Several messages in August of this year [mail to
Hoey@CMU-10A for copies] concerned short identities of the cube,
i. e. processes which return the cube to solved. Later in that
month I assisted David Plummer in a brute force attack on the
problem. We had plans to investigate all the positions up to eight
qtw, but unfortunately became busy on other projects. I have
finally come up with enough time to analyze and report the data
from the seven-qtw search.
There are 8,221,632 cube positions at a distance of seven
quarter-twists from solved, and 9,205,558 positions at seven or
fewer qtw. By recording cases of different seven-qtw processes
yielding the same position, a complete list of fourteen-qtw
identities is obtained. The task is then to reduce the list to
exclude multiple instances of equivalent identities.
We call two identities equivalent when one can be obtained
from the other by some combination of the following operations:
- uniformly relabeling the twists according to a rotation
or reflection of the cube,
- cyclically permuting the twists,
- reversing the order of the twists and inverting each one,
- substituting a sequence x for a sequence y, where xy' is
a shorter identity.
The first three criteria are easily implemented on a computer. The
fourth can be performed for the shortest identities, those
equivalent to F^4 and FBF'B', but I know of no algorithm to detect
all cases of equivalence due to substitution of the longer
identities. My strategy was to reduce the (several thousand)
identities by computer for the simple kinds of equivalence, and
then to look by hand for substitution equivalence between the
fourteen identities then remaining. I found three equivalences,
listed at the end of this note, but the possibility remains that
some of the following identities are equivalent. The list is,
however, complete (modulo bugs and cosmic rays).
Identities equivalent to the first six on this list were
independently discovered by Chris C. Worrell; I follow his
numbering for them. Identities I14-5 through I14-7 do not hold in
the Supergroup, because they twist face centers as noted in the
I14-1 BF' UB'U'F UL' BU'B'U LU' I14-2 B UBL' B'D'BD LB'U' L'B'L I14-3 BB U BB UD' RR U' RR U'D I14-4 BUB'U' L'FL UBU'B' L'F'L I14-5 (BB UD B U'D')^2 [Supergroup BB] I14-6 BF' U B'F LR' UD' F' U'D L'R [Supergroup UF'] I14-7 BF U B'F' LR' UD F' U'D' L'R [Supergroup UF'] I14-8 BF' UFRU'R'B'U'B'RBUR' I14-9 BF' UFRU'B' UD' F'U'R'FD I14-10 (BUBU'L'B') R (BLUB'U'B') R' I14-11 (BUBU'L'B') D'R'B' DLD'RD The twelve-qtw identity I12-2 = (BUBU'L'B') (B'D'B'DLB) can be substituted into identities I14-10 and I14-11 to yield: I14-10a (B'L'D'BDB) R (BLUB'U'B') R' I14-10b (BUBU'L'B') R (B'D'B'DLB) R' I14-10c (B'L'D'BDB) R (B'D'B'DLB) R' I14-11a (B'L'D'BDB) D'R'B' DLD'RD
Identity I14-10c can be obtained from I14-10 [by shifting seven
places and reflecting the cube through the UD plane] but I14-10,
I14-10a, and I14-10b are mutually inequivalent when twelve-qtw
identities are ignored. The same holds for I14-11 and I14-11a.
Strangely enough, I14-11a can also be transformed to I14-11 by
substituting with the identity (BDBD'R'B') (B'U'B'URB), which is
equivalent to I12-2.