From cube-lovers-errors@oolong.camellia.org Wed Jun 25 18:29:12 1997 Return-Path: cube-lovers-errors@oolong.camellia.org Received: from oolong.camellia.org (localhost [127.0.0.1]) by oolong.camellia.org (8.6.12/8.6.12) with SMTP id SAA09246; Wed, 25 Jun 1997 18:29:12 -0400 Precedence: bulk Errors-To: cube-lovers-errors@oolong.camellia.org 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