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

Hello all,

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
```