## 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.

## Clarification on how to contact the admin

Submitted by cubex on Sat, 08/13/2011 - 01:51.I've added a link to Martin's indexed cube lovers archive. Also searches for non-authenticated users will now work.

## Approximation formula for the lower bounds of nxnxn cube in slice turn metric

Submitted by kociemba on Wed, 08/03/2011 - 17:40.## Twenty-Four puzzle, some observations

Submitted by stannic on Sun, 07/24/2011 - 01:39.Hello all. I am new on this great forum. My first post is about Twenty-Four puzzle, larger version of classic Fifteen. I walked around sliding tile puzzles for quite some time. At some point I decided that what I have is too much for me alone, but enough to write about it here.

Many small puzzles have been solved long ago. There is some information in OEIS: A151944 (about MxN puzzles), A087725 (about NxN puzzle). AFAIK largest solved STP is 4x4 puzzle (classic Fifteen). It was known that 80 single-tile moves required and sufficient, and recently Bruce Norskog wrote on this forum about 43 multi-tile moves.