# Some thoughts about a proof, that 24 moves suffice

I thought about the number of cosets of H=<U,D,R2,L2,F2,B2> we need to
compute to show, that 24 moves suffice.

It is not difficult to show that the number of cosets needes for 24 moves is
at most 64430, provided that we get a maximum of 20 moves in each coset (which
is quite realistic).

In my way to enumerate the cosets, the flip of the edges and the positions of the four UD-slice corner give a symmetrized coordinate x with 0<=x<64430. The orientation of the corners give a coordinate y with 0<=y<2187 . Multiplying these numbers gives 140908410, which is slightly more than the actual number of symmetrized cosets, which is 138639780, so some pairs (x,y) describe the same coset, but this doesnt matter.

Now we take a look at the distribution at the distances of the corner-orientations, starting with the identity orientation y=0:

distance positions 0 1 1 4 2 34 3 186 4 816 5 1018 6 128

So if we show that for all 64430 cosets (x,0) we need at maximum 20 moves, this guaranties at most 26 moves, max. 6 moves to go to the identitiy orientation and max 20 moves to solve this position - nothing new at all.

I computed this distance tables with all possible start values for the orientation. The maximum distance is 6 in 147 cases, 5 in 1968 cases and 4 in 72 cases. Starting with an example position y0 with maximum distance 4 we have

distance positions 0 1 1 16 2 181 3 1087 4 902

So if we compute all 64430 cosets (x,y0), x<64430 and show for them <=20 moves indeed have shown that 24 moves suffice.

It is clear hat another choice of cosets might significantly reduce the number of cosets to compute, but computing coset-classes with 64430 cosets based on the corner-orientation has the advantage that the enumeration is quite easy.

I tried what is the best combination when generataing a distance table starting with two values for the orientation, and found some y1 and y2 with

distance positions 0 2 1 36 2 420 3 1575 4 154

So computing all the cosets (x,y1), x<64430 and (x,y2), x<64430 significantly increase the number of positions, for which 23 moves suffice.

Because I do not have a machine with enough RAM, I cannot tell how long it takes on average to show <= 20 moves for a single coset. Tom will tell us. But I think it is less than 1 hour and with 40 PC's it would take les that two months to make the 24 move proof in this way.