   Date: Sun, 02 Aug 92 20:00:00 -0400 ~~~ From: Mark Longridge <mark.longridge@canrem.com > ~~~ Subject: cube theory

I've been doing some research to try and figure out some things
about the cube. I've also tried (unsuccessfully) to develop a sort
of CRC or checksum for the cube. With this cube "signature" I could
then find out which depth each pattern requires. I'm puzzled how
some computer types managed to find God's Algorithm for the 3x3x3
squares group. How do you keep track of all the patterns without
repeating yourself? If it is by holding all patterns in an array
the array must become huge.

```Maximum Depth (using q and h moves)
-------------
```
```2x2x2 sq group       4 (24 total states)
Pyraminx            11 (or 14 including the 3 tips)
2x2x2               11 (14 using q turns only)
3x3x3 corners only  11
3x3x3 sq group      15 (half turns only, don't know if using q improves
this)
3x3x2 domino        18 (for 1 solution)
```

A local maxima is a state where any possible move will bring you closer
to a solution. This can occur on the 2x2x2 at depth 4 and on the 3x3x3
at
depth 3. Note that all possible patterns at maximum depth are local
maxima,
however it is surprising that local maxima may occur in patterns much
closer to the surface.

To date, no work has been done to determine the depth of the
dodecahedron
(megaminx) or square 1.

Some questions:

What pattern is an example of local maxima? e.g. 3x3x3 at depth 3
-> 12-flip, 12-flip 8-twist

```        q+h     Depth   Patterns
2x2x2             1        9
3x3x3             1       18
Dodecahedron      1       48
```
```Analysis of the full cube group
-------------------------------
```
```Moves Deep       arrangements (q+h)   arrangements (q only) *
```
```0                    1                   1
1                   18                  12
2                  243                 114
3                3,240               1,068
4            >  48,600              10,011
```

* Work by Zoltan Kaufmann

```Notes:  At 1 move deep each of the 6 sides can turn 3 ways (+ - 2)
giving 18 distinct patterns
```

At 2 moves deep it is redundant to turn the same side again
so 5 sides can turn 3 ways so 18x15=270
However, with opposite turns order is not significant,
e.g. T,D = D,T F,B = B,F L,R = R,L
since each of these can occur in 9 different ways there are 27
redundancies so 270 - 27 = 243

```     At 3 moves deep with the first 2 moves on opposite faces don't
turn the face used in move one since:
T,D,T = T2,D   F,B,F = F2,B   L,R,L = L2,R
This can occur in 3x3x3=27 ways for each case so 81 are dropped
(Remember the first 2 moves have already been weeded of
redundancy!)
Also when the 2nd and 3rd moves are of opposite faces
e.g. T,R,L = T,L,R  B,R,L = B,L,R   F,R,L = F,L,R   D,R,L = D,L,R
T,B,F = T,F,B  D,B,F = D,F,B   L,B,F = L,F,B   R,B,F = R,F,B
F,T,D = F,D,T  B,T,D = B,D,T   L,T,D = L,D,T   R,T,D = R,D,T
```
```since each of these can occur 27 different ways in each of the
cases this gives 27x12 = 324 redundancies
Thus 243x15 = 3645, removing the redundancies gives 3645-81-324=3240
```

>At 4 moves deep.... still working on this one! Zoltan Kaufmann
>has done 4 moves deep using quarter turns, but has anyone
>calculated farther using q and h turns? I'd be interested in
>the source code of any programs people have written on finding
>path-lengths. Also what is an example of a local maxima close
>to the surface, e.g. 4 moves. I believe Jim Saxe and Dan Hoey
>have done some work in this regard.

```        One more question: What is the maximum number of moves required
if you do the 3x3x3 one face last? The best results I've seen
are 19 q and h moves.
-> Mark Longridge
--
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```     