## Megaminx needs at least 45 moves

Submitted by Herbert Kociemba on Tue, 02/28/2012 - 17:56.## A Hamiltonian circuit for Rubik's Cube!

Submitted by Bruce Norskog on Mon, 02/20/2012 - 21:30.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

Submitted by stannic on Sat, 01/14/2012 - 15:26.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

Submitted by Bruce Norskog on Mon, 12/26/2011 - 13:33.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

## Number of canonical move sequences for nxnxn Rubik's cube in q-w metric

Submitted by kociemba on Mon, 12/26/2011 - 12:22.gfq[n_,x_]:=3/(6-4(x+1)^(2(n-1)))-1/2

and looks very similar to the generating function in h-w metric which is

gfh[n_,x_]:= 3/(6-4(3x+1)^(n-1))-1/2

## Number of canonical move sequences for nxnxn Rubik's cube in h-w metric

Submitted by kociemba on Sun, 11/20/2011 - 17:22.## Interchanging two faces

Submitted by brac37 on Fri, 11/11/2011 - 12:24.Just a question for fun. Suppose you have a Rubiks cube and you want to interchange two faces? How many stickers need to be moved?

Distinguish between opposite and adjacent faces and between using a screwdriver (for disassembling) or not, so you get four answers.

Next, do not read any further before adding those four answers to obtain a single answer.

## Some 3-color cube results

Submitted by Bruce Norskog on Wed, 11/02/2011 - 22:57.The Rubik's Cube can be simplified by using only 3 colors instead of the usual six colors. Generally, opposite faces would share the same color, and that is the convention I assume here in talking about a 3-color cube.

Kunkle/Cooperman showed that a scrambled cube can always be brought to a position within the squares group within 16 moves. This puts an upper bound for God's number for the 3-color cube at 16. It is also well-known that the cube can be put into the <U,D,L2,R2,F2,B2> group in 12 moves. That puts a lower bound on God's number for the 3-color cube at 12. The superflip equivalent for the 3-color cube requires 14 moves according to an optimal 3-color cube solver program I have written. (From solving a million random positions, it appears that about 1.4% of positions of the 3-color cube require 14 moves to solve.) This raises the lower bound for the 3-color cube to 14.

## Source Code for Face Turn Metric 20 Proof Released

Submitted by rokicki on Tue, 10/11/2011 - 12:03.http://cube20.org/src/

I spent a fair amount of effort documenting it. Any feedback is

welcomed.

## 5x5 can be solved in 109 MTM

Submitted by stannic on Mon, 08/15/2011 - 10:53.Any instance of the Twenty-Four puzzle (5x5) can be solved in 109m (multi-tile moves) or less. My proof consists of several steps. It is possible that there is logical error in this proof, so please check it thoroughly. However, I cannot find errors.