From cube-lovers-errors@curry.epilogue.com Thu Dec 26 17:07:53 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id RAA27424; Thu, 26 Dec 1996 17:07:52 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Message-ID: From: "joyner.david" To: "'acoles@fec.gov'" Cc: "'cube-lovers@ai.mit.edu'" Subject: RE: Rubik's Tangle Date: Thu, 26 Dec 1996 11:56:11 -0500 X-Mailer: Microsoft Exchange Server Internet Mail Connector Version 4.0.837.3 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Aaron Coles: The Rubik's tangle I bought is 3x3 and in that puzzle there are no tiles with a straight yellow. The version of the Rubik's tangle which that page you gave describes is not, as far as I know, marketed in stores. Here is my solution to the store version I bought: Notation: green=color 1, purple = color 2, red = color 3, yellow = color 4. Each tile will be denoted by a 4-tuple (a,b,c,d), where a is the color number of the straight rope, b is the color number of the quarter circle rope, c is the color number of the twisted rope (the one going from one side to the opposite which is not straight), d is the color number of the looped rope (the one going from one side to an adjacent side which is not in the shape of a quarter circle). The "orientation" of a tile will be 0 if the straight side is lined up vertically on the left, 1 if it is rotated 90 degrees clockwise from the orientation 0 position, 2 if it is rotated 180 degrees clockwise from the orientation 0 position (the straight side is lined up vertically on the right), 3 if it is rotated 270 degrees clockwise from the orientation 0 position. I labeled the (2-sided) tiles arbitrarily 1-9, with f for front and b for back. They are as follows: tile 1 f (1,3,4,2), b (1,4,2,3) tile 2 f (1,4,2,4), b (1,4,3,2) tile 3 f (1,2,4,3), b (1,2,3,4) tile 4 f (3,1,2,4), b (3,2,4,1) tile 5 f (3,4,2,1), b (3,4,1,2) tile 6 f (3,2,1,4), b (3,1,4,2) tile 7 f (2,4,3,1), b (2,1,3,4) tile 8 f (2,1,4,3), b (2,3,1,4) tile 9 f (2,3,4,1), b (2,4,1,3) Mathematics of puzzles: In general, let X be a collection of n interlocking puzzle pieces. Assume that there is a solution to the puzzle for X which uses every element of X. Call G a "subpuzzle graph on X" if there is a subpuzzle of interlocking pieces constructed from a subset Y of X such that G is a graph with vertices labeled by the subset Y of X and two vertices are connected by an edge if and only if the corresponding pieces fit or interlock in this subpuzzle. A "solution" to the puzzle will be a connected subpuzzle graph on X having n vertices associated to a solution. For almost every jigsaw puzzle I've seen and for the Rubik's tangle puzzle, a solution to a puzzle is a Hamiltonian graph. Algorithm for the Rubik's tangle: We shall construct a subpuzzle graph on the Rubik's tangle pieces as follows: Notation: Label the positions of the puzzle as 9-2-3 | | | 8-1-4 | | | 7-6-5 step 1: Pick a tile (18 possible choices) and put it in position number 1 with orientation 0. Draw a corresponding vertex and label it with this tile's 4-tuple. inductive step: Assume that k tiles have been placed in positions 1 through k and each tile fits with its neighboring tiles, k---------- >From: Aaron Coles[SMTP:acoles@fec.gov] >Sent: Monday, December 23, 1996 6:47 PM >To: cube-lovers@ai.mit.edu >Subject: Rubik's Tangle > >Does anyone know if the solution located at the below address is >valid?? > > http://www.math.uni-kiel.de/roesler/bruhn/tlsg.htm > >[ The moderator has taken the liberty of correcting a small typo here. > The original submitted message had an underscore ("_") in the URL >which > I have corrected to be a hyphen ("-"). - Alan ] >