[next] [prev] [up] Date: Sun, 24 Sep 95 00:31:55 +0200
[next] ~~~ [up] From: Michiel Boland <boland@sci.kun.nl >
[next] ~~~ [up] Subject: Order problems

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 <boland@sci.kun.nl>
University of Nijmegen
The Netherlands

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