# The Void Cube to 13q

Breadth First Enumeration 2014-09-27 19:58:23.094 VoidCubeClient[508:5903] 0 1 1 2014-09-27 19:58:23.095 VoidCubeClient[508:5903] 1 1 12 2014-09-27 19:58:23.099 VoidCubeClient[508:5903] 2 5 114 2014-09-27 19:58:23.101 VoidCubeClient[508:5903] 3 17 1,068 2014-09-27 19:58:23.105 VoidCubeClient[508:5903] 4 130 9,951 2014-09-27 19:58:23.133 VoidCubeClient[508:5903] 5 1,018 92,592 2014-09-27 19:58:23.333 VoidCubeClient[508:5903] 6 9,204 860,852 2014-09-27 19:58:25.033 VoidCubeClient[508:5903] 7 83,789 7,991,856 2014-09-27 19:58:40.300 VoidCubeClient[508:5903] 8 774,323 74,114,319 2014-09-27 20:01:03.984 VoidCubeClient[508:5903] 9 7,159,250 686,774,712 2014-09-27 20:25:47.908 VoidCubeClient[508:5903] 10 66,273,224 6,360,091,030 Coset Enumeration Void Cube Model 1.0 Group: R, U, F, TR, TU, TF Coset Base Subgroup: Subgroup with solved corner cubies and the UF and UR cubies in the solved position regardless of orientation. 484,989,120 cosets of size 7,431,782,400 Coset Symmetry Reduction: Oh+ Cosets solved since launch: 3,429,943 Average time per coset: 0:00:00.068 Server Status: Void Cube Enumerator Server Enumeration to depth: 13 Snapshot: Friday, October 3, 2014 at 6:56:34 PM Central Daylight Time Depth Reduced Elements 0 1 1 1 2 12 2 18 114 3 50 1,068 4 447 9,951 5 2,008 92,592 6 19,000 860,852 7 124,184 7,991,856 8 1,136,806 74,114,319 9 9,028,936 686,774,712 10 82,411,850 6,360,091,030 11 711,657,402 58,868,124,048 12 6,507,640,604 544,562,369,684 13 58,275,341,089 5,033,855,951,932 Sum 65,587,362,397 5,644,416,382,171 484,989,120 of 484,989,120 cosets solved Elapsed Time: 0:12:14:50

This work was performed using the <R, U, F, TR, TU, TF> group model. This model was discussed in a previous thread. It must be pointed out here that the <R, U, F, TR, TU, TF> metric is exactly the same as the <R, U, F, L, D, B> metric since TR = L , TU = D and TF = B on the void cube. It makes no difference if the left face is rotated holding the rest of the cube rigid or if the left face is held rigid and the rest of the cube is rotated in the opposite direction, the rearrangement of the cubies is the same. If a distinct state of the void cube is at say depth 13 in the <R, U, F, TR, TU, TF> metric it will be at depth 13 in the <R, U, F, L, D, B> metric.

To establish a benchmark, I first performed a breadth first expansion of the move tree, storing the states generated as 20 byte arrays and counting the new states produced with each added turn. Using Oh symmetry reduction and antisymmetry the calculation could be completed to depth 10 before exhausting computer memory. A note about the symmetry reduction. All states in the <R, U, F, TR, TU, TF> group model have the DLB cubie in its starting position and orientation. In general, except for the C3v symmetries (the three fold rotations about the UFR‐DLB axis and the three planes of symmetry containing that axis), the cubic group conjugates will not have the DLB cubie in the proper location. This necessitates reorienting the conjugate. So the equivalence classes are formed by ( c * m' * g * m ) where g is a group element, m is a cubic group symmetry, and c the whole cube rotation required to orient the conjugate with the DLB cubie in the solved position and orientation.

To proceed further I went over to a coset solver approach. One partitions the group into cosets of a subgroup of that group and finds the depth of the coset members independently. Here one only has to keep track of previously solved states for the current coset which reduces the memory required to a manageable figure. The subgroup chosen was the subgroup whose members have the corner cubies solved and the UF and UR cubies in the solved position regardless of orientation. This yields cosets characterized by having a particular configuration of the corner cubies, a particular location for the UF cubie and a particular location for the UR cubie. This gives the partition:

7! * 3^{6}* 12 * 11 = 484,989,120 cosets 10! * 2^{11}= 7,431,782,400 elements per coset

The number of cosets needing to be solved may be reduced by symmetry reducing the corner configurations, again using the (c * m' * g * m) expedient used in the breadth first calculation. The corner configurations are partitioned into symmetry equivalence classes. One member is chosen from each class and composed with the UF, UR position configurations, yielding 132 cosets per equivalence class. These go to the coset solver and the results are multiplied by the size of the equivalence class.

The depth 13 calculation required twelve hours running on 8 cores spread over two computers. If I were inclined to do so, with the hardware at my disposal I could extend the calculation out to depth 14 with a four to five day run. Beyond this would take weeks extending into months and years and you're getting into the realm of super computers.