From boland@sci.kun.nl Sat Sep 23 18:31:59 1995 Return-Path: Received: from wn1.sci.kun.nl by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA21232; Sat, 23 Sep 95 18:31:59 EDT Received: from canteclaer.sci.kun.nl by wn1.sci.kun.nl via canteclaer.sci.kun.nl [131.174.132.34] with SMTP id AAA29847 (8.6.10/2.13) for ; Sun, 24 Sep 1995 00:31:56 +0200 Message-Id: <199509232231.AAA29847@wn1.sci.kun.nl> To: cube-lovers@ai.mit.edu Subject: Order problems Date: Sun, 24 Sep 95 00:31:55 +0200 From: Michiel Boland Hello all, Had a great time reading the archives. What I haven't found there are order problems: what is the shortest (in terms of quarter turns or half- and quarter turns, whatever you prefer) transformation of the cube with a given order? Here is a list that my good old PC produced this afternoon. I hope some of you find this interesting. :) A couple of notes on the list: "Len" is the length of the transformation in terms of quarter twists. You will notice that I listed two transforms with order 3: one is minimal wrt quarter-turn metric, the other wrt half-turn metric. A notable absentee is number 11. I suspect that (U.R.F2B.D')2 is the shortest possible with order 11, but my comp just isn't fast enough to confirm this. Note that (U.R.F2B.D')2 yields an 11-cycle on the edges (see also Mark Longridge's mail from 15 Jul 1994.) (I use dots to maintain readability; personally, I do not like the U1F2L3 notation, but that's just a matter of taste :) Order Len 1 0 2 2 U2 3 6 U.R.U'D'R.D. 3 8 U2R2U2R2 4 1 U. 5 4 U.R.U.R' 6 4 U2R2 7 4 U.R.U'F. 8 4 U.R2D. 9 4 U.R.F2 10 4 U'R.U.F. 11 ? ???????????? 12 4 U.R.F.D' 14 6 U'R.U.R'F.D. 15 6 U.R2U.R2 16 5 U.R.U'F.D. 18 5 U.R.U'R'F. 20 5 U.R.U'L2 21 6 U2R.U2F. 22 6 U.R.F2B.D' 24 4 U.R2D' 28 4 U.R.U'L. 30 3 U.R2 33 4 U.R.F'D' 35 6 U2R.U2L' 36 4 U2R'F' 40 5 U.R.U2L. 42 6 U.R2U2R' 44 4 U'R.F'D. 45 4 U.R.U.L. 48 5 U2R.U.F. 55 6 U.R.F'U'B'L. 56 5 U2R.F'D. 60 3 U.R'F' 63 2 U.R' 66 6 U.R.U.F2L' 70 6 U.R'U.R.F.R' 72 4 U.R.U.F' 77 4 U.R'F'L' 80 3 U'R'F' 84 3 U.R.F. 90 3 U.R.D. 99 6 U.R2F.L2 105 2 U.R. 110 8 U.R.U2R'F.R.L' 112 6 U.R'U.F'R.D. 120 4 U.R.F.L' 126 4 U'R.F'L' 132 4 U.R.F'L. 140 4 U.R'U.F' 144 5 U.R'F'D2 154 6 U.R.U.F.L.D' 165 6 U.R'U.F2L' 168 4 U.R.D2 180 3 U.R.D' 198 6 U2R.F.D2 210 4 U.R'D.L' 231 4 U.R.F'D. 240 5 U'R.F'L2 252 4 U.R.F.L. 280 5 U'R'U'F.L' 315 4 U.R.D.L. 330 6 U2R.F'D'L' 336 6 U.R.U.F.D2 360 3 U.R.F' 420 4 U.R.D.L' 462 6 U'R.F'D2L' 495 6 U.R2U.F'L' 504 5 U.R2F.L' 630 6 U'R'U'F'L2 720 6 U'R'U'F'D2 840 5 U2R'F'D. 990 6 U'R'U'F'L.D. 1260 6 U.R'U.F'D2 -- Michiel Boland University of Nijmegen The Netherlands