From cube-lovers-errors@oolong.camellia.org Wed Jun 25 18:29:12 1997
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Date: Wed, 25 Jun 1997 23:28:16 BST
From: David Singmaster Computing & Maths South Bank Univ
To: cube-lovers@ai.mit.edu
Message-ID: <009B6547.EBD77100.328@vax.sbu.ac.uk>
Subject: 4^3 and 5^3
I've just seen a comment about the 5^3 saying the writer had problems
with the four pieces at distance 1 from the center. My approach treated
both these and the pieces ate distance 2 from the center in the same way.
We know that the commutator of two slice moves on the 3^3 produces two
3-cycles of the centres. Applying this idea to the 4^3, we find that
[f,r] gives two 3-cycles of central pieces, one on each face. By turning
one face and then inverting, we get a 3-cycle of central pieces, two being
in one face. E.g. [[r,b],U] = (Fur,Ubr,Ubl). A similar result holds if
we combine any two inner moves, so we can replace the b above by a central
slice on the 5^3 to obtain a 3-cycle of the pieces at distance 1 from the
central piece, while the process directly gives us a 3-cycle of the pieces at
distance 2 from the central piece. Although tedious, these moves mean that
once one has the corners and edges in place, the rest of the problem is
easy - though very tedious - it generally took me about an hour to do the
5^3, assuming I could get the corners and edges correct without making too
many mistakes.
DAVID SINGMASTER, Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411; fax: 0171-815 7499;
email: zingmast or David.Singmaster @sbu.ac.uk