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The void cube is the 3x3 without centers. For every legal 3x3 void position, there are 12 possible ways the centers can be inserted to yield a legal 3x3 cube position. (Pick any of the six colors for the top, then pick one of the adjacent four colors for the front; half of the time the result will have the wrong axis parity).

The "superflip" void position has a distance of 20. This can be shown by computing the optimal solution for all 12 axis insertions in the 3x3 cube; this yields only three unique positions (mod M), and all three have a distance of 20.

U1F1U2F1L2B1U2F1L3R3F2D1R2U2L2B1F3L1F2D1 (20f*) //superflip

D1L2F2R2B3D2L1R1U3R2U3F2D3R2U3B3F3D2R3U3 (20f*)

D1R2D2B3F1L1D3B1L2B2U1R2D1L2B2D2B3U2R2U3 (20f*)

Therefore the void cube has a Cayley graph in the half turn metric with a diameter of at least 20.

This is at present the only known distance-20 position on the void cube. I've evaluated my 84,161 unique-mod-M distance-20 positions in the 3x3 and found that there is no other distance-20 position in this set such that all of its legal axis rotations are still in this set. Since this set of 84,161 is of course far from complete, this does not mean there is not another distance-20 position in the void cube (or even possibley a distance-21 or greater position).

The real state space of the void cube is approximately 12x smaller than that of the normal 3x3; it may be substantially easier to prove a diameter of 20 for the void cube than it is for the normal 3x3.

The "superflip" void position has a distance of 20. This can be shown by computing the optimal solution for all 12 axis insertions in the 3x3 cube; this yields only three unique positions (mod M), and all three have a distance of 20.

U1F1U2F1L2B1U2F1L3R3F2D1R2U2L2B1F3L1F2D1 (20f*) //superflip

D1L2F2R2B3D2L1R1U3R2U3F2D3R2U3B3F3D2R3U3 (20f*)

D1R2D2B3F1L1D3B1L2B2U1R2D1L2B2D2B3U2R2U3 (20f*)

Therefore the void cube has a Cayley graph in the half turn metric with a diameter of at least 20.

This is at present the only known distance-20 position on the void cube. I've evaluated my 84,161 unique-mod-M distance-20 positions in the 3x3 and found that there is no other distance-20 position in this set such that all of its legal axis rotations are still in this set. Since this set of 84,161 is of course far from complete, this does not mean there is not another distance-20 position in the void cube (or even possibley a distance-21 or greater position).

The real state space of the void cube is approximately 12x smaller than that of the normal 3x3; it may be substantially easier to prove a diameter of 20 for the void cube than it is for the normal 3x3.