How many moves are enough to say we really scrambled the cube.

[swipymam]

xyzzy said:
The WCA regulations state the scrambling requirements in terms of uniform distributions (4b3: "equal probability for each state"), but there is good reason to prefer to use the stationary distribution instead of the uniform distribution if these happen to differ: the stationary distribution is what you get in the limit if you scramble with random moves forever. Why should the scramble distribution be uniform if doing random moves doesn't tend to a uniform distribution?

Interesting conclusions from Rubik's cubes and Markov chains:
If a solved cube undergoes N random turns , what is the probability p(d|N) that an optimal solver needs precisely d moves to solve it back?

At depths 3 < d < 18 , on average, 9.34 of the 12 moves (QTM) will make you go away from the solved state, and 2.66 will make you go closer.
After N = 31 or 32 random moves, p(d|N) is pretty close to the stationary distribution.
Can anyone clarify how close to stationary the probability distribution of the official WCA scrambler - TNoodle, is?

 

[Lucas Garron]

Can anyone clarify how close to stationary the probability distribution of the official WCA scrambler - TNoodle, is?

They are 100% the same, apart from filtering out the solved state and 1-movers. TNoodle is fully random state (except for puzzles that are too large and for whom optimal solutions are prohibitive to calculate: 5x5x5, 6x6x6, 7x7x7, Megaminx).

People have done various studies on the distribution. I was recently contacted by someone researching this; I'll see if I can get them to share their results here.

 

[swipymam]

People have done various studies on the distribution. I was recently contacted by someone researching this; I'll see if I can get them to share their results here.

This will dispel many doubts.
After filtering, there is no point in looking for an optimal solution, it is enough to find a fast suboptimal QTM or Time Robot Metric - TRM (t), defined as follows: quarter turn of any number of faces located on one axis is counted as 1 turn.