## Puzzle about the Cube: Coloring the Cayley Graph

Submitted by rokicki on Sat, 09/19/2009 - 14:22.Here's a slightly harder puzzle: What's the chromatic number of the Cayley graph for the half turn metric? If you can't figure it out, can you figure out an upper bound? A lower bound?

This was discussed on speedsolving.com before, but I think it's a good enough puzzle to present here as well.

## God's Algorithm out to 15q*

Submitted by rokicki on Sat, 09/19/2009 - 13:56.In any case, it is finally done; here are the results. First we have positions at exactly that depth:

d mod M + inv mod M positions

## Numerical formula

Submitted by mdlazreg on Tue, 09/15/2009 - 07:55.d positions I4 positions I4&I12 positions ALL -- ------------ ---------------- -------------- 0 1 1 1 1 12 12 12 2 114 114 114

## Drupal database corrupted

Submitted by cubex on Wed, 09/09/2009 - 17:12.I try my best to make sure everything is working but this one slipped through the cracks. Somehow the mysql database ballooned in size to over 2 gigabytes. After that happened the subsequent databases were not backed up correctly.

It would be a good idea for any posts to be buffered in some way before uploading to the forum, especially long ones.

## Watermelon Rubik's Cube

Submitted by Jerry Bryan on Mon, 08/10/2009 - 11:07.http://www.watermelon.org/FeaturedRecipe.asp

I am in no way connected with the National Watermelon Promotion Board.

## FTM Antipodes of the Edge Group

Submitted by Bruce Norskog on Tue, 07/21/2009 - 11:23.I have done my own independent breadth-first search of the edge group using the face-turn metric. I used symmetry/antisymmetry equivalence classes to reduce the number of elements in the search space. I confirm the "Unique mod M+inv" values for this group/metric that Rokicki reported in 2004.

I reduced the "coordinate space" for the search to 5022205*2048=10285475840 elements by using symmetry/antisymmetry equivalence classes of the edge permutation group. (This gives a much more compact overall coordinate space than using an edge orientation sym-coordinate, at a cost of more time required to calculate representative elements. This allowed me to keep track of reached equivalence classes with a ~1.3 GB bitvector in RAM and 5022205 KB disk files to keep track of distances.)

## God's Algorithm out to 13f*

Submitted by rokicki on Wed, 07/15/2009 - 14:51.First, the positions at exactly that distance:

d mod M + inv mod M positions -- ------------- -------------- --------------- 0 1 1 1 1 2 2 18 2 8 9 243 3 48 75 3240 4 509 934 43239 5 6198 12077 574908 6 80178 159131 7618438 7 1053077 2101575 100803036

## God's Algorithm out to 14q*

Submitted by rokicki on Wed, 06/24/2009 - 09:48.First, positions at exactly the given distance:

d mod M + inv mod M positions -- ------------ ------------- -------------- 0 1 1 1 1 1 1 12 2 5 5 114 3 17 25 1068 4 130 219 10011 5 1031 1978 93840 6 9393 18395 878880 7 86183 171529 8221632 8 802788 1601725 76843595

## God's Algorithm out to 12f*

Submitted by rokicki on Tue, 06/23/2009 - 10:22.The first table is the count of positions with exactly the given depth.

d mod M + inv mod M positions -- ------------ ------------ -------------- 0 1 1 1 1 2 2 18 2 8 9 243 3 48 75 3240 4 509 934 43239 5 6198 12077 574908 6 80178 159131 7618438 7 1053077 2101575 100803036

## Twenty-Nine QTM Moves Suffice

Submitted by rokicki on Mon, 06/15/2009 - 20:35.we have shown that there are no positions that require 30 or more

quarter turns to solve. All these sets were run on my personal

machines, mostly on a new single i7 920 box.

These sets cover more than 4e16 of the total 4e19 cube positions,

when inverses and symmetries are taken into account, and no new

distance-26 position was found. This indicates that distance-26

positions are extremely rare; I conjecture the known one is the

only distance-26 position.

In order to take the step to a proof of 28, I would need a couple