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From: verma@mahimahi.cs.ucla.edu (Rodent of Darkness)
Newsgroups: sci.math,comp.theory
Subject: Re: Rubik's Cube Problem
Message-ID: <30127@shemp.CS.UCLA.EDU>
Date: 20 Dec 89 00:28:08 GMT
References: <5310@garfield.MUN.EDU> <4330@rtech.rtech.com>
Sender: news@CS.UCLA.EDU
Reply-To: verma@mahimahi.cs.ucla.edu (Rodent of Darkness)
Organization: UCLA Computer Science Department
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In article <4330@rtech.rtech.com> mikes@rtech.UUCP (Mike Schilling) writes:

>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.

	Not exactly.  What you say is true for rearrangement of the cubbies
	not the stickers.  (i.e. if you take a rubik's cube apart, scramble
	the pieces then put it back together, you have one chance in twelve
	that the resultant cube is solvable.

	Random rearrangement of the stickers could, for example, make every
	center square red, or a corner could have three white stickers.  In
	eacher case the resultant cube is not solvable, but neither case is
	treated in your above analysis.  Note that you talk about sets that
	the cubes (I use the coined term cubbies for clarity) fall into; if
	you want to analyze the random sticker problem, you need to discuss
	the sets that stickers fall into and parity about stickers etc.

								--- TS