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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
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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 ]
>