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

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

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

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

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

There are three Web sites I consider to be the reference for these types of puzzles:

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

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

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

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

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

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. I have used that list, with modifications, as the basis of the table below. Jaap's Puzzle Page site was invaluable in providing combinations data for several puzzles, allowing me to add them to the table at the apropos rank. I have included a column for those puzzles I own, and one for pictures of those I do not, for reference. Various sources disagree on the number of combinations - I have included some alternative figures.

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.

Name and Notes	Combinations	Mechanism	Puzzles I Own	Reference Images
5x5x5 Rubik's Wahn Professor Cube	5.405747889*1082 2.8*1074	Udo Krell 6-armed spider		n/a
Dogic Jaap's page	2.1*1082	Zoltan and Robert Vecsei	Get new ones from Meffert's.	n/a
Megaminx	1068	Kersten Meier Ben Halpern 12-armed spider		n/a
Mozaika	6.27*1049	Rudolf Destics		n/a
4x4x4 Rubik's Revenge	7.4*1045	Peter Sebesteny grooved sphere		n/a
Alexander's Star	7.2*1034	Adam Alexander		n/a
Impossiball	2.4*1025	William O. Gustafson Wolfgang Kuppers	The Thomasball is essentially the same but with a different mechanism	n/a
3x3x3 Picture Cube 4 distinct orientations for all centers e.g. Hoey's Tartan Cube	8.8*1022	Erno Rubik Dan Hoey	n/a
Rubik's Perpetual Calendar (Kalender Kubus) O character on one center has only 2 distinct orientations	4.4*1022	Marvin Silbermintz	play with a virtual calendar cube here	n/a
Rubik's Cube 4th Dim. distinct center orientations	1.1*1022	Erno Rubik		n/a
Rubik's World	2.7*1021	Erno Rubik		n/a
3x3x3 Rubik's Cube Solutions here. Rubik's Ball and the Cuboctahedron are essentially the same.	4.3*1019	Erno Rubik 6-armed spider		n/a
Tantrix The Rock Tantrix Home Page Jaap's page	1*1018 "over a billion billion"	?		n/a
Mobius	?	?		n/a
Octagonal Prism Jaap's page	4.5*1017	?		n/a
Christoph's Magic Jewel (a Pyraminx Octahedron minus the tips)	2.0*1015 8.23*1018	Christoph Bandelow 6-armed spider	n/a	One of these auctioned for over $200
Magic Octahedron	2.0*1015	?	Regularly auctions for between $100-$200.	n/a
Square 1	1.2*1013	Dr. Vojtech Kopsky		n/a
3x3x2 Rubik's Domino	4.0*108	Erno Rubik	I have three knock-offs one is broken :-(	n/a
Rainbow Cube Jaap's page	2.4*108 239,500,800	Bethel Japan		n/a
Skewb Ultimate Jaap's page	1.0*108	Uwe Meffert		n/a
Pyraminx My operator to flip two edges in place: L T' R T R' T L' T'	7.6*107 75,582,720	Uwe Meffert 4-armed spider		n/a
Dino Cube #1 std coloring - each side diff.	1.9*107	?	Jaap's page says these are analogous
Halpern's Tetrahedron	3.7*106	Ben Halpern Kersten Meier	n/a
2x2x2 Pocket Cube Jaap's page	3.6*106 3,674,160	Erno Rubik 6-armed spider	The K-Ball is the same. Here is a site where you can play the K-Ball.	n/a
Skewb Jaap's page I like Meffert's solution.	3.1*106 25,194,240	Tony Durham		n/a
Diamond Cube Jaap's page	2.0*106 2,425,500	?		n/a
Bandaged Cube Jaap's page	1.0*106 1,108,800	?		n/a
Tetraminx (Snub Pyraminx) same as std w/ tips removed	9.3*105	Uwe Meffert		n/a
Skewb Diamond Jaap's page	138,240	Uwe Meffert		n/a
Pyramorphix Jaap's page I find I can solve the Pyramorphix using only four operators (beyond fiddling to get it into a tetrahedron shape and properly position the corners): Move the bottom face down: D2 R2 L2 R2 Exchange the left and right faces: R2 L D2 R L2 D' (this twists corners,too) Twist Up corner clockwise: (R L' R' L)x2 Twist Up corner counter-clockwise: (L' R L R')x2	136,080	Rubik, Barry Lockwood	The Starburst is a custom mod.	n/a
Dino Cube 4 color version I find this very easy to solve even without operators.	42,000	?		There are four versions of the Dino Cube: 2 colors 312.55/1 4 colors 528.99/1;535.55/1 6 colors, no dinos 337/1 6 colors, with dinos 667.96/1
Tony Fisher's Mental Block 3x3x1 aka Rubik's Layer	192 (??)	Custom-made by Tony, from a full-sized Skewb		n/a
Floppy Cube 3x3x1	??	Custom-made by Katsuhiko Okamoto This won First Prize at the IPP26 Design Competition.		n/a
Fisher's Cube / Diagonal Cube 8-color sticker variant	?	?		n/a
Tonne	?	?		n/a
2x2x3	?	?		n/a
Double and Triple Cubes	?	?		n/a
King Pillow Cube	?	?		n/a
Ultimate Cube	?	?		n/a
Truncated Rhombic Dodecahedron This cube-variant is almost a rhombic dodecahedron, except the four-color centers are flat, not pyramidal. The three-color corners are pyramidal.	?	?		n/a
Morph (1x2x2)	?	?		n/a

This section lists dihedral puzzles - puzzles whose halves can move relative to each other and permit the exchange of pieces between them.

Name and Notes	Combinations	Puzzle
Logi-VIP 1982 Logitoy AG, Austria Hubert Petutsching	?
Masterball Dr. Geza Gyovai	3.5*1026 6.7*1021
Smart Alex Dumitru A. Pop, patent on 26 May 1992 Jaap's page	6*1020
Tricky Disky Jaap's page	2.1*1013
Hungarian UFO	?
Brainball Andreas Unsicker Jaap's page Order one from The Rubiks Shop.	2*1012
Netblock UFO Wai K. Chan	2.0*108 200,121,075
Sando Ring	?
Rubik's UFO Erno Rubik	4*107
Brainbow Jaap's page	623,760
Gerdig UFO Gerhard Huncaga Jaap's page	130,040
Saturn Jaap's page	5040
Roundy (3-leaf/6-color version) Fritz Gruber 12/7/93	?
Roundy (3-leaf/3-color version) Fritz Gruber 12/7/93 Jaap's page	2880
Octo Bracelet	?
Clever Disk	?

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

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

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

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

One of these went for $150 on eBay.

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

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

Movable Gap, Rigid Frame
Pepsi Can Start with an idea as simple as mapping a 15-like puzzle onto a cylinder. This puzzle has advertised several popular drinks.	Billiards 9-Ball created by Joshua Frankel 3,628,800 positions Jaap's page	Massage Ball Otto Wu patent on 14 Feb 1995 6.1*1019 positions Jaap's page
Vadasz Cube (2x2x2 and 3x3x3 versions) (Also 4^3, which I don't have.)	Minus Cube (Russian)	Varikon black
Peter's Black Hole 5.4*1027 positions Jaap's page Twistypuzzles.com has an article by Ad van der Schagt titled "The History of Sliding Block Puzzles Before Peter's Black Hole" (PDF).	Clark's Cube	I-Qube
This is called the "Switch" or the "Knox Transposition Puzzle." It was issued by Mag-Nif in 1970 and also appears in their "Game Chest" set. The pegs slide in channels in the base. The object of the game is to exchange the sets of colored pegs in 24 moves or less. This actually borders on a non-jumping (exchange-only) type of Peg Solitaire.		Crossteaser 2.7*1011 positions [Crossteaser home page]
Inversion	Mad Marbles	Magic Jack
Tumbler - van Deventer	Pionir Cube	Panex Panex Puzzle resources page at Baxterweb Play a level-4 version online at cheesygames.com.
? (Hungarian barrel)	Diamond Bob's Billiards Eight Ball	Rubik's Brain Racker
Movable Gap, Movable Frame
Mind Twister aka Wisdom Ball Yang Ju-Hsun 1 June 1993 1.7*1075 positions Jaap's page	Saturn - LD Belgium white and black versions	Tomy Great Gears 1.46*1020 positions Jaap's page
This is called Entrapment. There are also some newer "clones" available. The clear plastic on the old ones is yellowed with age.	Atomic Chaos Christoph Hausammann 2.1*1012 positions Jaap's page	Pakovalec aka Xylinder 1.3*1010 positions Jaap's page
Missing Link 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 Ferdinand Lammertink 6.4*1028 positions Jaap's page
Tomy Rack 'Em Up Mizunuma Masanori and Watanabe Hiroyuki 1984 6.3*107 positions	Tomy Row By Row Mizunuma Masanori and Watanabe Hiroyuki 13 Nov 1984 2.8*1031 positions Jaap's page	SpongeBob Puzzlepants 10,080 positions
Russian Flower	Touchdown	Calendar/Bank
No Gap, Rigid Frame (Interlocking Orbits)
Equator, and Hungarian Globe 1.1*1025 positions Jaap's page	Hungarian Rings Endre Pap William Churchill, patent on 28 May 1891 7.5*1019 positions Jaap's page	Magic 8
Rubik's Rings 1.9*1014 positions Jaap's page a source	Circle Puzzle 369,600 [Jaap's page]	Rotascope Raoul Henrique Raba 1982 9.1*107 positions I obtained this Rotascope which is a souvenir of the sixth IPP at Jerry Slocum's house. The front contains invitation text and Jerry's home address and phone number, which I don't want to display here. This is a picture of the back - not very puzzling without a pattern to scramble.
Tsukuda Magic Puzzle (Turnstile) Douglas Engel 6.3*109 positions Jaap's page
Lotica	Turn Push	Whirligig
Mad Triad 3.1*1045 (symbols matter) Jaap's Page	Handy Mad Triad 8.3*1023 positions Jaap's page	Rubik's Shells 4.7*1014 positions Jaap's page
Cmetrick Too There are colored disks riding in "craters" in spheres embedded in the frame. The spheres rotate and can exchange disks.	Cmetrick Too Hard In this more difficult version, the centers of the disks are colored, too.
No Gap, Movable Frame
Topspin Ferdinand Lammertink 2.4*1018 positions Jaap's page	Trillion - red, black Gunpei Yokoi 1.0*109 positions Jaap's page	Nintendo Ten Billion Barrel and Club Nintendo Star Barrel Gunpei Yokoi 2.7*1014 positions Solution site here.
I have seen this design from several places. I believe it has been called "Sortospherical."	The Orb[-it] Christopher C. Wiggs and Christopher J. Taylor 7.4*1028 positions Jaap's page	Astrolabacus John D. Harris Pat. 8 Jul 1997 3.6*1016 positions Jaap's page
Port to Port Triple Cross Ferdinand Lammertink Pat. Aug 6 1996 5.9*109 positions Jaap's page	Gripple Murray J. Gould, patented 5 April 1988 2.0*1013 positions Jaap's page	Russian Gripple
Magic Sphere	Rotos Jaap's page	Magic Cross (Zauberkreuz)
Flip Side - Thinkfun	Swissmad 369,600 [Jaap's page]	Tsukuda's Square
Rubik's Fifteen	Binary Bisect 5 - Doug Engel
Overlapping Plates
Mind Lock	3-Level Puzzle Dollar Tree	Jushbox

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