Magic Polyhedra* Patents

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 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):

Get the puzzle as close to a solved state as you can by whatever means you like.
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).
Conceptually group the remaining moves and flips into a pair of two symmetric abstract operations (M and M').
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.
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.
Perform the sequence X' - that is, do the inverse of each of the steps you wrote down in reverse order.
Perform the single move s' which puts layer L back where we had it in step 3 - except for M and M'.
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.
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 Number:	US3637215
Popular Name:	Spinout
Inventor(s):	William Keister
Date Filed:	December 22, 1970
Date Issued:	January 25, 1972

Patent Number:	US3655201
Popular Name:
Inventor(s):	Larry D. Nichols
Date Issued:	April 11, 1972

PATTERN FORMING PUZZLE AND METHOD WITH PIECES ROTATABLE IN GROUPS

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

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

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

Patent Number:	BE887875
Popular Name:	Rubik's Cube
Inventor(s):	Ernö Rubik
Date Filed:	March 10, 1981
Date Issued:	July 1, 1981

The original patent for the 3x3x3 Rubik's Cube and one mechanism for the 2x2x2 Pocket Cube.

Patent Number:	EP0042695
Popular Name:	Pyraminx
Inventor(s):	Uwe Meffert
Date Filed:	June 10, 1981

The original Pyraminx - an order-3 tetrahedron with rotating verticies and trivial tips. 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 Number:	US4377916
Popular Name:	Rubik's Snake
Inventor(s):	Kenji Komiya
Date Filed:	July 20, 1981
Date Issued:	March 29, 1983

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

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

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

Patent Number:	US4410179
Popular Name:	Rubik's Cheese
Inventor(s):	Erno Rubik (Konsumex)
Date Filed:	November 9, 1981

A single layer puzzle with six pieces.

Patent Number:	DE3204033
Popular Name:	Thomas Ball
Inventor(s):	Dr. Gilbert Obermair
Date Filed:	February 5, 1982
Date Issued:	August 25, 1983

Another mechanism equivalent to the Impossiball. Twenty triangular caps twist in groups of five around a common vertex. The 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 Number:	US4402510
Popular Name:	Crossover
Inventor(s):	Gunpei Yokoi (Nintendo)
Date Filed:	February 12, 1982
Date Issued:	September 6, 1983

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

Patent Number:	GB2084471
Popular Name:	Rubik's Octahedron
Inventor(s):	Ernö Rubik
Date Issued:	April 15, 1982

An order-2 face-centered octahedron based on a Pyraminx-like central sphere. The face-centers are affixed to the sphere and the vertices float around.

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

Patent Number:	US4344623
Popular Name:
Inventor(s):	Toshiyuki Isobe
Date Issued:	August 17, 1982

A 2x2x2 mechanism (depicted is a vertical slice through a diagonal plane).

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

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

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

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

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

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

Patent Number:	US4378116
Popular Name:	Magic Domino
Inventor(s):	Ernö Rubik
Date Issued:	March 29, 1983

The 3x3x2 Rubik's Domino. Two mechanisms are suggested - one with a six armed spider (four of 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 Number:	US4378117
Popular Name:	Rubik's Pocket Cube
Inventor(s):	Ernö Rubik
Date Issued:	March 29, 1983

A 2x2x2 mechanism. Seven pieces are held in place against an eighth by three arms.

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

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

Patent Number:	US4405131
Popular Name:
Inventor(s):	Tibor Horvath
Date Issued:	September 20, 1983

An elaborate 2x2x2 mechanism utilizing nested shells. The cubies project arms into a central sphere with tracks to guide their movement.

Patent Number:	US4409750
Popular Name:	Rubik's Calendar
Inventor(s):	Marvin A. Silbermintz
Date Issued:	October 18, 1983

A calendar cube. The patent includes the sticker layout and a solution.

Patent Number:	US4421311
Popular Name:	Rubik's Revenge
Inventor(s):	Peter Sebesteny
Date Issued:	December 20, 1983

A 4x4x4 cube; pieces orbit a central sphere with projecting caps that both holds the face-centers and provides a track for their rotation.

Patent Number:	US4437667
Popular Name:
Inventor(s):	Ronald L. Miller
Date Issued:	March 20, 1984

Just a 3x3x3 cube, but the patent is for a variety of Scrabble-like games playable on the surface by the use of detachable stickers or markers for the cubie faces.

Patent Number:	US4441715
Popular Name:
Inventor(s):	Stephen C. Titus
Date Issued:	April 10, 1984

A sphere with 6 interchangeable segments. Like a Masterball without the latitudinal slices.

Patent Number:	JP60222076
Popular Name:	Master Rainbow Cube
Inventor(s):	Watanabe Yasunari, Ikegami Yuuji
Date Filed:	April 18, 1984

This puzzle is to the Rainbow Cube as the 5x5x5 is to the 3x3x3. Amazing!

Patent Number:	US4451039
Popular Name:
Inventor(s):	Clarence W. Hewlett, Jr.
Date Issued:	May 29, 1984

An order-3 face-centered magic octahedron. The tips are non-trivial and none of the exterior pieces are fixed in place. The central mechanism is a stella octangula with rotating tips, surrounded by six corner octahedra, twelve edge octahedra, and twenty four face tetrahedra.

Patent Number:	US4453715
Popular Name:	Halpern-Meier Pyramid
Inventor(s):	Benjamin R. Halpern
Date Issued:	June 12, 1984

The Halpern-Meier Pyramid. The edges and tips project feet into the center of the tetrahedron which combine to form a 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 what will later be known as "bandaging".

Patent Number:	US4473228
Popular Name:
Inventor(s):	John R. Hart
Date Issued:	September 25, 1984

A possible mechanism for an Impossiball - an icosahedron on which each "cap" of 5 faces may be rotated, allow the faces to migrate around the surface.

Patent Number:	US4474376
Popular Name:	Impossiball
Inventor(s):	William O. Gustafson
Date Issued:	October 2, 1984

The canonical Impossiball mechanism - an icosahedron on which each "cap" of 5 faces may be rotated, allow the 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 Number:	US4474377
Popular Name:
Inventor(s):	Jonathan J. Ashley
Date Issued:	October 2, 1984

A combination of a 2x2x2 cube and a Dino cube. The internal mechanism is a stella octangula from a Dino Cube with rotating tips (like an Alexander's Star), but deep-cut with an embedded 2x2x2 mechanism.

Patent Number:	US4496155
Popular Name:	Star of David
Inventor(s):	Israel Goldfarb
Date Issued:	January 9, 1985

The puzzle has the shape of a stella octangula. The core octahedron has a 2x2x2 mechanism like a modified Pyramorphix, but the tips can rotate as well.

Patent Number:	US4500090
Popular Name:
Inventor(s):	Antoliono Nieto
Date Issued:	February 19, 1985

Very similar to the Alexander's Star, but with a more complicated mechanism of grooves.

Patent Number:	US4506891
Popular Name:	Alexander's Star
Inventor(s):	Adam Alexander
Date Issued:	March 26, 1985

The popular Alexander's Star puzzle; a great dodecahedron where each vertex rotates. The central mechanism is a small stellated dodecahedron with a dodecahedral core; each tip rotates.

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

Patent Number:	US4513970
Popular Name:
Inventor(s):	Ovidiu Opresco; Jon D. Marinesco
Date Issued:	April 30, 1985

A 2x2x2 mechanism using 8 magnets on each plane (6 per piece). Non-cubical shapes are suggested, such as off-center or tiled cuts in cubes, off-center cuts in cubes, and visually attractive egg-shaped puzzles.

Patent Number:	US4522401
Popular Name:
Inventor(s):	William O. Gustafson
Date Issued:	June 11, 1985

A spherical 3x3x3 with no internal spider mechanism; the shell is held together in a manner similar to the design of US4441715.

Patent Number:	US4540177
Popular Name:
Inventor(s):	Tibor Horvath
Date Issued:	September 15, 1985

A 4x4x4 mechanism based around an embedded 2x2x2 mechanism. The face centers have feet embedded in 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.

Patent Number:	US4557484
Popular Name:
Inventor(s):	Benjamin F. Sherman, Jr.; Samuel Francis
Date Issued:	December 10, 1985

An elegant spherical 2x2x2 mechanism with the mobile octants riding around a hollow central sphere on tracks. Further, there is the suggestion that 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 Number:	US4558866
Popular Name:
Inventor(s):	William L. Alford
Date Issued:	December 17, 1985

A variety of designs: a Pyraminx without trivial tips and a Tetraminx; a face-centered order-2 octahedron; a PentUltimate mechanism with magnetic pieces around a metal sphere

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

Patent Number:	US4586713
Popular Name:
Inventor(s):	Ibrahim K. Abu-Shumays; Mary D. Abu-Shumays
Date Issued:	May 6, 1986

A generalized family of star-prisms puzzles; all are three layers tall.

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

Patent Number:	US4593907
Popular Name:
Inventor(s):	Ibrahim K. Abu-Shumays; Mary D. Abu-Shumays
Date Issued:	June 10, 1986

First, yet another 2x2x2 mechanism, followed by an application of it to various regular and irregular polyhedra. Next, a generalized 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 Number:	US4593908
Popular Name:
Inventor(s):	Baky B. Ibrahim
Date Issued:	June 10, 1986

A deep-cut order-2 vertex-centered octahedron based on a 2x2x2 embedded within an cuboctahedron with pyramidal tips that rotate on the underlying faces and split along their edges.

Patent Number:	US4600199
Popular Name:	Professor's Cube
Inventor(s):	Udo Krell
Date Issued:	July 15, 1986

Another hefty patent. This one starts with a deeply truncated cube and works its way up to the amazing 5x5x5 mechanism. It also suggests a deep-cut dodecahedron puzzle (which resembles but is distinct from the Pyraminx Crystal)

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

Patent Number:	US4817952
Popular Name:	Rubik's electronic 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 Number:	US4667961
Popular Name:
Inventor(s):	Ibrahim K. Abu-Shumays; Mary D. Abu-Shumays
Date Issued:	May 26, 1987

The name of the patent is "Star prism puzzles generalized" and it really is just a generalization of the earlier patent US4586713 by the same inventors.

Patent Number:	US4674750
Popular Name:
Inventor(s):	Ibrahim K. Abu-Shumays; Mary D. Abu-Shumays
Date Issued:	June 23, 1987

The outer puzzle is a rhombic dodecahedron where each vertex rotates. Internally, the mechanism is similar to the Alexander's Star - an internal cuboctahedron with rotating tips added to both the square and triangular faces, which trap the surface pieces in central notches.

Patent Number:	US4735417
Popular Name:	Gripple
Inventor(s):	Murray J. Gould
Date Filed:	June 25, 1987
Date Issued:	April 5, 1988

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

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

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

Patent Number:	US4706956
Popular Name:
Inventor(s):	Ibrahim K. Abu-Shumays; Mary D. Abu-Shumays
Date Issued:	November 17, 1987

Two octahedral variations on the 3x3x3 mechanism; one resembles the Trajber's Octahedron (essentially an extremely truncated cube), 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 Number:	US4871173
Popular Name:	Topspin
Inventor(s):	Ferdinand Lammertink (Binary Arts)
Date Filed:	September 2, 1988

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

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

Patent Number:	US4836549
Popular Name:
Inventor(s):	James T. Flake
Date Issued:	June 6, 1989

A stella octangula in the form of two intersecting order-four tetrahedrons. The tips are trivial (like the 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 Number:	EP0322085
Popular Name:	Rubik's Clock
Inventor(s):	Christopher Charles Wiggs
Date Issued:	June 28, 1989

Patent Number:	US4856786
Popular Name:	Masterball
Inventor(s):	Geza Gyovai
Date Issued:	August 15, 1989

The Masterball mechanism. Essentially Titus' six segment sphere (which this patent cites as prior art) with the addition of four bands (two polar, two equatorial) that rotate around the surface orthogonally to the underlying segments.

Patent Number:	US4865323
Popular Name:
Inventor(s):	Paul A. Heusinkveld
Date Issued:	September 12, 1989

Like a super-Masterball with six latitudinal segments vs. four on the Masterball. The mechanism for the surface bands is much more robust in appearance, however.

Patent Number:	US4872682
Popular Name:
Inventor(s):	Ravi Kuchimanchi; Madhukar N. Thakur
Date Issued:	October 10, 1989

A 3x3x3 combined with a "9-puzzle"; the faces on the cube can be slid around exactly as in a 15-puzzle; manipulating the 3x3x3 mechanism migrates the empty square between faces.

Patent Number:	US4957291
Popular Name:	Luminations
Inventor(s):	Donald C. Miffitt; Angelo Tortola; Charles S. Sebor; Robert L. Halliday
Date Filed:	November 6, 1989
Date Issued:	September 18, 1990

Patent Number:	US4889340
Popular Name:
Inventor(s):	Wilton R. Greene
Date Issued:	December 26, 1989

A variant of the Equator puzzle, where three intersecting bands on the surface of a cube allow square pieces to slide around and move between bands.

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

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

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

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

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

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

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

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

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

Patent Number:	US5116052
Popular Name:	Smart Alex
Inventor(s):	Dumitru A. Pop
Date Issued:	May 26, 1992

A cube with hexagons projecting from four sides. The hexagons are divided into six 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 Number:	US5271688
Popular Name:	Assembly Cube
Date Filed:	August 28, 1992
Date Issued:	December 21, 1993

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

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

Patent Number:	RO106960
Popular Name:
Inventor(s):	Florian Radut
Date Filed:	October 27, 1992

A 4x4x4 mechanism with a spherical core.

Patent Number:	RO106962
Popular Name:
Inventor(s):	Florian Radut
Date Filed:	December 14, 1992

The classic 3D marble variation of famous the 15-puzzle. A cube is packed with marbles (in a grid arrangement), and one marble is missing. By tipping the cube the marbles can be scrambled and (hopefully!) unscrambled.

Patent Number:	US5193809
Popular Name:	Square-1
Inventor(s):	Karel Hrsel; Vojtech Kopsky
Date Issued:	March 16, 1993

The canonical Square-1 patent. A simple dihedral mechanism lets the central slab trap any number of appropriately grooved pieces on either side.

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

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

Patent Number:	US5199711
Popular Name:
Inventor(s):	Zoltan Pataki; Geza Csomos; Janos Feher
Date Issued:	April 6, 1993

A solid mechanism for a puzzle sliced into six radial segments like Rubik's Cheese or the Masterball.

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

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

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

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

Patent Number:	RO110598
Popular Name:
Inventor(s):	Florian Radut
Date Filed:	June 14, 1994

Like the Hungarian Rings, this puzzle has six interlocking rings of eight marbles - the end result is similar to a 3x3x3 cube where the orientation of the pieces doesn't matter.

Patent Number:	US5338033
Popular Name:
Inventor(s):	Guillermo Nunez Serrano
Date Issued:	August 16, 1994

An extremely simple and elegant 2x2x2 mechanism, nominally in a spherical puzzle. Each of the 7 floating octants 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 Number:	US5358247
Popular Name:	Puzzle Ball
Inventor(s):	Uwe Meffert
Date Issued:	October 25, 1994

The Puzzle Ball - a spherical Skewb. A four-armed spider with triangular tips, four more floating triangular "faces", and six floating square "faces".

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

Patent Number:	US5386993
Popular Name:
Inventor(s):	Bernardo H. Apsan
Date Issued:	February 7, 1995

Another "Star of David" - a 2x2x2 mechanism embedded within an octahedron, decorated with trivially rotating tetrahedra to form a stella octangula.

Patent Number:	US5573245
Popular Name:	Lights Out 2000
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 Number:	US5564702
Popular Name:	Orbix
Inventor(s):	Uwe Meffert
Date Filed:	May 23, 1995

Patent Number:	US5603500
Popular Name:	Lights Out
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 Number:	US5645278
Popular Name:	Astrolabacus
Inventor(s):	John D. Harris
Date Filed:	February 2, 1996

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

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

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

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

Patent Number:	US5566941
Popular Name:	Mozaika
Inventor(s):	Rudolf Destics
Date Issued:	October 22, 1996

The Mozaika puzzle. The ingenious combination of the Equator puzzle (three bands of 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,

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

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

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

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

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

This 2x2x2 mechanism contains a complete 3x3x3 mechanism including edge pieces, all concealed by large caps on the corners. Puzzles using this mechanism have been found it is reportedly a great improvement over other 2x2x2 mechanisms.

Patent Number:	US5722657
Popular Name:
Inventor(s):	Dario Cabrera
Date Issued:	March 3, 1998

An irregular 3x3x3 variant. Each face-center has a trapezoidal projection and the corners are degenerate. The overall appearance is of three intersecting wedge-shaped slabs.

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

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

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

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

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

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

Patent Number:	DE19849279
Popular Name:	Master Pyraminx
Inventor(s):	Alexander Mack
Date Filed:	October 15, 1998

An order-4 tetrahedral puzzle, also known as the Master Pyraminx (not an order-3 Pyramix with 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 Number:	US6186504
Popular Name:	Maxim's puzzle
Inventor(s):	John Maxim
Date Filed:	July 9, 1999
Date Issued:	February 13, 2001

Patent Number:	US6209720
Popular Name:	Display box for the cube
Inventor(s):	Sen-Li Chen (East Sheen Industrial Co. Ltd., TW)
Date Filed:	January 4, 2000
Date Issued:	April 3, 2001

This redesign of the classic domed container for the 3x3x3 cube allows the cube to be positioned either flat on one face, balanced along an edge, or balanced on a single vertex.

Patent Number:	US6056290
Popular Name:	Dino Cube
Inventor(s):	James R. Holloway
Date Issued:	May 2, 2000

The canonical Dino Cube mechanism. An internal stella octangula's rotating tips have grooves at the base trap the outer edge pieces. A simplified version of Ashley's mechanism.

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

Patent Number:	US6158740
Popular Name:
Inventor(s):	Albert J. Hall
Date Issued:	December 12, 2000

A Dino Cube mechanism with metal tipped pieces around a central magnetic core, as well as generalized puzzles with magnetic cores.

Patent Number:	US6217023
Popular Name:	Darth Maul Cube
Inventor(s):	Thomas Kremer
Date Issued:	April 17, 2001

The Darth Maul cube; a 2x2x2 mechanism similar to the one used in Rubik's Pocket Cube™ but with 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.

* 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.