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Some Maths about the Cubominx


Feb 24, 2017
I ordered a QiYi Ivy Leaf cube the other day. I was excited to have it because it seems identical to Tony Fisher's Cubominx puzzle, which I've wanted since 2010. It's based on the Skewb and acts pretty similarly to it mathematically, although it's much simpler. Anyway, it came in today, and my roommate and I solved it pretty easily. So I got to thinking about the maths.

Any move will cycle 3 centers and change a single corner's parity by 1. The only invariant for this cube is that the edge permutation always has an even number of transpositions. There are 4 corners with 3 possible orientations and fixed permutation and 6 centers whose orientations are not discernible and whose permutations have an even number of transpositions. By that logic, there are (3^4) * (6! / 2) = 29160 permutations.

A breadth first search found that this was the correct number of permutations and that the number of states with a given minimum distance from solved is:

n | distance(n)
0 | 1
1 | 8
2 | 48
3 | 288
4 | 1640
5 | 7582
6 | 15262
7 | 4221
8 | 110
Interestingly, over half of the permutations take exactly 6 moves to solve. The expected number of moves is about 5.74, optimally.

I noticed that the centers can have orientations if you give them arrows like a supercube. I might find numbers about that later.