Discussions on the mathematics of the cube

## 2x2x2 Cube Antipodes

I have written a GUI NxNxN cube program to which I just added a 2x2x2 cube auto solve function. To test the performance of the solution algorithm I wanted try it on the 14 q-turn antipodes. So I did the depth-wise expansion of the group, found the 276 antipodes and reduced them with M symmetry. In the context of the fixed DBL cubie 2x2x2 model, that is the <R U F> group model, M symmetry classes are formed by ( c * m' * q * m ) where q is a group element, m ranges over the cubic symmetry group and c is the whole cube rotation needed to place the conjugate back in the group.

## How many 26q* maneuvers are there?

How many 26q* maneuvers are there?

Well, obviously we can't say for sure, as it hasn't yet been proved that the 3 known 26q* positions (which are symmetrically equivalent to each other) are the only 26q* positions. In another thread, Herbert Kociemba mentioned that there are "many" such maneuvers, but he did not attempt to generate them all (for the known 26q* positions).

I note that 26q* refers to a maneuver that is 26 quarter turns long and that is known to be optimal in the quarter turn metric. It may also refer to a position that requires a minimum of 26 quarter turns to solve. 26q (without the asterisk) refers to any maneuver 26 quarter turns long, but isn't necessarily optimal for the position it solves.

## 5x5 puzzle: Comparison between reduction chains (STM, 10000 instances)

The multi-chained approach used in kumi na tano allows to use multiple search chains at the same time. The main advantage is that the best chain can be choosen depending on the instance to be solved, rather than hard-coded into the search algorithm.

For example, the first of the following two 5x5 instances has its leftmost column solved, while second has solved four tiles in top-right corner.

```
1 17  9 10 18    18  3 16  4  5
6  0  2  3  8    11  7 17  9 10
11  5 22  7  4    19  2 23  0 21
16 15 20 23 13     6 20 14 12  1
21 19 12 14 24    13  8 15 22 24```

We cah use multi-chained approach here. The following two partitioning schemes:

## One Million Random Twenty-Four Puzzle Instances in the STM metric

I have solved sub-optimally 1,000,000 random instances of 5x5 sliding tile puzzle in STM metric (single-tile moves). The actual running time was about 18,5 hours. The minimum, maximum and average solution length were 73, 171 and 124.48 moves respectively. About 52% of 1,000,000 solutions were in range [118; 132]. There were only 32 instances with (suboptimal) solution length less than 81 (range [73; 80]). Only one instance was solved in 171 moves.

## Sliding tile puzzle suboptimal solver

Hello all.
I wrote a program capable to solve (MxN-1) sliding tile puzzles, such as the Fifteen puzzle. The program can solve puzzles from 2x2 to 11x11.
The main thread is on Speedsolving.com:
- Bulat

## Policy Change for New Accounts

Due to the constant spamming I have changed the access rules for new accounts. From now on new users must email cubexyz at gmail dot com and explain why they want an account here. A short note on your specific interests on Rubik's Cube and math should be sufficient.

Also the ban on gmail has been lifted. Sorry for the trouble, but deleting spam entries got tiresome.

Mark

## Megaminx needs at least 45 moves

Surprisingly, nobody seems to have done anything else as a rough analysis of the number of moves to solve the Megaminx puzzle, especially no analysis which includes the commutativity of some moves.

## A Hamiltonian circuit for Rubik's Cube!

I have found a Hamiltonian circuit for the quarter-turn metric Cayley graph of Rubik's Cube! In fact, it only uses turns of five of the six outer layers of the cube.

In more basic terms, this is a sequence of quarter moves that would (in theory) put a Rubik's cube through all of its 43,252,003,274,489,856,000 positions without repeating any of them, and then one more move restores the cube to the starting position. Note that if we have any legally scrambled Rubik's Cube position as the starting point, then applying the sequence would result in the cube being solved at some point within the sequence.

## Regularities in maximum WD values

Regularities in maximum WD values

This post is about any mathematical laws inside the Walking Distance heuristic. It seems like WD is not just puzzle to be computed. Maybe the whole WD heuristic is some math structure.

## A Hamiltonian Circuit for the 2x2x2

I have found a Hamiltonian circuit for the 2x2x2 cube group (3674160 elements). I have posted the solution on the speedsolving.com forum. Link: http://www.speedsolving.com/forum/showthread.php?34318