The Magic Polyhedra Patent Page 01/08/2007 03:34 PM

Magic Polyhedra* Patents

By Joshua Bell. Last updated May 07, 2006.

Background | Resources | Solution | Mechanics | Patents | Collection Background

In the early 1980's a fad briefly swept most of the civilized world - the Rubik's Cube®. This unassuming toy inspired books, competitions, songs, and even a cartoon series on television, along with several similar polyhedral puzzles. I briefly toyed with the puzzle but I was more fascinated with the mechanical aspects than the mathematical puzzle. The latter was answered handily with books (which I had) or persistence (which I did not), but the former was more mysterious. Fortunately it's easy to crack open a cube given a butter knife, with little harm done.

In late 2000 I ran across a discussion of different magic polyhedra beyond the 3x3x3 cube - within the larger family of sequential movement puzzles - and was briefly hooked. With a few hundred dollars spent via mail orders and on eBay I secured myself a nice collection. I even dabbled with making some custom puzzles. I'm still lousy at solving the darned things, but I appreciate them even more than I did in 1980. Resources

There are hundreds of cube-related pages out there. These are some of my favorites.

Hendrik Haak's Puzzle Shop & Museum - many rare puzzles on display, and several for sale TwistyPuzzles.com - an extremely extensive collection of pictures of twisty puzzles Jaap's Puzzle Page is one of the best collections of solutions and musings on the possibilities and mechanics of puzzles. Especially distinctive is his page of Ideas for other puzzles. Puzzler - a shareware Windows app that lets you play with nearly every twisty puzzle ever devised, including ones never constructed in the real world. Twisty Megasite - a puzzle collection and discussion forum. Cube-Lovers - an MIT-based mailing list running from 1980 through 2000. The archive is available at ftp://ftp.ai.mit.edu/pub/cube-lovers or via various web archives, such as http://www.math.rwth- aachen.de/~Martin.Schoenert/Cube-Lovers/ Magic Cube 4D - why stop at three dimensions?

Manufacturers and Suppliers:

Rubiks.com - home page for the Rubik's Cube. Meffert's Puzzles - home page for Uwe Meffert's range of puzzles, including the Skewb family, Pyraminx, Pyramorphix, and Megaminx. Hockey Puck Puzzle

Or search on eBay for related puzzles up for auction. But please check out the above suppliers first - you can order

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brand new 4x4x4 directly from Rubiks.com for about US$20 or a 5x5x5 directly from Meffert's for about US$30, so there's no need to pay high prices in an auction.

And some minor sites, worth visiting:

Puzzle Museum - Sequential Puzzles - has photos of some rare puzzles A Japanese collector's extensive collection of twisty and other puzzles and toys Georges Helm's puzzle collection Solution

There is really only one solution that you need for the majority of Rubik-like puzzles, especially if - like me - you're lousy at memorizing things and don't plan to solve the puzzle daily for fun and profit. Visit Alan Hensel's page, "How to Solve Almost Any Rubik-Like Puzzle" for a very detailed walk through the solution. Here's the short version:

What you need:

A sequential movement puzzle A piece of paper and pencil The ability to solve one layer (or face or slice or equivalent) of the puzzle. The ability to describe abstract operations necessary to solve the puzzle; e.g., "I need to cycle these three edges and rotate these two corners." A notation for moves on the puzzle (e.g. FBLRUD, F'B'L'R'U'D', where x' is the inverse of x) The magic formula: P XsX's' P'

Steps (places where you actually move the puzzle are underlined):

1. Get the puzzle as close to a solved state as you can by whatever means you like. 2. Invent, write down, and perform any sequence of moves, P, which brings all of the pieces you want to solve onto a single layer (L). 3. Conceptually group the remaining moves and flips into a pair of two symmetric abstract operations (M and M'). 4. Invent, write down, and perform any sequence of moves, X, which performs M but leaves layer L untouched at the end of the sequence; the rest of the puzzle can be messed up. 5. Perform the single move s which is a rotation of L which gets pieces corrected by M out of the way and gets the pieces to be corrected by M' into their place. 6. Perform the sequence X' - that is, do the inverse of each of the steps you wrote down in reverse order. 7. Perform the single move s' which puts layer L back where we had it in step 3 - except for M and M'. 8. Perform the sequence P' which puts everything back where you started.

And some comments:

You can do this piecemeal - if you have corners and edges that need work, you can simplify things by dealing with corners first then edges, or vice versa - the puzzle doesn't care how many steps you break it down into. When inventing P and X you may find it convenient to write the steps down as you go. To get through Step 3 you may need to break down a seemingly simple operation into two parts; e.g. if three edges (ABC) need to be cycled, consider it instead as two swaps (AB, BC). In this case, during Step 5 some pieces touched by M will be set up to be touched a second time by M'. This scheme won't work for some puzzles; when in doubt I always start with Jaap's Puzzle Page. Mechanics

Common features

Many puzzles are built around either a central "spider" mechanism (shafts joining at the center of the puzzle), a central sphere, or some intermediate combination. A spider is used in the 3x3x3 cube, the Skewb, and the Impossiball. The Pyraminx and 4x4x4 cube are based around a central sphere.

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In 2x2x2 puzzles, one octant must remain fixed to the central mechanism. If all eight were identical the internal mechanism could drift out of alignment with the rest of the puzzle, rendering it impossible to twist on two of the three axes. NxNxN puzzles where N is even are often based on a N+1 spider with the axis concealed or shared by the center pieces. A variant of the spider mechanism is the stellated polyhedron, where a core polyhedron is extended with pyramidal tips which rotate. The "cubie" analogs sit between these tips are attached to the mechanism by gaps in the base of the tips. When the tips rotate, the "cubies" are interchanged. This is used in the Alexander's Star (where the internal mechanism is a small stellated dodecahedron) and Dino Cube (where the internal mechanism is a stellated octahedron, or stella octangula). Magnets are rarely used in mass-produced puzzles, although often suggested as possible mechanisms.

Notation & Terminology

NxNxN - a puzzle in the family of Rubik's Pocket Cube (2x2x2), Rubik's Cube (3x3x3), Rubik's Revenge (4x4x4) and the Professor Cube (5x5x5). Each face or slice may be rotated 90°, 180° or 270°. Pieces of the same type (e.g. edges, corners, centers) can be swapped. XxYxZ - like an NxNxN a puzzle, but with different numbers of cubies on different axes. Includes the Magic Domino, with a 3x3x2 mechanism. Typically, sides which do not match other sides must be rotated 180° rather than 90° to permit the puzzle to continue to be manipulated. A trivial extension is one that does not affect the solvability of a puzzle, beyond rotations of the single extension piece which do not affect any other part of the puzzle. An example are the tips of a Pyraminx (which merely rotate around their single point of attachment to an otherwise external the puzzle), and the purely decorative, non-rotating tips of the extended 3x3x3 cubes (e.g. 3x3x"5" - wherein the top and bottom layer of cubies are decorated with another layer of fixed cubies). A non-trivial extension is a modification of an existing or more fundamental puzzle type with additional pieces that may be interchanged. Typically, these may only be interchanged with each other. The canonical example is the Abu-Shumays work on extending the 2x2x2 mechanism with an additional layer of 4 cubies. These extension cubies are not uniquely fixed to cubies within the base puzzle and may be rotated as a group. Adding extension cubies to more faces allows the extensions to be interchanged among the faces when the base 2x2x2 mechanism is twisted along a given axis, producing a new, non-decorative aspect to the puzzle. Higher order cubes (4x4x4 and 5x5x5) can be thought of as extreme non-trivial extensions of the 2x2x2 and 3x3x3 mechanism. (N+a)x(N+b)x(N+c) - a new notation (unique to this page) to describe an NxNxN puzzle with non-trivial extensions. Octahedrons with triangular cubies may be face-centered (a face rotates) or vertex-centered (a pyramid around a corner rotates). The order of an octahedral puzzle refers to the number of unit-triangles along one edge, regardless of triviality. A puzzle is said to be deep cut if there is a cut which bisects the entire puzzle. Thus, the 2x2x2 cube and Skewb are deep cut while the 3x3x3 cube is not. Patents

Thanks to the US Patent and Trademark Office and the European Patent Office it is possible to use the web to view patents for almost anything that has been patented within the last thirty-odd years. This is an attempt at a somewhat comprehensive list of interesting patents relating to magic polyhedra.

Details Description Representative Image Patent US3637215 Number: Popular Spinout Name: Inventor(s): William Keister Date Filed: December 22, 1970 Date Issued: January 25, 1972

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Patent US3655201 PATTERN Number: FORMING PUZZLE AND Popular METHOD WITH Name: PIECES Inventor(s): Larry D. Nichols ROTATABLE IN Date Issued: April 11, 1972 GROUPS

Patent US4413823 Number: Popular Entrapment Name: Inventor(s): Jeffrey D. Breslow Date Filed: November 3, 1980 Date Issued: November 8, 1983

Patent US4376537 Number: Popular Billion Barrel and Name: Trillion Inventor(s): Gunpei Yokoi (Nintendo) Date Filed: December 30, 1980 Date Issued: March 15, 1983

Patent US4373729 Number: Popular Missing Link Name: Inventor(s): Steven P. Hanson; Jeffrey D. Breslow Date Filed: January 12, 1981 Date Issued: February 15, 1983

Patent BE887875 The original patent Number: for the 3x3x3 Rubik's Cube and Popular Rubik's Cube one mechanism Name: for the 2x2x2 Inventor(s): Ernö Rubik Pocket Cube. Date Filed: March 10, 1981 Date Issued: July 1, 1981

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Patent EP0042695 The original Number: Pyraminx - an order-3 Popular Pyraminx tetrahedron with Name: rotating verticies Inventor(s): Uwe Meffert and trivial tips. Date Filed: June 10, 1981 The core is a central tetrahedron, sphere or four- armed spider. Attached to this are four octahedrons. Each octahedron has a trivial rotating tip and between the octahedrons are the six mobile tetrahedral edge pieces. The patent also describes the stellated pyraminx. Patent US4377916 Number: Popular Rubik's Snake Name: Inventor(s): Kenji Komiya Date Filed: July 20, 1981 Date Issued: March 29, 1983

Patent US4392323 Number: Popular Rubik's Snake Name: Inventor(s): Erno Rubik (Politoys) Date Filed: July 24, 1981

Patent US4471959 Number: Popular Rubik's Fifteen Name: Inventor(s): Erno Rubik Date Filed: September 1, 1981

Patent US4415158 Number: Popular Puzzler Name: Inventor(s): Douglas A. Engel (General Symmetrics, Inc.) Date Filed: September 21, 1981 Date Issued: November 15, 1983

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Patent US4410179 A single layer Number: puzzle with six pieces. Popular Rubik's Cheese Name: Inventor(s): Erno Rubik (Konsumex) Date Filed: November 9, 1981

Patent DE3204033 Another Number: mechanism equivalent to the Popular Thomas Ball Impossiball. Name: Twenty triangular Inventor(s): Dr. Gilbert caps twist in Obermair groups of five Date Filed: February 5, 1982 around a common vertex. The Date Issued: August 25, 1983 popular puzzle has a cap on each vertex to make the puzzle decorated and shaped like a soccer ball, but the patent proposes other outer shapes such as a dodecahedron or icosahedron. Patent US4402510 Number: Popular Crossover Name: Inventor(s): Gunpei Yokoi (Nintendo) Date Filed: February 12, 1982 Date Issued: September 6, 1983

Patent US4553754 Number: Popular Orb-it Name: Inventor(s): Christopher C. Wiggs & Christopher J. Taylor Date Filed: February 16, 1982

Patent GB2084471 An order-2 face- Number: centered octahedron based Popular Rubik's on a Pyraminx-like Name: Octahedron central sphere. Inventor(s): Ernö Rubik The face-centers Date April 15, 1982 are affixed to the Issued: sphere and the vertices float around.

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Patent US4452454 Number: Popular Equator Name: Inventor(s): Wilton R. Greene Date Filed: July 30, 1982 Date Issued: June 5, 1984

Patent US4344623 A 2x2x2 Number: mechanism (depicted is a Popular vertical slice Name: through a diagonal Inventor(s): Toshiyuki Isobe plane). Date Issued: August 17, 1982

Patent DE3144834 Number: Popular Hungarian Globe Name: Inventor(s): Molnar Ferenc Date Issued: October 14, 1982

Patent DE3110834 Number: Popular Megaminx Name: (Bandelow) Inventor(s): Dr. Christoph Bandelow; Helmut Corbeck Date Issued: October 14, 1982

Patent WO8203564 Number: Popular Hungarian Name: Supernova Inventor(s): Szlivka Ferenc Date October 28, 1982 Issued:

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Patent DE3104021 Number: Popular Babylon Tower Name: Inventor(s): ARXON SPIEL & FREIZEIT GMBH (DE) Date Issued: December 2, 1982

Patent GB2100134 Number: Popular Inversion Cube Name: Date December 22, Issued: 1982

Patent US4526372 Number: Popular Puzzle Toy (like Name: equator with double rows, no orientations) Inventor(s): Evangelos T. Kikis Date Filed: February 3, 1983

Patent US4378116 The 3x3x2 Rubik's Number: Domino. Two mechanisms are Popular Magic Domino suggested - one Name: with a six armed Inventor(s): Ernö Rubik spider (four of Date Issued: March 29, 1983 which are sliced in half vertically and terminate within the body), and one that dispenses with the spider in favor of a tab-and- groove mechanism. Patent US4378117 A 2x2x2 Number: mechanism. Seven pieces are held in Popular Rubik's Pocket place against an Name: Cube eighth by three Inventor(s): Ernö Rubik arms. Date Issued: March 29, 1983

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Patent DE3245341 Number: Popular Pyramorphix Name: Inventor(s): Manfred Fritsche Date Issued: August 11, 1983

Patent DE3206560 Number: Popular Magic Name: Octahedron Inventor(s): Karl Rohrbach Date Issued: September 1, 1983

Patent US4405131 An elaborate Number: 2x2x2 mechanism utilizing nested Popular shells. The cubies Name: project arms into a Inventor(s): Tibor Horvath central sphere with Date Issued: September 20, tracks to guide 1983 their movement.

Patent US4409750 A calendar cube. Number: The patent includes the Popular Rubik's Calendar sticker layout and Name: a solution. Inventor(s): Marvin A. Silbermintz Date Issued: October 18, 1983

Patent US4421311 A 4x4x4 cube; Number: pieces orbit a central sphere with Popular Rubik's Revenge projecting caps Name: that both holds the Inventor(s): Peter Sebesteny face-centers and Date Issued: December 20, provides a track 1983 for their rotation.

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Patent US4437667 Just a 3x3x3 Number: cube, but the patent is for a Popular variety of Name: Scrabble-like Inventor(s): Ronald L. Miller games playable on Date Issued: March 20, 1984 the surface by the use of detachable stickers or markers for the cubie faces.

Patent US4441715 A sphere with 6 Number: interchangeable segments. Like a Popular Masterball without Name: the latitudinal Inventor(s): Stephen C. Titus slices. Date Issued: April 10, 1984

Patent JP60222076 This puzzle is to Number: the Rainbow Cube as the 5x5x5 is to Popular Master Rainbow the 3x3x3. Name: Cube Amazing! Inventor(s): Watanabe Yasunari, Ikegami Yuuji Date Filed: April 18, 1984

Patent US4451039 An order-3 face- Number: centered magic octahedron. The Popular tips are non-trivial Name: and none of the Inventor(s): Clarence W. exterior pieces are Hewlett, Jr. fixed in place. The Date Issued: May 29, 1984 central mechanism is a stella octangula with rotating tips, surrounded by six corner octahedra, twelve edge octahedra, and twenty four face tetrahedra. Patent US4453715 The Halpern-Meier Number: Pyramid. The edges and tips Popular Halpern-Meier project feet into Name: Pyramid the center of the Inventor(s): Benjamin R. tetrahedron which Halpern combine to form a Date Issued: June 12, 1984 sphere. The face- centers connect in the center of the sphere and hold everything together. Also includes a Megaminx variant and a truncated tetrahedron with face centers, corners and edges, and a suggestion for http://www.calormen.com/TwistyPuzzles/twisty.htm Page 10 of 30 The Magic Polyhedra Patent Page 01/08/2007 03:34 PM

what will later be known as "bandaging". Patent US4473228 A possible Number: mechanism for an Impossiball - an Popular icosahedron on Name: which each "cap" Inventor(s): John R. Hart of 5 faces may be Date Issued: September 25, rotated, allow the 1984 faces to migrate around the surface.

Patent US4474376 The canonical Number: Impossiball mechanism - an Popular Impossiball icosahedron on Name: which each "cap" Inventor(s): William O. of 5 faces may be Gustafson rotated, allow the Date Issued: October 2, 1984 faces to migrate around the surface. With one piece removed the toy still holds together and the game turns into a variant of the 15- Puzzle; adjacent triangular pieces may be slid into the gap. Patent US4474377 A combination of a Number: 2x2x2 cube and a Dino cube. The Popular internal Name: mechanism is a Inventor(s): Jonathan J. stella octangula Ashley from a Dino Cube Date Issued: October 2, 1984 with rotating tips (like an Alexander's Star), but deep-cut with an embedded 2x2x2 mechanism.

Patent US4496155 The puzzle has Number: the shape of a stella octangula. Popular Star of David The core Name: octahedron has a Inventor(s): Israel Goldfarb 2x2x2 mechanism Date Issued: January 9, 1985 like a modified Pyramorphix, but the tips can rotate as well.

Patent US4500090 Very similar to the Number: Alexander's Star, but with a more Popular complicated Name: mechanism of Inventor(s): Antoliono Nieto grooves. Date Issued: February 19, 1985

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Patent US4506891 The popular Number: Alexander's Star puzzle; a great Popular Alexander's Star dodecahedron Name: where each vertex Inventor(s): Adam Alexander rotates. The Date Issued: March 26, 1985 central mechanism is a small stellated dodecahedron with a dodecahedral core; each tip rotates.

Patent DE3333914 Number: Popular Skewb Diamond Name: Inventor(s): Michael Richter Date Issued: April 18, 1985

Patent US4513970 A 2x2x2 Number: mechanism using 8 magnets on Popular each plane (6 per Name: piece). Non- Inventor(s): Ovidiu Opresco; cubical shapes are Jon D. Marinesco suggested, such Date Issued: April 30, 1985 as off-center or tiled cuts in cubes, off-center cuts in cubes, and visually attractive egg- shaped puzzles.

Patent US4522401 A spherical 3x3x3 Number: with no internal spider mechanism; Popular the shell is held Name: together in a Inventor(s): William O. manner similar to Gustafson the design of Date Issued: June 11, 1985 US4441715.

Patent US4540177 A 4x4x4 Number: mechanism based around an Popular embedded 2x2x2 Name: mechanism. The Inventor(s): Tibor Horvath face centers have Date Issued: September 15, feet embedded in 1985 a central sphere. The edge pieces lock under adjacent centers, and corner pieces lock under adjacent edges and centers just like on the 3x3x3 cube.

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Patent US4557484 An elegant Number: spherical 2x2x2 mechanism with Popular the mobile octants Name: riding around a Inventor(s): Benjamin F. hollow central Sherman, Jr.; sphere on tracks. Samuel Francis Further, there is Date Issued: December 10, the suggestion that 1985 at each intersection of four pieces the tips may be swapped by rotation, thus yielding a (2+2)x(2+2)x(2+2) mechanism. Patent US4558866 A variety of Number: designs: a Pyraminx without Popular trivial tips and a Name: Tetraminx; a face- Inventor(s): William L. Alford centered order-2 Date Issued: December 17, octahedron; a 1985 PentUltimate mechanism with magnetic pieces around a metal sphere

Patent US4708345 Number: Popular Manipulative Name: amusement device (hexagonal ring) Inventor(s): David B. Ayers Date Filed: January 6, 1986

Patent US4586713 A generalized Number: family of star- prisms puzzles; all Popular are three layers Name: tall. Inventor(s): Ibrahim K. Abu- Shumays; Mary D. Abu-Shumays Date Issued: May 6, 1986

Patent SU1232171 Number: Popular Russian Master Name: Pyraminx Inventor(s): Alexsandr A. Ordynets Date Issued: May 23, 1986

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Patent US4593907 First, yet another Number: 2x2x2 mechanism, followed by an Popular application of it to Name: various regular Inventor(s): Ibrahim K. Abu- and irregular Shumays; Mary polyhedra. Next, a D. Abu-Shumays generalized Date Issued: June 10, 1986 extension mechanism for making a 2x2x(2+1) - and the application of that to polyhedra. Then duplicating the extension to produce a 2x2x(2+2), and the logical follow-on of 2x(2+2)x(2+2) and (2+2)x(2+2)x(2+2) puzzles, and suggestions for polyhedra to embed the mechanism within. It goes on to suggest a possible true 4x4x4 mechanism, but does not give details. Wow! Patent US4593908 A deep-cut order- Number: 2 vertex-centered octahedron based Popular on a 2x2x2 Name: embedded within Inventor(s): Baky B. Ibrahim an cuboctahedron Date Issued: June 10, 1986 with pyramidal tips that rotate on the underlying faces and split along their edges.

Patent US4600199 Another hefty Number: patent. This one starts with a Popular Professor's Cube deeply truncated Name: cube and works its Inventor(s): Udo Krell way up to the Date Issued: July 15, 1986 amazing 5x5x5 mechanism. It also suggests a deep- cut dodecahedron puzzle (which resembles but is distinct from the Pyraminx Crystal)

Patent US4685680 Number: Popular Rubik's Magic Name: Inventor(s): Erno Rubik Date Filed: July 28, 1986 Date Issued: August 11, 1987

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Patent US4817952 Number: Popular Rubik's electronic Name: cube Inventor(s): Gabor Biro; Robert Breitner; Gabor Fazekas; Istvan Hejjas; Peter Niklai; Erno Rubik; Karoly Szabo; Istvan Tothmatyas Date Filed: December 9, 1986 Date Issued: April 4, 1989 Patent US4667961 The name of the Number: patent is "Star prism puzzles Popular generalized" and it Name: really is just a Inventor(s): Ibrahim K. Abu- generalization of Shumays; Mary the earlier patent D. Abu-Shumays US4586713 by the Date Issued: May 26, 1987 same inventors.

Patent US4674750 The outer puzzle Number: is a rhombic dodecahedron Popular where each vertex Name: rotates. Internally, Inventor(s): Ibrahim K. Abu- the mechanism is Shumays; Mary similar to the D. Abu-Shumays Alexander's Star - Date Issued: June 23, 1987 an internal cuboctahedron with rotating tips added to both the square and triangular faces, which trap the surface pieces in central notches. Patent US4735417 Number: Popular Gripple Name: Inventor(s): Murray J. Gould Date Filed: June 25, 1987 Date Issued: April 5, 1988

Patent DE8707783U Number: Popular Pyraminx Crystal Name: Date July 16, 1987 Issued:

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Patent US4782676 Number: Popular Padlock Name: Inventor(s): Ira Friedman Date Filed: October 5, 1987 Date Issued: November 8, 1988

Patent US5176382 Number: Popular Atomic Chaos Name: Inventor(s): Christoph Hausammann (Joker SA/AG Date Filed: November 9, 1987

Patent US4706956 Two octahedral Number: variations on the 3x3x3 mechanism; Popular one resembles the Name: Trajber's Inventor(s): Ibrahim K. Abu- Octahedron Shumays; Mary (essentially an D. Abu-Shumays extremely Date Issued: November 17, truncated cube), 1987 the other the Star Puzzler but with fixed rather than trivial tips. Then an icosahedral puzzle where the vertex-caps are fixed but overlap to create mobile edge pieces. Patent US4871173 Number: Popular Topspin Name: Inventor(s): Ferdinand Lammertink (Binary Arts) Date Filed: September 2, 1988

Patent DE3711368 Number: Popular Cross Name: Inventor(s): Margaritis Athanasios Date Filed: October 20, 1988

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Patent US4881738 Number: Popular Manipulative Name: amusement device (hexagonal ring) Inventor(s): David B. Ayers Date Filed: January 11, 1989

Patent US4836549 A stella octangula Number: in the form of two intersecting order- Popular four tetrahedrons. Name: The tips are trivial Inventor(s): James T. Flake (like the Date Issued: June 6, 1989 Pyraminx). Sub- pyramids of size 2 (just like on the Pyraminx) may be rotated 120°, cycling three pieces. Moving one layer deeper, another rotation of 120° may be performed which moves the sub- pyramid plus the "base" of the other tetrahedron. Patent EP0322085 Number: Popular Rubik's Clock Name: Inventor(s): Christopher Charles Wiggs Date Issued: June 28, 1989

Patent US4856786 The Masterball Number: mechanism. Essentially Titus' Popular Masterball six segment Name: sphere (which this Inventor(s): Geza Gyovai patent cites as Date Issued: August 15, 1989 prior art) with the addition of four bands (two polar, two equatorial) that rotate around the surface orthogonally to the underlying segments. Patent US4865323 Like a super- Number: Masterball with six latitudinal Popular segments vs. four Name: on the Masterball. Inventor(s): Paul A. The mechanism Heusinkveld for the surface Date Issued: September 12, bands is much 1989 more robust in appearance, however.

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Patent US4872682 A 3x3x3 combined Number: with a "9-puzzle"; the faces on the Popular cube can be slid Name: around exactly as Inventor(s): Ravi in a 15-puzzle; Kuchimanchi; manipulating the Madhukar N. 3x3x3 mechanism Thakur migrates the empty Date Issued: October 10, 1989 square between faces.

Patent US4957291 Number: Popular Luminations Name: Inventor(s): Donald C. Miffitt; Angelo Tortola; Charles S. Sebor; Robert L. Halliday Date Filed: November 6, 1989 Date Issued: September 18, 1990 Patent US4889340 A variant of the Number: Equator puzzle, where three Popular intersecting bands Name: on the surface of Inventor(s): Wilton R. Greene a cube allow Date Issued: December 26, square pieces to 1989 slide around and move between bands.

Patent DE3821297 Number: Popular Tricky Disky Name: Date December 28, Issued: 1989

Patent US5267731 Number: Popular Roundy (4 faces) Name: Inventor(s): Fritz Gruber Date Filed: April 28, 1990

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Patent US5184822 Number: Popular Rubik's Dice Name: Inventor(s): Jozsef Bognar; Erno Rubik Date Filed: December 28, 1990 Date Issued: February 9, 1993

Patent WO9102574 Number: Popular Puck Name: Inventor(s): Zoltan Pataki; Geza Csomos Date March 7, 1991 Issued:

Patent US5074562 Number: Popular Equator Name: Inventor(s): Daniel E. Green Date Filed: May 14, 1991

Patent US5114148 Number: Popular Puzzle Device Name: (like Mozaika but different mechanism) Inventor(s): Ming-Zen Liu Date Filed: July 9, 1991

Patent US5116053 Number: Popular Puzzle Can Name: Inventor(s): Karl Blankenburg, Karl V. Blankenburg, Duane Rode Date Filed: August 5, 1991 Date Issued: May 26, 1992

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Patent US5172912 Number: Popular Back Spin Name: Inventor(s): Ferdinand Lammertink (Binary Arts) Date Filed: January 15, 1992

Patent US5215305 Number: Popular Wisdom Ball Name: Inventor(s): Yan-Ju Hsun Date Filed: May 18, 1992

Patent US5116052 A cube with Number: hexagons projecting from Popular Smart Alex four sides. The Name: hexagons are Inventor(s): Dumitru A. Pop divided into six Date Issued: May 26, 1992 triangular segments and rotate around their centers. The entire cube is split at the equator and rotates, interchanging pieces of adjacent hexagons. A very simple mechanism but a nice puzzle. Patent US5271688 Number: Popular Assembly Cube Name: Date Filed: August 28, 1992 Date December 21, Issued: 1993

Patent US5271688 Number: Popular Assembly cube Name: Inventor(s): Hank Chang (Famwealth International Inc., TW) Date Filed: August 28, 1992 Date Issued: December 21, 1993

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Patent EP0502261 Number: Popular Pyrix Name: Inventor(s): Jaap Koops; Jan Essebaggers Date Issued: September 9, 1992

Patent RO106960 A 4x4x4 Number: mechanism with a spherical core. Popular Name: Inventor(s): Florian Radut Date Filed: October 27, 1992

Patent RO106962 The classic 3D Number: marble variation of famous the 15- Popular puzzle. A cube is Name: packed with Inventor(s): Florian Radut marbles (in a grid Date Filed: December 14, arrangement), and 1992 one marble is missing. By tipping the cube the marbles can be scrambled and (hopefully!) unscrambled.

Patent US5193809 The canonical Number: Square-1 patent. A simple dihedral Popular Square-1 mechanism lets Name: the central slab Inventor(s): Karel Hrsel; trap any number of Vojtech Kopsky appropriately Date Issued: March 16, 1993 grooved pieces on either side.

Patent US5370394 Number: Popular Gerdig Ufo Name: Inventor(s): Gerhard Huncaga Date Filed: March 26, 1993 Date Issued: December 6, 1994

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Patent HU207233 Number: Popular Rubik's Triamid Name: Inventor(s): Ernoe Rubik; Emil Visnyovszky Date Issued: March 29, 1993

Patent US5199711 A solid mechanism Number: for a puzzle sliced into six radial Popular segments like Name: Rubik's Cheese or Inventor(s): Zoltan Pataki; the Masterball. Geza Csomos; Janos Feher Date Issued: April 6, 1993

Patent RO106962 Number: Popular Mad Marbles Name: Inventor(s): Florian Radut Date Issued: August 30, 1993

Patent US5389063 Number: Popular Massage Ball Name: Inventor(s): Otto Wu Date Filed: October 28, 1993

Patent US5417425 Number: Popular Lights Out Name: Inventor(s): Revital Blumberg; Michael Ganor; Avish J. Weiner Date Filed: April 8, 1994 Date Issued: May 23, 1995

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Patent US5437454 Number: Popular Switchback Name: Inventor(s): Ira Friedman Date Filed: April 29, 1994 Date Issued: August 1, 1995

Patent RO110598 Like the Hungarian Number: Rings, this puzzle has six interlocking Popular rings of eight Name: marbles - the end Inventor(s): Florian Radut result is similar to Date Filed: June 14, 1994 a 3x3x3 cube where the orientation of the pieces doesn't matter.

Patent US5338033 An extremely Number: simple and elegant 2x2x2 mechanism, Popular nominally in a Name: spherical puzzle. Inventor(s): Guillermo Nunez Each of the 7 Serrano floating octants Date Issued: August 16, 1994 clips into the axes and is held in place by virtue of the shape. Contrast this with Rubik's 2x2x2 mechanism which falls apart when any single piece is removed. Patent US5358247 The Puzzle Ball - Number: a spherical Skewb. A four-armed Popular Puzzle Ball spider with Name: triangular tips, four Inventor(s): Uwe Meffert more floating Date Issued: October 25, 1994 triangular "faces", and six floating square "faces".

Patent US5542673 Number: Popular Triple Cross Name: Inventor(s): Ferdinand Lammertink (Binary Arts) Date Filed: December 30, 1994

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Patent US5386993 Another "Star of Number: David" - a 2x2x2 mechanism Popular embedded within Name: an octahedron, Inventor(s): Bernardo H. decorated with Apsan trivially rotating Date Issued: February 7, 1995 tetrahedra to form a stella octangula.

Patent US5573245 Number: Popular Lights Out 2000 Name: Inventor(s): Avish J. Weiner; Avraham Y. Olti; Gyoray M. P. Benedek; Michael Ganor Date Filed: May 5, 1995 Date Issued: November 12, 1996

Patent US5564702 Number: Popular Orbix Name: Inventor(s): Uwe Meffert Date Filed: May 23, 1995

Patent US5603500 Number: Popular Lights Out Name: Inventor(s): Avraham Y. Olti; Gyoray M. P. Benedek; Zvi Herman; Revital Blumberg; Michael Ganor; Avish J. Weiner Date Filed: January 18, 1996 Date Issued: February 18, 1997 Patent US5645278 Number: Popular Astrolabacus Name: Inventor(s): John D. Harris Date Filed: February 2, 1996

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Patent US5622368 Number: Popular Puzzle Disc Name: (Topspin variation) Inventor(s): Chiu-Huang Chang Date Filed: February 13, 1996

Patent US5622368 Number: Popular Puzzle disc Name: (Super Top Spin) Inventor(s): Chin-Huang Chang Date Filed: February 13, 1996 Date Issued: April 22, 1997

Patent WO9608297 Number: Popular King Ring Name: Inventor(s): Zoltan Pataki; Istvan Varadi; Attila Kovacs Date March 21, 1996 Issued:

Patent US5628512 Number: Popular Netblock Ufo Name: Inventor(s): Wai K. Chan Date Filed: April 15, 1996 Date Issued: May 13, 1997

Patent US5566941 The Mozaika Number: puzzle. The ingenious Popular Mozaika combination of the Name: Equator puzzle Inventor(s): Rudolf Destics (three bands of Date Issued: October 22, 1996 square on the surface of a sphere) with a 2x2x2 mechanism allowing the corners of the puzzle to shift position. Appropriate colorings of the puzzle can vary the difficulty significantly,

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Patent US6003868 Number: Popular Brain Ball Name: Inventor(s): Andreas Unsicker Date Filed: February 4, 1997 Date Issued: December 21, 1999

Patent US5857673 Number: Popular Inside Out Name: (Vadasz Kocka) Inventor(s): Daniel Ablonczy, Jozsef Vadasz Date Filed: March 12, 1997 Date Issued: January 12, 1999

Patent US5816571 Number: Popular Chromo Ball Name: Inventor(s): Tsun Ding Chen Date Filed: July 8, 1997

Patent US5826871 Number: Popular Eastsheen 2x2x2 Name: Inventor(s): Chen Sen Li Date Filed: December 23, 1997 Date Issued: October 27, 1998

Patent US5826871 This 2x2x2 Number: mechanism contains a Popular 2x2x2 complete 3x3x3 Name: mechanism mechanism Inventor(s): Chen Sen Li including edge Date Filed: December 23, pieces, all 1997 concealed by large caps on the Date Issued: October 27, 1998 corners. Puzzles using this mechanism have been found it is reportedly a great improvement over other 2x2x2 mechanisms.

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Patent US5722657 An irregular 3x3x3 Number: variant. Each face- center has a Popular trapezoidal Name: projection and the Inventor(s): Dario Cabrera corners are Date Issued: March 3, 1998 degenerate. The overall appearance is of three intersecting wedge-shaped slabs.

Patent US5992850 Number: Popular Eastsheen 4x4x4 Name: Inventor(s): Chen Sen Li Date Filed: March 25, 1998 Date Issued: November 30, 1999

Patent US5992850 Number: Popular 4x4x4 Name: mechanism Inventor(s): Chen Sen Li Date Filed: March 25, 1998 Date Issued: November 30, 1999

Patent US6129356 Number: Popular Eastsheen 5x5x5 Name: Inventor(s): Chen Sen Li Date Filed: May 12, 1998 Date Issued: October 10, 2000

Patent US6129356 Number: Popular 5x5x5 Name: mechanism Inventor(s): Chen Sen Li Date Filed: May 12, 1998 Date Issued: October 10, 2000

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Patent US6022021 Number: Popular Mad Triad Name: Inventor(s): Heng-Chun Ku Date Filed: July 17, 1998 Date Issued: February 8, 2000

Patent HU214709 Number: Popular Dogic Name: Inventor(s): Robert Vecsei; Zsolt Vecsei Date Issued: July 28, 1998

Patent DE19849279 An order-4 Number: tetrahedral puzzle, also known as the Popular Master Pyraminx Master Pyraminx Name: (not an order-3 Inventor(s): Alexander Mack Pyramix with Date Filed: October 15, 1998 rotating edges, which also goes by that name). The central mechanism is similar to a Skewb - a four armed spider with four fixed triangular faces (which form the face-centers), four floating triangular faces (which connect to the tips) and six floating square faces which are built up into octahedral "edge" pieces. Patent US6186504 Number: Popular Maxim's puzzle Name: Inventor(s): John Maxim Date Filed: July 9, 1999 Date Issued: February 13, 2001

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Patent US6209720 This redesign of Number: the classic domed container for the Popular Display box for 3x3x3 cube allows Name: the cube the cube to be Inventor(s): Sen-Li Chen positioned either (East Sheen flat on one face, Industrial Co. balanced along an Ltd., TW) edge, or balanced Date Filed: January 4, 2000 on a single vertex. Date Issued: April 3, 2001

Patent US6056290 The canonical Number: Dino Cube mechanism. An Popular Dino Cube internal stella Name: octangula's Inventor(s): James R. rotating tips have Holloway grooves at the Date Issued: May 2, 2000 base trap the outer edge pieces. A simplified version of Ashley's mechanism.

Patent WO0025874 Number: Popular K-Ball Name: Inventor(s): Saleh Khoudary Date May 11, 2000 Issued:

Patent US6158740 A Dino Cube Number: mechanism with metal tipped Popular pieces around a Name: central magnetic Inventor(s): Albert J. Hall core, as well as Date Issued: December 12, generalized 2000 puzzles with magnetic cores.

Patent US6217023 The Darth Maul Number: cube; a 2x2x2 mechanism similar Popular Darth Maul Cube to the one used in Name: Rubik's Pocket Inventor(s): Thomas Kremer Cube™ but with Date Issued: April 17, 2001 axis protruding directly from the fixed octant, and embedded in a bust of a popular character.

NOTE: The information contained in this list should not be relied upon for reference purposes. Dates, names and numbers may be incorrect. This information is provided with no warranty of any kind, express or implied.

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* Why "magic polyhedra"? Rubik® is a trademark, and refers to Professor Ernö Rubik, inventor of a specific set of puzzles including the wildly popular 3x3x3 cube that will forever bear his name. Similar puzzles created by other inventors should not be referred to as "Rubik's xyz" as that is both a misuse of the trademark and a disservice to the hard work by these other inventors; as the above patents show, extending the concept of a sliced polyhedron beyond the 3x3x3 cube is not trivial. A neutral name for this class of puzzles should be used instead. "Magic Polyhedra" and "Twisty Puzzles" are both in use to serve this role.

Rubik® and Rubik's Cube® are registered trademarks of Seven Towns Limited.

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