Discussions on the mathematics of the cube

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

In h-w metric, a move of the nxnxn cube is a 90 or 180 degree turn of a face together with 0..n-2 adjacent slices. When counting the canonical move sequences the commutativity of the moves on one axis has to be taken into account. The number of canonical move sequences can be computed quite elegantly using matrices

Interchanging two faces

Hello all,

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

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

You can find the source code used for the "20" proof at:


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

5x5 can be solved in 109 MTM

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

To contact the admin of this site send emails to: cubexyz at gmail dot com

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

I tried to derive some analytic approximation formula for the lower bound in h-s metric, that is half-turm metric with slice moves for large n. There are 9n possible slice moves for an nxnxn cube, and without using any other relations where would be (9n)^k move sequences of length k. In my simplified model I only used the relations that the 3n slice moves of one axis commute and that there are never two successive moves with the same slice (the latter does not hold in quarter turn metric).

Twenty-Four puzzle, some observations

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.

Optimal Void Cube Up Face Odd Parity Maneuvers

Optimal Void Cube Up Face Odd Parity Maneuvers

When solving the void cube one encounters odd position parity positions as the cube approaches being solved. The last step for me is to solve the Up face corners and this is where I encounter odd permutations. Corners first solvers may encounter an odd permutation of the Up face edges. The question is what is the shortest maneuver which will convert an odd permutation of the Up face corners into an even permutation of the Up face corners leaving everything else unchanged. This is regardless of the effect on the corner twist. Likewise, what is the shortest maneuver which will convert an odd parity up face edge permutation to an even parity permutation regardless of the edge flip?

Lower Bounds for n x n x n Rubik's Cubes (through n=20) in Six Metrics

In January 1981, Dan Hoey posted to cube-lovers a description of a
technique to compute a lower bound on God's Number. This technique
considered the maximum number of positions that can be reached by what
is called "syllables"---consecutive moves on the same axis, possibly
turning distinct faces. Since all the moves that make up a syllable
commute, we can select a single canonical move sequence to represent
every syllable, and then determine how many move sequences of a given
total length can be made out of only these syllables. This gives a
more accurate bound on God's Number because it eliminates many