## Linear formula

Submitted by mdlazreg on Mon, 09/28/2009 - 08:13.We have then the following formula:

P(n) = sum(R(k)*P(n-k)) for 1 <= k <= n

Calculating R(k) using rokiki's results we deduce :

1 12 = 12*1 2 114 = 12*12 -30*1 3 1068 = 12*114 -30*12 +60*1 4 10011 = 12*1068 -30*114 +60*12 -105*1

## Algorithm for Counting Identities

Submitted by Jerry Bryan on Sat, 09/26/2009 - 16:45.I've been thinking about writing a program to calculate and count duplicate positions - roughly speaking, those positions that are half way through an identity. What I have in mind will probably be a more time consuming program to write than I would prefer. So I wonder if I could ask Herbert Kociemba and/or mdlazreg to post a little something about the programs they have already written to find identities. It may well be that there is a much simpler approach to calculating duplicate positions than what I have in mind.

What I have in mind is an iterative deepening depth first search beginning at the Start position. If that's all I did, the search would simply count 12^{n} maneuvers for each distance from n, and it would not extract any useful information about how many duplicate positions there are for each n. To solve these problems, I propose to store all the duplicate positions and not to store those positions that are not duplicate. This would be for the quarter turn metric. The program I have in mind would not be able to handle the face turn metric.

## God's algorithm for FTM mod 48, 2. Try

Submitted by kociemba on Sat, 09/26/2009 - 07:32.distance positions mod 48 0 1 1 18 2 3 3 24 4 39 5 12 6 22 7 12 8 40 9 3 10 4 11 20

## 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