# 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.

I got the following distribution optimally solving a million randomly scrambled 3-color cubes.

moves count 7 1 8 37 9 357 10 4,421 11 49,818 12 378,592 13 552,604 14 14,170

I note that the 3-color cube has 2^{11}*3^{7}*(12 choose 4)*(8 choose 4)^{2} = 10,863,756,288,000 configurations
of corners and edges with respect to fixed centers.
Because each center can not be distinguished from the opposite center,
there are really only about 1/4 that many positions.
I get the following counts for symmetry classes.

class size count 1 2 2 8 3 14 4 6 6 222 8 352 12 8,086 16 8,566 24 5,982,162 48 226,325,259,954 --------------- total 226,331,259,372

With under a quarter trillion symmetry-reduced positions, a complete breadth-first search should be a doable task.

## The centers of the 3-color 4x4x4 cube can be solved in 11 moves.

I have completed a breadth-first for solving the the center pieces of a 3-color 4x4x4 cube.
The results are given below.
As the 4x4x4 does not have any fixed centers to serve as a reference,
positions are counted under the assumption that the orientation of the cube matters
(24!/8!^{3} = 9,465,511,770 total positions).

symmetrically distance positions distinct 0 6 1 1 18 1 2 432 4 3 7,722 45 4 130,608 520 5 2,041,152 7,404 6 30,316,656 106,399 7 365,166,768 1,271,430 8 2,664,549,816 9,259,729 9 5,689,455,696 19,762,729 10 713,840,016 2,479,705 11 2,880 12 ------------- ---------- Total 9,465,511,770 32,887,979

I have also broken down the numbers by the sizes of the symmetry classes.

class size 6 12 18 24 36 48 72 96 144 288 dist - -- -- -- -- -- -- -- --- --- 0 1 1 1 2 2 2 3 1 4 7 17 16 4 4 1 18 1 96 400 5 4 6 2 38 6 547 6801 6 8 82 14 2110 104185 7 2 6 19 189 31 6609 1264574 8 1 7 13 150 87 15263 9244208 9 2 12 119 222 14851 19747523 10 6 10 158 1952 2477579 11 4 8 Total 1 1 6 2 37 53 615 519 41451 32845294 Pos 6 12 108 48 1332 2544 44280 49824 5968944 9459444672