# Kilominx can be solved in 34 moves

Submitted by Ben Whitmore on Sun, 02/11/2018 - 12:58.

Last night, I found this thread on the speedsolving forums which proves an upper bound of 46 moves. First, the puzzle is separated into two halves, which takes at most 6 moves. Each half is then solved in at most 20 moves (= 7 moves for orientation + 13 moves for permutation, after orientation is solved), for a total of 6+2*(7+13) = 46. xyzzy writes

The ⟨U,R,F⟩ subgroup, while much smaller than G_0, is still pretty large, having 36 billion states. It's small enough that a full breadth-first search can be done if symmetry+antisymmetry reduction is used, but I will leave this for another time.I just completed this BFS. No symmetry reduction was necessary, just a standard BFS was used, and it took just over 11 hours to run.

Depth New Total 0 1 1 1 12 13 2 96 109 3 768 877 4 6144 7021 5 49152 56173 6 392364 448537 7 3117359 3565896 8 24649511 28215407 9 192551113 220766520 10 1438993775 1659760295 11 8766794158 10426554453 12 21419138541 31845692994 13 3866926287 35712619281 14 215919 35712835200 15 0 35712835200So the diameter of <U,R,F> is 14 which gives an upper bound on the full puzzle of 6+2*14 = 34.