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 http://www.calormen.com/TwistyPuzzles/twisty.htm Page 1 of 30 The Magic Polyhedra Patent Page 01/08/2007 03:34 PM 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. http://www.calormen.com/TwistyPuzzles/twisty.htm Page 2 of 30 The Magic Polyhedra Patent Page 01/08/2007 03:34 PM 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.