You have been down voted, but I think this is a reasonable point and a reasonable question. It seems to me that the method of iterated transformation you sugge

You have been down voted, but I think this is a reasonable point and a reasonable question. It seems to me that the method of iterated transformation you suggest will indeed yield a random draw. My analogy is to Markov chain Monte Carlo. It is well known that MCMC on a discrete space will converge to the desired distribution (we’ll get to that in a second) if the chain is irreducible and aperiodic. Irreducible means any state can eventually transition to any other state using the permutations allowed. This is easy to validate if we just use the various face rotations. Aperiodic essentially means that there is no fixed “loop” from one state back to itself that occurs with probability one. (The even parity issue that @someone raised comes in here. An absolute fix in the lack of full understanding of all possible such issues is to allow null moves with some low probability.). For a simple state space like this, the main thing is to make the permutations chosen in such a way that all states are reachable. OK, iterated permutations converge in distribution, but to what? All the states in the target distribution here have equal probability, so (in MCMC terms) the posterior is totally flat: P(s) / P(s’) = 1 for all states s, s’. This means that a reversible Metropolis-Hastings sampler that just accepts every proposed move will converge to a flat posterior. OK, so we just need to make the permutations we propose reversible. What is reversible? It means that, if you are at state s, you are just as likely to go to s’ as if you are at s’ and going to s. [1] The easy way to ensure this is to make “clockwise twist of left face” chosen with the same probability as “counterclockwise twist of left face”. This is the obvious strategy anyway, which is why reversible samplers are popular. So in sum, the recommended algorithm is to start from any legal state and perform “N” permutations randomly chosen from clockwise/counterclockwise (50/50 chance) rotations of a randomly chosen face. To accomplish the null moves, just subtract a small random number from N. On a discrete state space, you will get exponentially fast convergence to the desired distribution. The space is strongly connected, because you can get from any state to any other state with at most 40 moves. I don’t see the problem here. [1] https://en.m.wikipedia.org/wiki/Metropolis–Hastings_algorithm#Formal_derivation https://en.m.wikipedia.org/wiki/Metropolis–Hastings_algorith..., A(x, x’) = 1.