> Intuitively, it isn't obvious to me why this would be so, but it's nerd sniped me into thinking about how combinations work in general and the geometry of the

> Intuitively, it isn't obvious to me why this would be so, but it's nerd sniped me into thinking about how combinations work in general and the geometry of the Rubik's cube particularly. My guess would be that this can be hand-waved by looking at it like a sort of discrete space. Then a solved cube is a single point in that space (let's assume we already normalize for orientation etc.). The moves we can make on a cube connect each point in the space to a set of adjacent points. The shortest path from A to B would be the space's metric. Now we know that the maximum distance from the origin / solution point is 20. Let's call all the points with a given distance n S_n (like "stratum"). To explain the distribution we only need to look at paths that either go from S_n to S_n+1 or to S_n-1, because "lateral moves" are never relevant to the shortest/optimal paths. So for small n the number of points grows, because each stratum we go away from the solution gives us more possible moves to go even further away. For S_20 there are no moves that go further away, for S_19 each combination has at most one move that goes further away etc., so essentially I think the limitation of the path length at the edge reduces the possible combinations you can get in those stratums, i.e. close to the center a random move will bring you farther away, so there has to be more points farther away, but when you are close to the edge, a random move either doesn't move you at all (stratum-wise) or closer to the center, so there have to be fewer points as you approach the edge. The really interesting thing however is how these two balance each other to place most points in S_17 and S_18.