From:

~~~ ~~~ Subject:

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