Xref: utzoo sci.math:9036 comp.theory:125
Path: utzoo!attcan!utgpu!jarvis.csri.toronto.edu!cs.utexas.edu!samsung!think!ames!pacbell!rtech!mikes@rtech.UUCP
From: mikes@rtech.UUCP (Mike Schilling)
Newsgroups: sci.math,comp.theory
Subject: Re: Rubik's Cube Problem
Message-ID: <4330@rtech.rtech.com>
Date: 19 Dec 89 18:35:18 GMT
References: <5310@garfield.MUN.EDU>
Sender: news@rtech.rtech.com
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From article <5310@garfield.MUN.EDU>, by chris2@garfield.MUN.EDU (Chris Paulse):
> 
> If I had a solved Rubik's cube, and the colors on each face were
> just stickers on the black plastic surface, if I exchanged some
> of the stickers, would the cube still be solvable in the normal way?
> 
No, there's three kinds of *parity* preserved by any legal move:

1.	The positions of the cubes fall into two sets; a legal move
	can't move between sets.  This is much like Sam Loyd's 15-16 puzzle.
2.	The orientations of the edge cubes fall into two sets.  Reversing
	a single edge cube moves from one set to another, but a legal
	move reverses an even number.
3.	The orientations of the corner cubes fall into three sets.  Turning
	a corner cube clockwise or counter-clockwise moves into another set.
	A ninety-degree legal move turns two clockwise and two
	counter-clockwise.

So if you randomly rearrange the stickers, you only get a solvable position
one twelfth of the time.