   Date: Wed, 11 Jan 95 15:04:43 -0500 (EST)   From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ ~~~ Subject: When is 4x4x4 Solved in GAP?
```On 01/08/95 at 13:47:00 Martin Schoenert said:
```
```Mark Longridge wrote in his e-mail message of 1995/01/03
```
```                        ...  I don't know how a normal 4x4x4 could
be represented though.
```

I fail to see the problem. Just number the facelets. The only
problem would then lie in deciding what the generators are -- i.e.,
which kind of slice moves do you accept. You would also have to
decide whether to model the invisible 2x2x2 inside, but again if you
did, just number the invisible facelets and include their movements

The problem is that many different positions all look solved. For
example, you can permute the 4 center facelets of one face or exchange
two adjacent edges, and the cube still looks solved (of course you cannot
do all this independently). So if we take the obvious permutation group
on the 6*16 points, then a whole subgroup would correspond to what a
puzzler would consider solved states. If by a model we mean a group
whose elements correspond to the different states a puzzler would see,
and whose identity corresponds to what a puzzler would consider solved,
then I have no good idea how to model the 4x4x4 cube as a permutation
group.

Start by numbering all the facelets and defining your generators (about
which there might be some controversy), and call the resulting group G.
Decide which permutations you consider equivalent and call this set K.
K would probably include such things as the whole cube rotations C,
as well invisible permutations of the four center facelets, etc.
In most reasonable choices for K, K would certainly be a group. Your
model then becomes the set of cosets G/K (which is *not* a factor
group! I am learning.). The questions then become: 1) can you define
an operation on the cosets G/K such that G/K is a group, and 2) can you
find a mapping from G/K onto a subgroup of G such that the mapping
respects costs? If the answer to both questions is "yes", then it is
this subgroup of G which you would want to put into GAP.

By the way, I am sensitive to the distinction between G and CG, but in
the case of any Face centerless cube such as 2x2x2 or 4x4x4, it would
seem to me that the distinction is less important than in cubes with
Face centers such as 3x3x3 and 5x5x5.

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Robert G. Bryan (Jerry Bryan)                        (304) 293-5192
Associate Director, WVNET                            (304) 293-5540 fax
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