Discussions on the mathematics of the cube

## Blockbuilding analyses

I've done a few more analyses that may be of some interest to the speedcubing community. I'm guessing the first two may have been done before. Such an analysis has been talked about on speedcubing forums (such as in this thread http://games.groups.yahoo.com/group/speedsolvingrubikscube/message/13163), but I haven't located any actual results. I'll be happy to give credit for any prior result, if I'm made aware of it.

The goal in these analyses is to build a 2x2x2 sub-block from a scrambled cube state. These analyses do not consider choosing the easiest of eight possible such blocks, but rather one such block is picked, and the distance distribution for all possible scrambles is determined for building that block. Only the three edges and the one corner for that block need to be considered. There are 10560 edge configurations and 24 corner configurations, for a total of 253440 positions. The analysis was carried out in both FTM and QTM.

## 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).

## Non trivial identities

As you know there is a recurrent formula based on the cube trivial identities that holds true up to level 5 in QTM. It breaks at level 6:

P[0] = 1
P[1] = 12
P[2] = 114
P[3] = 1,068
P[4] = 10,011
P[5] = 93,840
P[6] < 879,624

The real number at level 6 is 878,880 which is 744 shorter than what the formula predicts.

This discrepancy is obviously caused by some identities other than the trivial ones.

Does anyone have the list of all those identities?

The only ones I know about are the ones listed in Jaap's website:

## How Close Are We to God's Algorithm?

Tom Rokicki's message about 25 moves as a new upper limit in the face turn metric got me to thinking about how close we might be to God's Algorithm.

First, I think a distinction must be made that I really hadn't thought about very much. It occurs to me that Tom's method or something akin to it might eventually be able to determine the diameter of the cube group in the face turn metric without actually enumerating the complete God's algorithm. Which is to say, Tom's method is a near-optimal solver for cosets. Generally speaking, it doesn't determine optional solutions and it doesn't determine solutions for individual positions. But that's ok for determining the diameter. If Tom's program could reduce the overall upper limit for the diameter down to some N and if at least one position could be found for which the optional solution required N moves, then the diameter would be proven to be N. And the best candidate we have for N right now is 20. So I think it's possible or even likely that the diameter problem will be solved before the full God's Algorithm problem will be solved. That possibility hadn't occurred to me until recently.

## Twenty-Five Moves Suffice

On March the 3rd I finally finished the last coset required to prove that 25 moves suffice.

I have finally finished a draft version of a paper detailing the technique and result:

http://tomas.rokicki.com/rubik25.pdf

The technique uses the coset solver I have described previously here (over the past two years)
but I have also made it faster, and also gotten a new machine that is faster and has more memory to run these cosets more quickly.

In the end, it required about 4,000 cosets to be solved to prove a bound of 25.
By solved, I mean an upper bound on the largest distance in that coset was found;

## Prime numbers and Rubik cube

The most fascinating thing about prime numbers is how to predict one.

For example in the following list:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

only 2, 5, 7, 11, 13, 17 and 19 are prime numbers.

Now if we look at the Rubik cube's positions, we can also count them, something like:

U D F B L R U' D' F' B' L' R' UU UD UF UB UL UR UU' UD' UF' UB' UL' UR'....

Similar to the prime number concept we can come up with the prime cube position concept. So in the above list UU' is not a prime position.

In prime numbers theory, the following theorem has been proved:

P(x)~x/ln(x) where P(x) is the prime numbers counting function.

## 2 questions

Hi,

I am asking for help to understand the second phase of the 2-phases algorithm. I do understand the first phase.

My question is once we bring the scrambled cube to a position in the G1 = subgroup, how is this position taken to the Identity?

Kociemba says the following in his website:

In phase 2 the algorithm restores the cube in the subgroup G1, using only moves of this subgroup. It restores the permutation of the 8 corners, the permutation of the 8 edges of the U-face and D-face and the permutation of the 4 UD-slice edges. The heuristic function h2(a,b,c) only estimates the number of moves that are necessary to reach the goal state, because there are too many different elements in G1.

## Some "F2L" computer analyses

I have done some "restricted" God's algorithm calculations for solving two layers of Rubik's Cube, given that the four edges of the face layer are already solved. I say it's a "restricted" calculation because it only considers certain moves or sequences of moves. For convenience, I will consider the two layers to be solved are the bottom layer (generally referred to as the D layer), and the layer above that (typically referred to by speedcubers as the E layer).

Given that four edges of the D layer are taken to be solved, the remaining four edges being considered (those that belong in the E layer) must be distributed among 8 possible edge locations. So the number of configurations for those four edges is 8*7*6*5*(24) = 26880 (including orientations). The four corner cubies being considered (those that belong in the D layer) have 8*7*6*5*(34) = 136080 configurations (including orientations). This makes for a total of 26880*136080 = 3,657,830,400 configurations.

## Efficient test for equivalence of bandaged 4x4x4s ?

I know that bandaged cube (regardless of their size) are often ignored, which gave me my niche to live in...

I tried to write a program to generate all valuable bandagings of a 4x4x4. Here valuable means:
1. No single gluing can be added without restricting at least one turn which was possible before.
2. From every set of equivalent bandagings, which are transformable into each other by turning and/or generating symmetries, at most one is in the final list.
3. No bandagings which could be realized by a 2x2x2 or a 3x3x3.

Only the 108 bondings between the surface cubies are considered.

## New hex-axes search 2x faster than triple-axes search

One interesting option in Kociemba's Cube Explorer is the Triple Search option.
This option searches along three axes at once, instead of just one.

How much does Triple Search help? For random cubes, when searching for a length
20 solution, it helps tremendously. I implemented my own Kociemba solver with
single-axis search, triple-axis search, and a new "six-axis" search, and compared
the amount of time needed to find length 20 or better solutions for 3,000 random
cubes. This program was single-threaded (Kociemba's cube explorer is multi-threaded),
runs only from the command line, and is brand new code (so it has not been heavily