Discussions on the mathematics of the cube

## Linear formula

Let's assume that there is a linear formula that gives us P(n) using P(k) where 0 <= k <= n-1. P(n) is the number of positions at depth n.

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

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 12n 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

In FTM the complete knowledge of the distribution of the symmetric subgroups (first table of http://kociemba.org/symmetric2.htm ) lets us not only give the odd and even entries in the distance table but also the distances mod 48, because all unsymmetric position contribute with a multiple of 48. So we get
```
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

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.

## God's Algorithm out to 15q*

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`

## Numerical formula

I wrote a program that counts cube positions by taking into account only the identities of length 4 and the identities of length 12. The results of this program are in the second column below:

``` d                  positions I4     positions I4&I12   positions  ALL
--                  ------------     ----------------   --------------

0                             1                    1                1
1                            12                   12               12
2                           114                  114              114```

## Drupal database corrupted

Sorry folks, I've been very busy lately and I just noticed the mysql database was badly corrupted on Sept. 6th, 2009. The database is updated daily, but all the backups on Sept. 6th and after are unusable. I think the only post lost was Tom Rokicki's.

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

I trust that I may be forgiven for being slightly off topic. After all, a watermelon Rubik's cube is not very mathematical. But still, it's an interesting concept.

http://www.watermelon.org/FeaturedRecipe.asp

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

## FTM Antipodes of the Edge Group

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*

Just finished running out to a distance of 13 in the face turn metric.

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```