Discussions on the mathematics of the cube

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

Number of 4x4x4 positions for up to 5 moves

After hearing about this paper which talks about the size of God's number for nxnxn cubes, I was thinking about what we currently know about God's algorithm for the 4x4x4 and realized that I couldn't seem to find any partial distance distributions on the 4x4x4 on the web.

So I've run my own analyses to get the number of positions up to 5 moves from the solved state. I have done this for six metrics. First, I have "single-slice" metrics where a move is only allowed to turn a single layer. Second, I have twist metrics where a move is only allowed to twist the cube along one plane. This is sometimes called face-turn metric because a face layer (possibly along with additional adjacent layers) is (are) turned with respect to the rest of the cube. Finally I also used what I termed "block turns" where some block of one or more adjacent layers (not necessarily including a face layer) are turned with respect to the rest of the cube. For each of these, I also considered whether or not a move must be restricted to quarter-turns only.

An Alternate Universe of the Cube

I've been meaning to explore new variations on the 3x3x3 cube for a while and I think I've come up with something new.

If we consider the rotation of a 2x2x2 block as one move, say the UFR block, and call it the z-move for lack of a better name. Now all sorts of weird stuff becomes possible, e.g. (z, C_U2)^6 will generate a 5-spot pattern! The centres cycle (U,L,R,B,F) in befuddler notation.

So in this particular universe of the 3x3x3 cube we are considering 8 corner moves which move a 2x2x2 block rather than the usual 6 face moves. I won't spoil all the fun, but a 3-spot, 4-spot and 6-spot are possible and these spots patterns are quite different from what is possible in the 'normal' universe of the cube.