

Truncated Corners 3x3x3 Cube
2x2x2 Cuboctahedron

The 2x2x2 Cuboctahedron (a truncated 2x2x2) - called the "Friki Cube" made by juanan [T]
3x3x3 Cuboctahedron

The 3x3x3 Cuboctahedron
4x4x4 Master Cuboctahedron

The 4x4x4 Master Cuboctahedron. Made by Jürgen Brandt, Sandy. [T]
5x5x5 Cuboctahedron

The 5x5x5 Cuboctahedron. Made by Jürgen Brandt.

Jade Club Pillowed 3x3x3 Cube - Meffert
Venus Cube 3x3x3

Venus Cube - a 3x3x3 shape variation with overlapping - designed by Evgeniy Grigoriev (he called it the "Fluffy Cube") - produced by Meffert

The Brain Cube, designed by Jason Freeny. A 3x3x3 clad with a squishy material called Kraton, and textured to resemble a brain. Solve by aligning all the fissures. Comes in a glass jar, formaldehyde not included.
Purchased from Marbles The Brain Store.

King Pillow Cube - 3x3x3

King Pillow Cube
A commercially produced shape variation.
Confused Pillow Cube - 3x3x3

Confused Pillow cube from "Socube"
Hexagonal Prism 3x3x3

Hexagonal Prism 3x3x3
Rhombohedron 3x3x3

Rhombohedron 3x3x3
Truncated Hexagonal Dipyramid 3x3x3

A "Blue Diamond" (Truncated Hexagonal Dipyramid shape mod to a 3x3x3).
These are being mass-produced in China.
Hexagonal Dipyramid 3x3x3

A cheap way to make a Hexagonal Dipyramid - combine the parts from two Guo Jia diamonds.

Now Hexagonal Dipyramids are mass-produced by Dian Sheng.
3x3x3 core

Hexagonal Dipyramid 4x4x4

Super Dipyramid
(hexagonal dipyramid from 4x4x4 core)
Dian Sheng "Tank Diamond"

Dian Sheng "Tank Diamond"
Dian Sheng Pyramid (aka "aXe")

Dian Sheng Pyramid (aka "aXe")
designed by John Lin
Various 3x3x3 Shape Variants

Various 3x3x3 Shape variants, including: Heart, Apple, Star, Egg/Potato, Cake, Concave Cube, Ingots (3 colors)

Various 3x3x3 Sticker Variants, Color Variants, and Size Variants

Extended and Multicore Cubes
3x3x3 Extended to 3x3x4

3x3x4 Extended Cube
This simple extended cube-variant has an extra piece glued to each of the nine facelets of one face.
3x3x3 Extended to 3x3x5

3x3x5 Extended Cube
This simple extended cube-variant has an extra piece glued to each of the nine facelets of two opposite faces.
3x3x3 Extended to 4x4x4

3x3x3 Extended to 4x4x4
This is a cheap and simple extended cube-variant from Hong Kong, not a 4x4x4 Evil Twin as the description led me to believe. Caveat Emptor!
Extended 3x3x3 with Deleted Corners

An Extended Cube.
Mini Evil Twin - Mike Grimsley

Mini Evil Twin
Siamese Cube chains

Double and Triple Cubes
Available commercially from various sources, made from keychain 2x2x2 cubes. Two cores share a corner in "Siamese" configuration.
Bandaged and Otherwise Constrained
Bandaged Cube - Meffert

1.0*106 = 1,108,800

Bandaged Cube
Jaap's page
Andreas Nortmann has investigated bandaged cube variations - Andreas says there are 7356 different bandaged 3x3x3 cubes, of which 5705 are (subjectively) non-trivial. Read his articles in the TwistyPuzzles forum: [T] [T] [T] [T] [T] ; also Hidetoshi [T]

Bandaged Cube Kit - Cubetwist

A Cubetwist Bandaged Cube Kit, ordered from Lightake
Nightmare Cube - Tanner Frisby

Nightmare Cube from Tanner Frisby [T] [Y] [Y]

A Nightmare Cube is a bandaged 3x3x3 hidden inside a 2x2x2 shell.

Before the first move, all normal 2x2x2 twists are permitted. After a few turns, however, the bandaging comes into play so various moves become blocked, and then solving becomes a nightmare! Tanner told me the YBR corner has no bandaging.

In October 2008, Adam Cowan issued free STL files for the Nightmare Cube, in the TP Forums, based on an idea mentioned by Noah Hevey in a post from March 2008. Tanner's version is made from a different core, though.

TP Forum member "sublime" made one from wooden corner pieces and a modified keychain 3x3x3 core, and then posted about his copy of the printed version.

Also see: [T] [T]

Folks have noted that solving a Nightmare cube is more akin to navigating a hidden maze, than applying conventional operators. The solution methodology is discussed at Jaap's page.

New Spring 2x2x2 Cube with Bandaged 3x3x3 Core

A "New Spring" clear 2x2x2 with internal colored bandaged 3x3x3
The New Spring version is not a "Nightmare" cube, since all 2x2x2 moves always work. But restoring the internal 3x3x3 is not trivial!
Various other Bandaged Cubes

Just FYI, there are a slew of other bandaged cubes available, including 4x4x4 cubes.
Camouflage Cube 3x3x3 - WitEden

Camouflage Cube
A bandaged and extended 4x4x4
Ordered from WitEden
Various versions are available.
Bermuda Cube - DaYan

DaYan Bermuda Cube Neptune (black)
One of a series of eight types.
Latch Cube - Okamoto

Latch Cube - Okamoto
Quarter Cube - Okamoto

The Quarter Cube, designed by Katsuhiko Okamoto and Takafumi Haseda, produced by Chronos Co. Ltd.
Constrained Cube - Tom van der Zanden

The Constrained Cube designed by Tom van der Zanden. Several versions available that have different side turning constraints (90,180,270, and Ultimate). I ordered the "Ultimate" version.
Pocket Cube - Justin Eplett - Meffert

Pocket Cube - designed by Justin Eplett,
produced by Meffert.
A 3x3x3 with clever bandaging and extensions. Two different color schemes.
Axised Cubes
If the standard 33 cube is rotated 45° about one face's axis (e.g. z axis) then built up and cut down to be re-formed into a cube, one obtains the Fisher's Cube; approx. 30° around z [T] gives the Windmill Cube; 45° around z and x (or 90° about an edge-to-edge axis) gives the Slice Cube; combining Fisher's and Windmill gives a "normal-sized" Greenhill's Cube (which is actually larger - Anthony says [T] it is "a 'Truncated Cube' (corners trimmed down to triangles), stood on one corner then built out to a Cube shape. This basically determined the edge length - 77mm."); 60° about a corner-to-corner axis gives the Axis Cube [T] [T]. An "axised" Cube with twists, reformed into a cube gives the Ghost Cube [T].

Fisher's Cube

Fisher's Cube - originally designed by Tony Fisher, now mass-produced
An axis-rotated 3x3x3 (single axis x 45 degrees)
Solve as a 3x3x3, but also has four of six face centers that can be rotated by 90, 180, or 270 degrees.

These are the Supercube face algorithms - algorithms exist to rotate a single face center by 180, or a pair - one by 90 and another -90.

U180: ( (U R L U2) R' L') x2

U90 & F-90: F B' L R' - U D' F' U' D - L' R F' B U

U90 & D-90: R L' F2 B2 R L' U R L' F2 B2 R L' D'

Master Fisher Cube - Moyu

Moyu Master Fisher Cube
Fisher's Cube / Diagonal Cube - 8-color sticker variant

Fisher's Cube / Diagonal Cube
8-color sticker variant
Windmill Cube

Windmill Cube
Slice Cube

At the 2012 New York Puzzle Party (NYPP) hosted by Tom Cutrofello, I bought this hand-made Slice Cube from fellow attendee and twisty puzzle enthusiast "Zhewei." He had posted about this puzzle on the Twisty Forums here.
Axis Cube

Axis Cube

Now mass-produced.

Ghost Cube

Ghost Cube
Purchased from Jason Smith.
Possibly the most difficult of the "axised" designs.
Ghost Cube - Meffert

Meffert has mass-produced the Ghost Cube
I got examples in purple metallic, and black with white stickers.
Yong Jun (YJ) Moyu Crazy Yileng Fisher Cube

Yong Jun (YJ) Moyu Crazy Yileng Fisher Cube - another "axised" 3x3x3
Skewb Shapemods
Skewb X-treme

Skewb X-treme - 10-color version, designed by Tony Fisher,
produced by Meffert.
Golden Cube - Tony Fisher

Meffert's Fisher's Golden Cube
Perhaps the most famous of the Skewb mods, produced commercially by Meffert.
I bought the "Lunar New Year" set of three, and also a black one.
Mental Block - Tony Fisher

Tony Fisher's Mental Block aka Rubik's Layer
Custom-made by and purchased from Tony. A buildup of a full-sized Skewb using add-on prosthetics. First made by hand by Tony back in 1996. [W]
Curvy Rhombohedron Skewb - John Lin

Curvy Rhombohedron Skewb - designed by John Lin
Squished Skewb - John Lin

Squished Skewb - designed by John Lin
This is pretty large, and a very nice turning puzzle!
Edgematching
Einstein Cube

12^6 = 2,985,984 (Einstein w/ 12 pos./face)
3*106
24^6 = 191,102,976 (Turn 12)
1.9*108

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.

Cuboids

This chart is a 3-D scatter plot showing fully functional cuboid puzzles that have been made either commercially or as custom creations. A cuboid has a cubic or brick shape, with dimensions k x m x n. "Fully functional" cuboids allow turns through every cut - "extended" and multi-core (e.g. chaos, evil-twin, fused, and siamese) cuboids are not included here.

The puzzles in the chart are positioned such that k <= m <= n. In the chart, the left-to-right (x) axis is k, the front-to-back (y) axis is m, and the bottom-to-top (z) axis is n. Each axis except m (y) runs from 1 to 7 - puzzles where a dimension exceeds 7 are few and are shown off the chart. I omitted m=1 to reduce clutter. This prevents the inclusion of the so-called 1x1x1 (no big loss IMHO), and forces the 1x1x2 to be located at 1x2x1, the only violation of the m<=n rule. 1.1.n for n!=2 are omitted. A red "post" beneath each item gives a hint as to z scale.

In the table below, mass-produced commercially available puzzles are highlighted like this. Cuboids I have are highlighted like this. Cuboids I don't have and would like to see mass-produced are highlighted like this.

When available, links are given to YouTube videos (shown as [Y]), websites (shown as [W]), TwistyPuzzles forum threads (shown as [T]), and Shapeways (shown as [S]).

 Kevin Sadler has posted a nice overview of cuboid puzzles on his "Puzzlemad" blog. Kevin specifies subcategories for the cuboids: Domino Cuboids - of the form N.N.(N+O) or N.(N+O).(N+O) where O is odd. No shapeshifting - oblong sides only turn 180°. e.g. 1.1.2, 1.2.2, 1.2.4, 1.4.4, 2.2.3, 2.3.3, 2.3.5, 2.5.5, 3.3.4, 3.4.4, 4.4.5, 4.5.5, etc. Shapeshifting Cuboids - of the form N.N.(N+E) where E is even, N > 1. e.g. 2.2.4, 3.3.5, 4.4.6, etc. Brick Cuboids - N.(N+O).(N+E) or N.(N+O).(N+O+2) - shapeshifts in only 2 of 3 axes. e.g. 1.2.3, 2.3.4, 3.4.5, etc. Floppy Cuboids - N.(N+E).(N+E) e.g. 1.3.3, 2.4.4, 2.4.6, 3.5.5, 3.5.7, etc. The 3.5.7 shares characteristics of the Brick and Floppy classes.

Joshua Bell has a nice photo of his cuboid collection on Flickr.

Table last updated February 22 2015.

Another way to organize the cuboids is by cross-section versus number of layers,
though this format contains redundant information.
Also, a few puzzles (most not yet made) are omitted as I have compressed the table.
The table below summarizes which have been mass-produced, custom-made, and not yet made.
P means proportional. C means cubic. N means non-proportional.
The heavy squares outline square cross-sections and cubes. Asterisks indicate puzzles I have.

 cross-sec. => layers 2x1 2x2 2x3 2x4 3x1 3x3 3x4 3x5 3x6 4x4 4x5 5x5 5x6 6x6 7x7 1 * * * * * * * 2 * * * * * * * * * 3 * * * * * * * * P C * P N * * * * 4 * * * * * * * * * * 5 * * P C * * * * * * 6 * P N * * * 7 P C * * 8 9 P C *

Puzzle Name and Notes Positions Mechanism
1x1x3
Rubik's LED Flashlight 1x1x3
1x1x4
1x2x2
The Morph, and a black cuboid
Only 6 positions possible!
I find this harder to mix up than to solve.
?
1x2x3
by Scott Bedard.
Scott Bedard
1x2x5
IQube blue - www.hanayamatoys.co.jp
and IQube red - www.hanayamatoys.co.jp
1x2x9
The Jade Chopsticks, a 1x2x9. [T] The ambigram on it was designed by John Langdon.
Based on the 1x2x13 designed by Oskar van Deventer and the 1x2x9 designed by Ola Jansson.
Mass-produced by Meffert.
1x3x3
Floppy Cube
I have an original custom-made by Katsuhiko Okamoto, and the commercial version now available from Gentosha.
This won First Prize at the IPP26 Design Competition.
Katsuhiko Okamoto
The Scramble Cube
Okamoto's follow-up to his Floppy Cube -
originally known as the Super Floppy when it won the Puzzle of the Year award in the 2009 IPP Puzzle Design Competition. Knock-offs were promptly produced but did not function in the same way - the Scramble Cube does not allow naked edge centers to be rotated.
3,041,280 positions Katsuhiko Okamoto
1x3x4
Designed by Ola Jansson, made by Tanner Frisby
Ola Jansson
1x4x4
Designed by Ola Jansson, made by Karl-Heinz Diekmann
Ola Jansson
1x5x5
Murilo - Shapeways
Murilo
2x2x2
aka the Pocket Cube
3.7*106
3,674,160
Erno Rubik
6-armed spider.
2x2x3
- this mod is known as the "Slim Tower." I got a hand-made version a while ago but I forget from whom. Okamoto's version is now commercially available from Gentosha.

Solution Algorithms:

1) Gather top 2x2 face
2) Solve top face - swap adj. UFL + UFR:   F U' F U F  R U R U' R
(Do twice for a diag. swap.)
3) Flip & solve the other 2x2 face as top

4) If not done, solve the middle layer - 0, 1, or 2 will be correct...
M is middle layer CW seen from top

4a) if 0, do   R M2 R  (FL<->FR + BL<->BR)
if not done, do 4b or 4c

4b) if 1 (at BL) -

CW 3-cycle:   M' R M R

CCW 3-cyc:   R M' R M  (inverse of above)

4c) if 2:

swap adj. FR + BR:  (R U2)3

swap diag. FL + BR:  F (R U2)3 F  (F converts to adj. swap)

(See Robert Munafo's site.)
241920 positions. Okamoto
2x2x4
I bought one of Garrett Ong's hand-made 2x2x4 puzzles (before I knew they'd be mass-produced). Later, I received a copy of the new Rubik's 2x2x4, signed by designer Hidetoshi Takeji.
Garrett Ong, Hidetoshi Takeji

2x3x3
Rubik's Domino

This is available both vintage and new in various versions. The most common vintage version is known as a Groove Domino (its internal mechanism relies on grooves - turning is very rough - I have several); a smoother turning version is known as the Spindle Domino (I finally acquired one); there is also a Russian Domino which has a more complex internal mechanism and turns more smoothly (I found one!); there are also recent black and white versions, and a reproduction of the old design with pips, based on the new 2x3x3, made by Smaz. QJ has produced a 2x3x3 cylinder.

Also shown are the smaller vintage clone dominos, and a version in the shape of a Chinese Knot.

Solution Algorithms (by Stefan Pochmann) from Jaap's page, and from www.cubeinfo.co.cc:

 1) Solve the edges in one 3x3 layer (intuitively) - you can avoid parity fixes if you remember your cube's color scheme. 2) Hold this layer D, and drop its corners into place - put a prospect in UFR and drop it into DRF using R U R U' R (NOTE - if the colors don't line up, you need to swap the L and R edges. If a D corner is in the wrong place, drop something else on it.) 3) Permute the U corners (they won't need orientation) Swap adj. UFR and URB: (R U R U' R) U' D -- (R U' R U R) D' Swap diag. UFR and UBL: RU' RU' RU RD' RU RU' RDR 4) Permute the U edges (they won't need orientation) Swap adj. UF and UR: (RU)x2 (RU2)x2 RU RU' R Swap opp. UF and UB: (RU2)x3 H-perm: MUM U2 MUM Z-perm B<->R & F<->L: MU (MF)x2 U'M

4.0*108 Erno Rubik

2x3x4
Designed by Garrett Ong [T] [S]
Garrett's 2x3x4 won the Summer Puzzle Building Contest. It's a great achievement at its price point (under \$60).
Garrett Ong
2x3x4
mass-produced commercial version from DaYan / Mf8
I modified mine with the 3 extra split edges that eliminate hidden bandaging and restore full functionality
2x4x4
by Ola Jansson (Olz). I have a version 3.25, made by Ola. Here is a video of version 3.1 which looks similar: [Y]; The original was much larger: [Y]; [T] [S] Solve video: [Y]

This is a very nicely finished, stable and incredibly smooth-turning puzzle, and it shape-shifts.

Ola Jansson
2x4x6
A black and a blue copy of the 2x4x6 cuboid twisty puzzle -
designed by Hunter Palshook, produced by Calvin's Puzzle

The comparison photos show the 2x5x5 with the KHD/Olz 1x4x4, and the original hand-made 1x3x3 from Okamoto.

2x5x5
Designed by Ola Jansson, made by Karl-Heinz Diekmann
Ola Jansson
3x3x3
Rubik's Cube
4.325*1019 Erno Rubik
6-armed spider
3x3x4
by James Lee at Cube4you (also Cubefans).
Previously only available as an expensive hand-made custom creation. This is based on Jin Kim's design - get STLs by Tom van der Zanden at TwistyPuzzles.com Forums, thread #12134.
4.13 x 1016 Jin Kim
3x3x5 (Proportional)
3x3x5 (Cubic)
From Cube4You.
Note: this design was first sold by Adam Cowan and Jason Smith.
3x3x6 (Witeden)
3x3x7 (Cubic)
From Cube4You.
3x3x9 (Cubic)
Also 3x3x9 Roadblock I - black
3x4x4
Ayi's 3x4x4
3x4x5
Designed by Tom van der Zanden.

I have a Shapeways print in black, put together by Tom. I also have a transparent instance of the mass-produced version by Mf8. Both turn very nicely, and both shape-shift.

3x5x5
A custom FDM 3D-printed 3x5x5 cuboid twisty puzzle
Not loose and turns great.
I love the purple material and the custom hollow stickers!
4x4x4
Rubik's Revenge
7.4*1045 Peter Sebesteny
grooved sphere
4x4x5
Ayi's Toy [T] [W]
4x4x6
Tom van der Zanden designed a 4x4x6 cuboid [T],
which has now been mass-produced by Calvin Fan and marketed under his Calvin's Puzzle line [T].
4x5x5
Ayi's 4.5.5
ayistoy.com
5x5x5 2.8*1074 Udo Krell
6-armed spider
6x6x6
V-Cube 6
from Verdes Innovations.
1.57*10116 Panagiotis Verdes
7x7x7
V-Cube 7
from Verdes Innovations.
1.95*10160 Panagiotis Verdes

Octahedral Twisties

Dodecahedral Twisties

Face-turning Order 1
Pentultimate - GB 1.1.7

For the last puzzle in 2010, I received Tom van der Zanden's excellent Pentultimate.

You can buy one, and several other great twisty puzzles, at Tom's Shapeways shop.

This is the order-1 face-turning dodecahedron. It has six cuts, all of which pass through the center of the puzzle, midway between pairs of opposing faces, and are great circles on the circumscribed sphere. Each divides the puzzle into two halves.

It is an engineering design masterpiece and employs a sophisticated "shells" mechanism. The shells build upon a Megaminx, through a Pyraminx Crystal, Master Pentultimate, to the outer Pentultimate. In a shells mechanism, the pieces of an inner shell hold in the pieces of the next shell out. For example, the Pyraminx Crystal has two shells - an inner Megaminx and the outer Crystal. The faces of the inner Megaminx hold in inner edges, which in turn hold in the outer Crystal corners, which hold in the outer Crystal edges.

The Pentultimate is 25mm (1") on an edge, and is the same size as a QJ 3x3x3 dodecahedron. The design explores the limits of economical miniaturization within the 3D printing process, yet the puzzle is not fragile and is quite comfortable to hold and manipulate. It was announced on the TP Forums here. You can see an image of the complicated internal mechanism in that thread.

For information on the earlier impressive albeit fragile so-called "knucklehead" mechanism pioneered by Jason Smith, who designed and constructed the first working version of this puzzle, see an article at Jason's Puzzle Forge website.

Face-turning Order 2
Flowerminx / Kilominx - GB 1.1.12

2.36*1025 - Megaminx core

Flower Minx - from Mefferts, designed by David Litwin. This is a corners-only Megaminx, aka "Kilominx."
Equiv. to Impossiball.

Megaminx - GB 1.1.1

1.0*1068 (12 color version); 6.144*1063 (6 color version)
Kersten Meier, Ben Halpern - 12-armed spider

Megaminx
(Tomy version, Meffert's tiled version, tiled Chinese version, stickered black Hong Kong version, stickered white Hong Kong version, DaYan Megaminx version with solid colored plastic pieces and corner ridges.)
Also the Holey Megaminx from Mefferts (in black and white), designed by Lee Tutt.

Hungarian Supernova

Original Hungarian Supernova in package, Hungarian Supernova re-issue
Holey Megaminx / Void Megaminx - Lee Tutt / Meffert

The Holey Megaminx from Mefferts (in black and white), designed by Lee Tutt.
Pyraminx Crystal - GB 1.1.3

1.68*1066 - A build-up of the Megaminx, without centers.

Meffert's Pyraminx Crystal (black tiled version, and black and white stickered versions)
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.

Starminx I - GB 1.1.5 - Tom van der Zanden

In 2011, I had the pleasure of meeting Tom van der Zanden [S] at IPP 31 in Berlin.
Tom made me a slightly larger version of his Starminx I [T]. The Starminx is shown in comparison to Tom's Mini-Pentultimate. Previous versions of the Starminx have been made by Drew Cormier [T] and Aleh [T].
Starminx I - GB 1.1.5 - Mf8

Mf8 Starminx II - in translucent purple
This is a mass-produced version of what the twisty forum knows as the Starminx I, previously custom-made by Aleh [T] , Drew Cormier [T] , and Tom van der Zanden, in mini [T], and larger size [T] . (Mf8 called their Dino-Dodecahedron a Starminx I, hence the naming confusion.)
I finally stickered my purple Mf8 Starminx.
As people have noted, it does not turn as smoothly as Tom's (admittedly much more costly) 3D printed version.
Curvy Starminx - Mf8

Mf8 Curvy Starminx
Master Pentultimate - GB 1.1.6 - Eric Vergo

I received a Master Pentultimate designed and made by Eric Vergo. It turns very smoothly. This is a great design! Shown along with Tom van der Zanden's Pentultimate.
(Void) Master Pentultimate - GB 1.1.6 - mass produced by Mf8

Mf8 (Void) Master Pentultimate
(Shown with optional center panels in place.)
Multidodecahedron - Tom van der Zanden

A Multidodecahedron by Tom van der Zanden [T] The theory behind this puzzle was discussed by Carl Hoff here.
This is a custom 3D printed puzzle, but the pieces of my copy have been through a "tumbling" process that makes them smooth. [T]
The Multidodecahedron is an outer Master Pentultimate, where the face centers expose an inner Megaminx shell having its own stickers. A full solution entails solving both layers. [T]
Face-turning Order 3+
Gigaminx - GB 1.1.9

3.65*10263

Gigaminx

Originally realized by Tyler Fox. Subsequently made by others. Commercially released by James Lee at Cube4you (also Cubefans).

Master Kilominx / Hyperminx - Mf8

Mf8 Master Kilominx (solid colors)
Also known as the Hyperminx
This is essentially a Gigaminx where the central stars have been hidden. The pieces overlap during turns.
Teraminx - GB 1.1.41

1.16*10525

Teraminx

Originally designed by Drew Cormier. According to Drew, the Teraminx contains 555 pieces! Produced commercially by Cube4you and Mf8. I got a C4U because of a misrepresentation by a vendor. The MF8 version is superior, and I got one from Meffert.

Petaminx

3.16*10996

Petaminx
This had been a very expensive custom-made puzzle (in the \$3000 price range), designed by Drew Cormier. However, it has been mass-produced (at the \$200 price range) by Mf8.
(Included for reference to show number of positions - I don't have this.)

Vertex-turning Order 1
Halfminx aka Drew's Chopasaurus - GB 1.2.9 - Drew Cormier

The order-1 vertex-turning dodecahrdon is the "Halfminx" - made by Drew Cormier, who called it the "Chopasaurus." [T]
I don't have this. The closest commercially available puzzle is the Skewb Ultimate - see below.
Skewb Ultimate

1.0*108 - Uwe Meffert

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 toAntipode 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

Vertex-turning Order 2
Dino Dodecahedron - GB 1.2.1 - Mf8 (their Starminx I)

A Dino Dodecahedron (aka Dinominx) by Mf8 (they call it a Starminx I), purchased from hknowstore. This puzzle was first proposed by Lukeharry then made by Kevin Uhrik, now mass-produced by Mf8. [T]
I solve this with no algorithms - it's a fun puzzle. Also quite large - larger than a Pyraminx Crystal.
Pentagram - GB 1.2.3 - Eric Vergo

I bought the 1st copy of Eric Vergo's Pentagram puzzle - first announced at the TwistyPuzzles forums. It is an order-2 vertex-turning dodecahedron, designed by Eric and 3D printed by Shapeways. It is the same size as the Meffert's Pyraminx Crystal. You can buy a copy at Eric's Shapeways shop.

There is an online video review of this puzzle on YouTube.

Pentagram - GB 1.2.3 - Eric Vergo

Pentagram designed by Eric Vergo
produced by Mf8
Once again an expensive 3D printed twisty gets mass produced.
Bauhinia - Mf8

Bauhinia - produced by Mf8
A "Rex Dodecahedron."
Edge-turning Order 1
Big Chop - GB 1.4.3 - Oskar van Deventer, Jason Smith

The order-1 edge-turning dodecahrdon is the "Big Chop." As of this writing (June 2013) there is no satisfactory mechanism for this puzzle. Oskar made one using magnets. Jason Smith attempted another interesting mechanism. [T]
I don't have this.
Edge-turning Order 2
Helicopter Dodecahedron - Mf8

Mf8 Helicopter Dodecahedron, in black
Geared
Gear Minx I

Gear Minx I - gear twisty mechanism designed by Oskar van Deventer, offered by Meffert.
Gear Minx II

Gear Minx II - gear twisty mechanism designed by Oskar van Deventer, offered by Meffert.
Circle / Crazy
Crazy Megaminx Plus - Saturn - Mf8

DaYan/Mf8 Crazy Megaminx Plus Saturn

The fit and turning on the copy I got are very good, and I like the stickerless design and brightly colored plastic. It is about the same size and weight as a Meffert's Pyraminx Crystal. Purchased at the HK Now Store.

Shapemods
2x2x2 Dodecahedron

A 2x2x2 in the shape of a dodecahedron. Produced by LanLan.
3x3x3 Dodecahedron

3x3x3 Dodecahedron. Produced by QJ.

More difficult than the regular 3x3x3, since all six centers must be properly oriented.

Edgematching
Dodeca Nona

3.99*1018 - 1122 solutions.

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.

Mobius

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.
Related
2x2 Megaminx

2x2 Megaminx - designed by Gregoire Pfennig and Felixouchon
Produced by WitEden [T]
Very difficult to align and turn.
Alexander's Star

Alexander's Star
(Equal to a Megaminx with no corners and no centers.)

GloBall

GloBall [T]
See GloBall variants here.
Kytka (Flower) GloBall

Kytka (Flower)
A derivative of the Globall. This is a corners-only Megaminx or Pyraminx Crystal, same as Litwin's Kilominx / Meffert's Flower Minx.
Logic Star GloBall

Logic Star
A derivative of the Globall.

Rhombic Dodecahedral Twisties

Since Rhombic Dodecahedra are not regular polyhedrons, they are difficult to classify using the scheme I have employed for the regular polyhedral puzzles.

 Face-turning Order 2 Rhombic-18 - David Pitcher The Rhombic-18, by David Pitcher. [T] [Y] [S] This is a twisty puzzle in the shape of a Rhombic Dodecahedron. All faces turn, and all 4-fold corners turn. It is similar to Matt Shepit's Rua puzzle [T], which is also a face-turning RD, but does not allow vertex turns of any sort. Vertex-turning Order 2 DRD (Dino Rhombic Dodecahedron) - Drew Cormier I bought the Dino-Rhombic Dodecahedron (DRD) DIY from Drew Cormier. This puzzle is a vertex-turning rhombic dodecahedron where all 4-part and 3-part vertices turn. It turns well, but due to a design issue the 3-part corners turn only counter-clockwise. Rex Rhombic Dodecahedron - William Kretschmer This is a Rex Rhombic Dodecahedron (RRD), designed by William Kretschmer. It was announced on the TwistyPuzzles forums, and is available from Will's Shapeways shop. As Will says, the turning is nearly flawless. It's about the same size as the LanLan 4x4x4 RD. This is a great puzzle! Vertex-turning Order 3 Mini-Mini-Rhombiminx - Eric Johnson A Mini Mini Rhombiminx from Eric Johnson. It's a vertex-turning Rhombic Dodecahedron, but unlike the similar-looking DRD, the Rhombiminx is order-3 and only the 4-part vertices turn. It's hand-made using cast custom parts and an Eastsheen mini 2x2x2 core. Mini Rhombiminx This puzzle is the same size as the Dino-Rhombic Dodecahedron (DRD) I got from Drew Cormier - here are some comparison photos: Here is a group shot with various Rhombic Dodecahedra twisty puzzles. The black Mini-Rhombiminx is in the center. The white Mini-mini Rhombiminx is below it; clockwise from there is a 2x2x2 Rhombic Dodecahedron made by Karl-Heinz Diekmann, a Kite Skewb, a truncated 4x4x4 RD, a Lanlan 4x4x4 RD, a QJ 3x3x3 RD, and the custom DRD. I received a black Mini Rhombiminx. This is an order-3 vertex-turning rhombic dodecahedron, built around a 2x2x2 using custom parts. (The first Rhombiminx was built around a cut-down 4x4x4.) Every 4-part vertex turns, and there are 3 mutually perpendicular cuts through each square cross-section (i.e. the 2x2x2 cuts). It is larger than the white Mini-mini Rhombiminx I got a while ago, which is built around a mini-Eastsheen 2x2x2. Devil's Eye / Evil Eye - Closed Moyu Devil's Eye (closed) a rhombic dodecahedral form of the cross cube Devil's Eye / Evil Eye - Open Moyu Devil's Eye (open) Devil's Eye / Evil Eye - Closed, solid Colored pieces Moyu Devil's Eye (closed) - with solid colored pieces Geared Gear Change - Meffert Gear Change pair Shapemods Rhombic Dodecahedron Skewb 1.0*108 Rhombic Dodecahedron Skewb Since the orientation of every piece matters, this is similar to the Skewb Ultimate. 2x2x2 Rhombic Dodecahedron - Karl-Heinz Diekmann A 2x2x2 Rhombic Dodecahedron, made by Karl-Heinz Diekmann. 3x3x3 Rhombic Dodecahedron - QJ, LanLan Rhombic Dodecahedron (3x3x3) in black (QJ), and white (LanLan) also truncated version 4x4x4 Rhombic Dodecahedron - LanLan Lanlan 4x4x4 Rhombic Dodecahedron also truncated version Diamond Cube Truncated Rhombic Dodecahedron This vintage cube-variant is almost a rhombic dodecahedron, except the four-color centers are flat, not pyramidal. The three-color corners are pyramidal.

Icosahedral Twisties

Icosahedral twisties are few and far between - most of them are custom-made. The seminal icosahedral puzzle is the Dogic, but it is out of production and unlikely to come back.

Spheres, Eggs, and Pucks

 Mozaika 6.27*1049 - Rudolf Destics Mozaika The three interlocking equatorial bands of square tiles can be moved around the sphere. In addition, you can rotate a group of four triangular "corner" pieces with the associated nine squares, in increments of 90 degrees. Impossiball 2.36*1025 - William O. GustafsonWolfgang Kuppers Impossiball (Equal to a Megaminx with no edges and no centers.) Traiphum's Megaminx Ball Traiphum's Megaminx Ball Produced by Calvin Fan at Hknowstore. Thomasball The Thomasball is essentially the same as the Impossiball, but with a different mechanism. Ball.B 7.12*1040 The Ball.B - from Poland. 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. The version with flags is equal to the Megaminx but with face centers orientation visible - i.e. a "Super-Megaminx." That has 2.5*1076 positions. Masterball 4.1*1020 - Dr. Geza Gyovai - patent 4856786 Masterball (Geomaster, aka Rainbow version), Duo B&W, Dragon, Circus, Soccer See other versions at Les Casse-Tete de Chantal Marusenko Sphere Marusenko Spheres Dioctipoid Dioctipoid 1 and 2 www.dioctipoid.com These are face-turning octahedra in spherical form. Twistball 4 Color and Flower Ball versions Twist Ball The Twistball is a spherical Dino Cube, but where the corners are visible. It is available with many different colorings/patterns. The 4-Color version is very simple. The Flower Ball requires all pieces to be correctly placed and oriented, so is slightly more difficult. As with the Dino, both can be solved intuitively without algorithms. Geert Hellings wrote up a nice post on the Tp forums about the history of this puzzle. Intellect Ball Intellect Ball 9 cm (handy) and 13 cm (large!) versions. 3x3x3 Sphere A standard 3x3x3 cube in spherical form. From various vendors, in various sizes. Rubik's World 3x3x3 Sphere 2.7*1021 - Erno Rubik Rubik's World Rubik's World 2x2x2 Sphere Rubik's World 2x2x2 Sphere Finder Ball The Finder Ball was available in various colors/patterns. It is a 2x2x2 sphere. My version is a glode, in Spanish. Hungarian Zodiac Machball (2x2x2 Sphere) A vintage Hungarian Zodiac Machball. D Ball The D Ball was available in various colors/patterns. It is a 2x2x2 sphere. Venus Dreamball The Dreamball was available in various colors/patterns. It is a 2x2x2 sphere. My copy is a Venus Ball. Quarks Quarks is from Fourier Idea, Inc. It is a hollow 2x2x2 sphere. Magic K-Ball The K-Ball is a 2x2x2 sphere. It was offered in various colorings/patterns. Spherical Skewb 2160 for the 4-color Various Skewb-core balls including a Mefferts Beach Ball (4 color) and a Beijing Olympics ball Skewb Egg - Tony Fisher - Meffert A white Skewb Egg from Meffert, designed by Tony Fisher. Morph Egg - Adam Cowan - Meffert Morph Egg Produced by Meffert, designed by Adam Cowan. Based on a 3x3x3, but the subtle curves make this mod surprisingly difficult to solve! Brainbow 623,760 Brainbow Jaap's page Rubik's Cheese (Sajt) 96 Rubik's Cheese (Sajt) Hungarian patent, 9 November 1980, HU 2679 Ayi's 3-Layer Cheese Ayi's 3-Layer Cheese Gear Ball Gear Ball - designed by Oskar van Deventer, produced by Meffert

Dihedral Twisties

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

 Quartet The Quartet from the Shapeways shop of "RubixFreakGreg" -- designed by "Lykwid" [T], the Quartet is a square version of the triangular Grimace made by Smaz. 2-Layer Grimace 2-layer Grimace 3-Layer Grimace 3-layer Grimace Smart Alex 6*1020 Smart Alex Dumitru A. Pop, patent on 26 May 1992 Jaap's page Rubik's UFO 4*107 - Erno Rubik Rubik's UFO Original in gray, newer version in green. Logi-VIP 2.7*1025 - 1982 Logitoy AG, Austria - Hubert Petutsching Patent WO8101638 Logi-VIP Bulgarian Barrel The Bulgarian Barrel Olidjus Olidjus - a rare vintage original twisty from Russia Square 1 4.36*1011 - Dr. Vojtech Kopsky Square 1 Square 2 Square 2 First designed by Dave Litwin, "Jake," and "Noda" back in 2003 [T], then mass-produced [T]. Super Square 1 1.19*1022 Super Square 1 (4-layer) Produced by cube4you. Watch a video. Super Square 1 Star Mod A Super-Square-1 Star mod - Brett made it in all white then I swapped in the black pieces. You can find on-line [dis]assembly instructions here. Rainbow Nautilus - Selkirk - Meffert The Rainbow Nautilus, designed by Tim Selkirk [T] and mass-produced by Meffert. Octo Bracelet 3.7*1018 (?) Octo Bracelet Sando Ring 4.97*1014 Sando Ring (aka King Ring) Tricky Disky 2.1*1013 Tricky Disky Jaap's page Hungarian UFO 2.1*1013 Hungarian UFO Brainball 2*1012 Brainball Andreas Unsicker Jaap's page Sphere XYZ 3.37*1011 Sphere XYZ Offered by Lori Powers and Adam Giemek, and LA1 Products Netblock UFO 2.0*108 = 200,121,075 Netblock UFO Wai K. Chan Gerdig UFO 130,040 Gerdig UFO Gerhard Huncaga Jaap's page Saturn - MagNif 5040 SaturnJaap's page Clever Disk Clever Disk Turbo Mind Twister Turbo Mind Twister Snow Mystery Snow Mystery

Strange Twisties

This section is a catch-all for other twisty puzzles that aren't easily categorized above. In no particular order.

Unusual Permutation Puzzles

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. I found a second version, a promo for the University of Connecticut Health Center, on auction. [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 See U.S. Design Patent D283523 awarded to Margit C. Balint in Apr. 1986. 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. I have a white version, and a black one in its package. Labyrint Gear Up - designed by Oskar van Deventer made by George Miller Eggcentric - designed by Oskar van Deventer Writer's Block - designed by Oskar van Deventer, purchased from Bits and Pieces. Produced by RecentToys. Use an included "key" to find a set of moves that extends all the pens, allowing the box to be opened and the pens to be reset. Reminds me of "Lights Out." A set of Bishop Cubes This is the Jugo Flower, made from metal, from William Strijbos. The Jugo Flower (aka Yugo Flower or Game Jugo) is one of the most rare twisty puzzles - I have read that only seven prototypes were made. You can see examples of the original plastic versions at Hendrik Haak's website. Wil has had the puzzle reproduced in metal. The fifteen petals can each be flipped over around the long axis. There are four marks on the top of the aluminum hub, and only those four petals positioned at the marks are able to flip, simultaneously. All the petals can be rotated around the hub, provided they are properly aligned (the mechanism is somewhat "catchy"), and a new set of four petals can be positioned at the marks. The goal is to scramble the petals, then restore them to all face-up. This puzzle is similar in principle to "Lights Out."

I have had a Columbus' Egg puzzle since I was a kid. U.S. Patent 4489944 - Hatakeyama 1984. I also still have the instructions, though the packaging is long gone. I have found very few references to it on the web (TwistyPuzzles has a version with different branding listed with no info) and have had to wade through a lot of unrelated material because of the name. The instructions say it was issued by HirschCo at 2633 Greenleaf Ave. Elk Grove Village, IL 60007.

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.

Slider
Position
Cylinders
which move
Down1, 4, 5
Middle3, 5
Up2, 4, 5

3-Dimensional Sliding Piece Puzzles

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

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

Inversion

Mad Marbles by Lakeside (2 instances)

Magic Jack

Tumbler - van Deventer

Pionir Cube

Panex
Panex Puzzle resources page at Baxterweb
Play a level-4 version online at cheesygames.com.

A cross between the Pionir Cube and Munroe's Marbles, from China.

Orbo, by Popular Playthings; and an Asian clone - the Magic Rainbow Ball by Yong Jun

Rainbow Black Hole

? (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.

The Bloxbox is notable since the design by Piet Hein is one of the first examples of sliding cubes in a cube. (The first U.S. patent, 416344, for a puzzle like this was awarded to Charles Rice in 1889.)

Cubedron and Cybedron
Pantazis Houlis at Mindstrat Puzzles has invented a series of what he calls "Gravity Puzzles." These are edge-matching puzzles encloszed in transparent spheres, where the pieces must be tilted into position so that patterns along the edges match, and a piece flips as it moves from one position to another.

Equal-7 - issued by Recent Toys
Tilt the cube to slide the dice - four successively harder objectives - make the total on all sides: 10, 11, 12, 7.

Pionir Pyramid, designed and exchanged at IPP32 by Roxanne Wong; made by Mf8

A vintage sliding piece puzzle in the shape of a bottle of Glenlivet Scotch.
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

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 and Milton Bradley 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 Festival Flower
A vintage 3D sliding puzzle
I have examples of the 1-petal, 5-petal, and 6-petal versions.

Touchdown

Calendar/Bank

Heartache - Kohner

Da Vinci's Mona Lisa Codebreaker, Twisted Mind - another version, using numbers, and with a transparent case, and Twist O Mania

Double Sliding - Dario Uri
Here are three different 3-D sliding piece puzzles by Doug Engel: Blocked Barrel 15, Barrel Slide 121, and Barrel Shuttle 11.

The Mini and Braille Eni Puzzles

Capzule

Instant Insanity II - by Winning Moves.
This reminds me of the Pakovalec (aka Xylinder).
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

3.1*1045
(symbols matter)
Jaap's Page

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.

The Arusloky Puzzle

3-layer and 4-layer Leesho (Liso)

Hungarian Olympic Rings
This puzzle is called Moeraki, a name chosen since the shape of its pieces is reminiscent of a formation of spherical boulders found in New Zealand.

It was designed by Kasimir Landowski and won an IENA gold medal at the 2008 Nuremberg Trade Fair. You can read about the history of the puzzles and order them online at the Casland Games website. Various virtual examples are available at the website and each physical puzzle includes a disk offering some virtual puzzles.

Moeraki is a type of sliding piece puzzle, which I would categorize as of the No Gap, Rigid Frame (Interlocking Orbits) variety. Moeraki No. 3, the one I have, has a square tray and two interlocking oval tracks of pieces in five colors. Moeraki No. 4 has a triangular tray and three interlocking circular tracks of pieces in four colors.

I received my No. 3 as a gift at IPP32, on the condition that I review it. Normally, I don't blog or review puzzles per se, but I accepted it since I had planned on buying one anyway. A policy of mine in general is to try not to say anything if I don't have something nice to say (in print, at least). I don't always succeed in keeping to my policy, but happily I will be in no danger here since I honestly like the Moeraki puzzle.

The concept of a set of markers riding in interlocking circular or oval tracks, which can be mixed by alternate rotations of the groups of pieces in the different tracks, is certainly not new. An example called "Race War Puzzle Between Gold & Silver" designed by William Churchill was awarded U.S. patent 507215 in October of 1893. More recent examples include the Hungarian Rings puzzle by Endre Pap et al (EP0050755) , and the Magic 8. On the Casland website it is mentioned that Ivan Moskovich also received a patent for a similar idea in 1979 (see U.S. patent 4509756), and is now collaborating with Landowski. Original concept or not, the Moeraki puzzle stands out as a very nice implementation and is probably among the most user-friendly of this class of puzzles in my collection.

You can find analysis and solutions of the puzzles at Jaap's Puzzle Page. Jaap gives the total number of possible arrangements for No. 3 as 3,969,069,923,590,200, which, surprisingly, is still larger than the U.S. national debt. Jaap notes that any two diametrically opposed beads will always remain diametrically opposed no matter what moves you make, which I find non-obvious on initial inspection, and fascinating.

Dieter Gebhardt has also published some analysis about this type of puzzle. Dieter's article "Rotational Puzzles with Two Tracks and Two Intersections" can be found in the March 2002 issue #57 of the journal of Cubism For Fun (CFF). Dieter's article supplies a notation convention and a general theoretical basis for deducing useful move sequences, and would be of interest to one attempting to gain a more than superficial understanding of such puzzles, although it deals with types of only two intersections.

These puzzles may appear to be intrinsically simpler than puzzles such as Rubik's Cube and its ilk, and in truth it is possible to attack them without extensive knowledge or analysis of solution procedures or "operators." And, as with twisty puzzles in general, the large number of possible states is not a reliable indicator of difficulty. However, it should be noted that some piece swaps may require upwards of 100 moves to solve, so patience and perseverance will definitely be assets to the recipient of these puzzles. At his website, Diniar Namdarian gives instructions for accomplishing a last swap of two pieces.

When I played with mine, I found the very first challenge to be simply opening the package! The puzzle ships in a cardboard box, which contains a clear plastic case for the puzzle, secured by four clips. A clip can be removed by prying its rounded end away from the case, but the force required makes one leery of breaking something. The puzzle is 135mm square and about 15mm thick. Production values are high - the plastic is good quality and brightly colored. A thoughtful touch is a removable block on the perimeter of the base, which will allow you to extract the sliding pieces from the puzzle in order to effect a brute-force restoration of the solved state, should you deem that necessary. The case also holds a CD-ROM containing additional puzzle software.

For me, assuming one enjoys this class of puzzle, two criteria determine whether a puzzle of this type succeeds or fails as a mechanical puzzle worth playing. First, do the pieces remain securely in their tracks? I have had copies of Hungarian-Rings style puzzles where the beads just fell out. I can report that the puzzle is well-engineered and well-built, and the Moeraki beads will not come out by accident. Second, is it easy to rotate the pieces along their tracks, preferably while holding the puzzle in two hands and using only one's thumbs to slide the pieces? Here I can also give the Moeraki a good grade. As with all interlocking-orbit type puzzles, the orbits must be properly aligned so that the pieces do not catch and impede movement. Out of the box, the movement of the Moeraki beads is somewhat stiff, but a benefit is that the tracks of beads do not tend to overshoot on a move, and a simple tilt of the puzzle will not cause unwanted movement. I gave it a shot of CRC Industrial Food-Grade Silicone spray and now the action is silky smooth!

I think the Moeraki would make good Christmas presents for the puzzler in your life (or for yourself). See what other puzzlers have had to say about the Moeraki puzzles here, here, and here.

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

369,600
[Jaap's page]

Tsukuda's Square / "it"

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

Super Brain Spinner, from FoxMind
DieN Logical Toys

One Circle Two Circles, designed, made, and exchanged at IPP32 by Diniar Namdarian
Overlapping Plates

Mind Lock

3-Level Puzzle
Dollar Tree

Jushbox

Here are some interesting sites: