From mschoene@math.rwth-aachen.de Fri Oct 28 17:54:41 1994 Return-Path: Received: from samson.math.rwth-aachen.de by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA22088; Fri, 28 Oct 94 17:54:41 EDT Received: from hobbes.math.rwth-aachen.de by samson.math.rwth-aachen.de with smtp (Smail3.1.28.1 #11) id m0r0ycG-000MPgC; Fri, 28 Oct 94 22:13 MET Received: by hobbes.math.rwth-aachen.de (Smail3.1.28.1 #19) id m0r0ycB-0000PrC; Fri, 28 Oct 94 22:13 PST Message-Id: Date: Fri, 28 Oct 94 22:13 PST From: Martin.Schoenert@math.rwth-aachen.de To: cube-lovers@life.ai.mit.edu In-Reply-To: Mark Longridge's message of Thu, 27 Oct 1994 21:56:00 -0400 <60.825.5834.0C1B76FC@canrem.com> Subject: Re: Shift Invariant Part 2 Mark Longridge writes in his e-mail message of 1994/01/27 ...or (U1 R1)^35 ? And indeed, (U1 R1)^(35 * 40) is shift invariant. Mark kindly points out, that my process (UR)^140 for the ``odd'' element is a strange choice, given that (UR)^140 = (UR)^35. I can't recall how I arrived at this process. Somehow I simply missed that (UR)^140 = (UR)^35, which is especially strange since I know that (UR) has order 105 since 1982. Mark continues Equivalent to (U1 R1)^35= (R1 U1)^35 & Shift Invariant UR11 = U2 R1 U1 R1 U1 R3 U1 R3 U1 R3 U2 R1 U1 R1 U1 R3 U1 R3 U1 R3 (22 q or 20 h moves) Is UR11 the shortest process effecting the ``odd'' element in ? Mark continues Is this odd due to ( U1 R1 )^35? Actually everything about the above description appears even. It is an even number of quarter turns... The ``odd'' element o has odd order as element of the cube group, i.e., o^3 = id. All other shift invariant elements e have even order, i.e., either e^2 = id or e^4 = id (for some ``abelian'' elements). Mark continues I actually did use GAP on the < U, R > group but I couldn't resolve the resulting position (can GAP use letters? I should have used letters). I assume you wonder whether GAP can find a process for a given element. In fact GAP can do this (you define a homomorphism from the free group on U,D,L,R,F,B to the cube group and then ask for a preimage of the element). But the process is usually extremly long, e.g., for the ``central'' element GAP finds a process that has length > 2*10^6 (don't try this at home ;-). There is an improved algorithm by Philip Osterlund, which is a lot better, but still not good enough to help in the quest for god's algorithm. For example it finds a process for the ``central'' element of length 228. Mark continues Martin, you will be pleased to hear that I like GAP, but I need a bigger hard drive for that beast! Look at it this way: The system costs you $200, and you even get a hard drive for free! Seriously, you don't need the full distribution (32 MByte), but only the executable and the library (5 MByte). However, you should have enough real memory; 8 Mbyte is the minimum, 16 MByte is better, and the 64 MByte that I have in my workstation don't hurt. Have a nice day. Martin. -- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany