# All 3x3x3 involutions solved

Submitted by Andrew Skalski on Tue, 05/28/2024 - 16:14.

I optimally solved all 170,911,549,184 involutions in half-turn metric. The entire process took about 2.5 weeks to generate the 37GB solution file. Source code is available on GitHub.

Here are the total and 48-way symmetry reduced counts by distance:

d Total Unique by symmetry -------------------------------------- 0 1 1 1 6 1 2 3 1 3 72 2 4 39 4 5 960 25 6 886 41 7 12708 292 8 19526 506 9 160059 3501 10 349368 7741 11 2133822 45543 12 6903354 146698 13 33277628 700019 14 167193510 3500419 15 932736959 19478862 16 6251752591 130385528 17 37364321274 778842829 18 104805995145 2184417694 19 21346235247 445145591 20 456026 10842 Sum: 170911549184 3562686140

Solutions were found in four phases:

- IDA* search to depth 18 for each of the 336,004 symmetry reduced edges-only involutions
- Neighbor search (extend known solutions
`s`by a move/antimove pair:`m s m'`) - Twophase solver
- Optimal solver for the remaining 6-7 million positions

The neighbor search was very effective in finding distance 19 solutions. That involutions can be extend in this manner is why I think the distance distribution is so top-heavy compared to all cubes.