From @mail.uunet.ca:mark.longridge@canrem.com Mon Oct 3 05:48:56 1994 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA07705; Mon, 3 Oct 94 05:48:56 EDT Received: from portnoy.canrem.com ([198.133.42.251]) by mail.uunet.ca with SMTP id <102162-2>; Mon, 3 Oct 1994 05:49:15 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA01841; Mon, 3 Oct 94 05:46:17 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1B2945; Mon, 3 Oct 94 05:23:07 -0400 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: < U, R > Processes From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.808.5834.0C1B2945@canrem.com> Date: Mon, 3 Oct 1994 01:13:00 -0400 Organization: CRS Online (Toronto, Ontario) Alas, no antipodes yet, but some interesting results nonetheless. Process UR8 improves on the best known process for a certain quad-twist in the U layer at 20 q turns. Table 3 in Winning Ways gives a 22 q turn process. The following results should be particularly interesting to the physical cube solver as it is easier to execute a sequence of 2 adjacent sides compared to a sequence using 3 or more sides, which may require some re-orienting of the cube. I will measure the "face index" of a process by the number of different sides used in a certain cube sequence. Such a measure could be used to evaluate the relative elegance of two equally long processes with respect to their face indices. Jerry Bryan mentions: > Also, the global maxima are of length 25. > Does this tell us anything about the Q-turn length of the global > maxima for the full cube group? Well, that reminds me of one of the hardest patterns that Dik Winter tried to find an optimal sequence for: p141a alternate method F1 R1 L2 U3 R2 L3 U3 D2 R2 F1 D1 B1 D1 F2 U3 of Superfliptwist + 6 X R3 D3 F2 D2 L2 **This process was one of the hardest ever to reduce to 20 moves, requiring over 19 hours on an SGI R4K Indigo, 28 q turns** My own $.02 worth is that an antipode for the full group of the 3x3x3 cube is probably deeper than an antipode for the < U, R > group. Optimal Sequences for < U, R > group elements (positions) --------------------------------------------------------- Edge 3-cycle UR1 = U3 R1 U2 (R1 U1)^2 R2 U3 R3 U3 R2 U1 (16 q, 13 h) Double adjacent edge swap UR2 = U3 R3 U3 R2 U1 R1 U1 R3 U3 R1 U1 (R1 U3)^2 R3 U3 (18 q, 17 h) Diagonal Corner twist UR3 = U1 R1 U3 R1 U3 R2 U1 R1 U1 R3 (U3 R1)^2 U2 R3 U3 R3 (20 q, 18 h) Double opposite edge swap, also in sq group 24 q, 12 h UR4 = R2 U2 R3 (U2 R2)^2 U2 R3 U2 R2 (20 q, 11 h) Edge 7-cycle, equivalent to (U1 R1)^15 UR5 = U3 R1 U3 R3 U3 R1 U2 R3 U1 R3 U2 R1 U3 R3 (U3 R1)^2 (20 q, 18 h) Corner Tri-Twist UR6 = (U3 R3)^2 U1 R1 U3 R3 U3 R2 U1 R2 U3 R3 U3 R1 U1 R3 (20 q, 18 h) Corner Quad-Twist, Flat style UR7 = R1 U3 (R1 U1)^2 (R3 U3)^2 R2 U3 R1 U1 R3 U3 R1 U3 R3 (20 q, 19 h) Corner Quad-Twist, Arms & Legs style (20 q, 20 h) UR8 = R1 U1 R3 U1 R3 U3 R1 U1 R1 (U3 R3)^2 (U1 R1)^2 U3 R3 U3 ML Doodle Position UR9 = (U2 R2)^2 U2 R3 U1 R2 (U3 R2)^2 U1 R1 (22 q, 14 h) Same position found by hand: (a non-optimal 24 q, 15 h) (U2 R2)^3 U1 R1 (U2 R3)^2 U2 R1 U1 4 Opp Corner Swap, also in sq group at 26 q, 13 h UR10 = U3 R3 (U1 R1)^2 U2 R3 U1 R1 (U2 R2)^2 U1 R3 U1 (22 q, 17 h) Other Subgroups within reach ---------------------------- 11. || = 2^12 3^4 5^2 7 = 58060800 12. || = 2^12 3^4 5^2 7 = 58060800 17. || = 2^8 3^5 5^2 7 = 10886400 21. || = 2^13 3^4 5^2 7 = 116121600 22. || = 2^15 3^4 5^2 7^2 = 3251404800 I welcome any proposed < U, R > group antipodes. I haven't really looked for anything exotic like < U, R > positions which are shift invariant, or even if such a beast is possible! Of course I already mentioned that... U2 R2 U2 R2 U2 R2 = R2 U2 R2 U2 R2 U2 ...but aside from that nothing comes to mind. Generally when there are elements which occur in both the square's group AND the < U, R > group the latter is the shorter in q turns. -> Mark <- Email: mark.longridge@canrem.com