# Gear cube extreme can be solved in 25 moves

Submitted by Ben Whitmore on Thu, 02/15/2018 - 23:07.

Write the puzzle as the group <R,F,U,D> where R and F are 180 degree moves. We use a two-phase algorithm to first reduce the state of the puzzle to the subgroup <R3,F3,U,D>, and then finish the solve in the second phase. The subgroup <R3,F3,U,D> is the group of all positions where all of the gears are oriented, because R3 is the same as R' except the gear orientation remains unchanged.

The first phase is easy to compute. There are only 3^8 = 6561 positions because each gear has only 3 different orientations, despite having 6 teeth.

Phase 1 distribution:

Phase 2 distribution:

The cancellation between phases was Michael Gottlieb's idea, and all the computations were done by me.

The first phase is easy to compute. There are only 3^8 = 6561 positions because each gear has only 3 different orientations, despite having 6 teeth.

Phase 1 distribution:

Depth New Total 0 1 1 1 4 5 2 8 13 3 78 91 4 102 193 5 1064 1257 6 920 2177 7 3576 5753 8 592 6345 9 216 6561 10 0 6561The second phase is harder. The number of positions is 24*8!^2/2 = 19,508,428,800, since it turns out that the permutation of the 3 unfixed edges on the E slice is completely determined by the permutation of the centres. This phase was solved with a BFS and took around 7 and a half hours to complete.

Phase 2 distribution:

Depth New Total 0 1 1 1 12 13 2 98 111 3 774 885 4 5909 6794 5 42535 49329 6 287933 337262 7 1814194 2151456 8 10497612 12649068 9 52654764 65303832 10 231184420 296488252 11 856533794 1153022046 12 2617727430 3770749476 13 5732703244 9503452720 14 7032541792 16535994512 15 2833285424 19369279936 16 139100392 19508380328 17 48472 19508428800 18 0 19508428800This gives an upper bound of 9+17 = 26. To reduce this to 25, notice that U and D moves do not affect the orientation of the gears, so that every optimal phase 1 solution will end with some rotation of the R or F face. Now we can check that every one of the 48472 depth 17 positions in <R3,F3,U,D> has a 17 move solution beginning with a rotation of the R or F face. It turns out that all of these positions have a large number of solutions, and in particular, they all have at least one solution beginning with R3. Each of these positions and solutions can be reflected in the plane through LB and RF to get another position which is also at depth 17, and a 17 move solution beginning with F3'. Hence all of these positions have a solution beginning with a rotation of the R or F face, and so we can always force a cancellation between the two phases whenever phase 2 requires 17 moves, giving an upper bound of 25 moves.

The cancellation between phases was Michael Gottlieb's idea, and all the computations were done by me.