From hoey@aic.nrl.navy.mil Wed Nov 29 12:18:52 1995
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Date: Wed, 29 Nov 95 12:14:40 EST
From: hoey@aic.nrl.navy.mil
To: mschoene@math.rwth-aachen.de (Martin Schoenert),
frb6006@cs.rit.edu (Frank R Bernhart), Cube-Lovers@life.ai.mit.edu
Newsgroups: sci.math
Subject: Generating Rubik's Cube
Message-Id: <9511291210.Hoey@AIC.NRL.Navy.Mil>
References: <1995Nov29.054118.9651@cs.rit.edu>
Distribution:
About generating the cube's group with arbitrary elements of that
group, mschoene@Math.RWTH-Aachen.DE (Martin Schoenert) writes:
> ... Rubik's cube can be generated by 2 elements.
> Moreover almost any random pair of elements will do the trick....
Actually, I think it's more accurate to say that a random pair of
elements has nearly a 75% probability of generating the cube. At
least, I'm pretty sure that's an upper bound, and I don't see any
reason why it shouldn't be fairly tight. That's for the group where
the whole cube's spatial orientation is irrelevant. I think it's more
like 56% (9/16) if you also need to generate the 24 possible
permutations of face centers.
About the minimal presentation of the cube group on the usual generators,
frb6006@cs.rit.edu (Frank R Bernhart) writes:
> The answers may be in SINGMASTER, et.al.
> "Handbook of Cubic Math" or BANDEMEISTER (sp?) "Beyond R. Cube"
I recall Singmaster wanted to know if anyone found a reasonably-sized
presentation; I don't know if any have been found in the intervening
fifteen years. The best I know of is a few thousand relations, some
of them several thousand letters long. I've been meaning to try
chopping that down a bit.
Dan posted and e-mailed
Hoey@AIC.NRL.Navy.Mil