   Date: Tue, 07 Aug 84 19:24:00 -0400 (EDT)   From: Dan Hoey <hoey@NRL-AIC >
~~~ ~~~ Subject: The pocket cube: correction, calculation, and conjectures

Well, maybe this list is dead after all, if I can tell you there are
(7!/2)(3^6) positions of the pocket cube, and have it stand for a week.
The correct number is of course (7!)(3^6) = 3674160, since the
generators are odd.

But not being one to eat crow with a straight face, I have hacked the
good hack, so I can give you the exact number P(N) of pocket cube
positions exactly N quarter-turns from solved. (This was done in
September 1981 for the half-twist metric; see the archives.) I have
also computed the number L(N) of local maxima at each distance. These
numbers are given below.

``` N     P(N)     L(N)
0        1        0
1        6        0
2       27        0
3      120        0
4      534        0
5     2256        0
6     8969        0
7    33058       16
8   114149       53
9   360508      260
10   930588     1460
11  1350852    34088
12   782536   402260
13    90280    88636
14      276      276
```

An approach for dealing with these numbers (suggested to me by Dale
Peterson) is to form the Poincare polynomial

```p(x) = SUM P(i) x^i
i
```

in hopes that it can be factored nicely. Unfortunately, this doesn't
work out--with the exception of the obvious factor (x+1), p(x) is
irreducible. I have also tried to decompose p(x) using the power
(1+x)^2
series for ------------, which agrees with the first five terms of p(x)
3 - 2(1+x)^2
due to the lack of non-trivial identities. I haven't found any good
ways of expressing p(x), but there may be something there. The point
of all of this is that it could conceivably lead to a conjecture--or
even a derivation--of God's number for the 3^3 puzzle.

I might pass along another fuzzy recollection from a year and a half
ago, in hopes that it is more informative than incorrect. Dale
mentioned another classical method for dealing with group diameters.
It seems there is a class of groups, called reflection groups, for
which tight diameter bounds can be derived. A reflection group is a
group of matrices with eigenvalues of plus and minus one. Some
properties generalize to pseudo-reflection groups, where the
eigenvalues all have complex magnitude one. We managed to construct
isomorphisms between the 3^3 edge group and a reflection group, and
between the corner group and a pseudo-reflection group. As I recall,
he was fairly certain that the full cube group did not qualify, but
that was beyond my depth.

So if you think cubes are dead, remember it's not because the
results are all in.

Dan     