   Date: Fri, 16 Dec 94 01:32:00 -0500   From: Mark Longridge <mark.longridge@canrem.com > ~~~ Subject: Cyclic Decomposition

I've been working on a new algorithm to find move sequences to reach
certain positions on the 3x3x3 cube. The basic idea is to find a
sequence such that:

(S1 S2 S3... SX) ^N = Goal State
where X is the number of moves in the fragment and
N is the number of times the fragment is repeated.

I call such a process to be "Cyclicly Decomposable".

Certain states, such as the 12-flip, require quite a few moves, in
fact more moves than possible to search using brute force even when
using high-order computers. The best results using the Kociemba
algorithm need 28 q turns or 20 q+h turns for the 12-flip.

While the cyclic decomposition algorithm (henceforth the CD algorithm
for short) usually requires more moves than the Kociemba algorithm
it does have an mnemonic advantage. The following is the best
result using the CD algorithm to date:

```p192 2 Flip in face     (F1 B1 L1 T1 D1)^6           (30 q)
p195 12 Flip            (T1 B3 T1 L1 F3 R3)^6        (36 q)
```

Note that both of these processes use 5 of the 6 generators.

Some cube positions are extremely resistant to CD but flip and
twist patterns are no problem. In particular, the 6 X order 3
pattern does not yield to CD with values of X up to 7.
Naturally with N = 1 we can always find one of the optimal paths
but I am more interested in cases where N > 1. One may note that
with N > 1 using CD processes we are still free to use any number
of q turns except a prime number, for which N must be 1, but this
should not be too constraining.

By relaxing the conditions somewhat we can conceive of sequences
which are near-CD, that is CD with a small suffix or prefix:

```p169 4 Y's Rot + 2 X    (F2 B2 D1 L2 R2) ^2 + T1      (11 q+h)
```

By looking at the best sequence the Kociemba algorithm can
find for a position, we can count the number of q turns and
use this as a starting point for an attempt with CD...

```p1 6 X order 3   R2 L3 D1 F2 R3 D3 F1 B3 U1 D3 F1 L1 D2 F3 R1 L2 (20 q)

Looking at p1 we can infer that possibly X=5 and N=4 may lead to
finding a CD process or    X=4, N=5
X=10, N=2
X=20, N=2 etc.
```

At the very least we can discount X * N = odd number.

-> Mark <-
Email: mark.longridge@canrem.com     