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I've finally managed to compute God's Algorithm out to 15q*. This took longer than I expected; I had difficulties using multiple cores because occasionally the memory consumption of the concurrently-calculated cosets would exceed my physical RAM; even though this was rare, it happened frequently enough to completely stall the computation. Also, the way memory was allocated and freed led to pretty intense memory fragmentation.

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

Here's an easy puzzle for Rubik's Cube: What's the chromatic number of the Cayley graph for the quarter turn metric?

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.