Looping or Repeating Algorithms?

[k-punkt]

Your D moves are buggy. Look carefully at how the D face changes the first time you do L R D'.

Ok, seems there must be a serious bug in my cube program.
So everything I wrote was sort of nonsense.

Would be best all my posts are deleted. Sorry!

Edit:
At least my program is working now correctly. Seems 1260 is indeed the max number of repeats for an algorithm before it's running in a loop on a 3x3 cube and R U2 D' B D' is a legit example for this case.

 

[Lucas Garron]

An interesting question!

It's definitely possible to compute an exact answer by breaking down all the possible combinations of orbit cycles and tallying the possible orders ("order" is the group theory term for "loop length").

...

This gives me an average order of very roughly ≈120. But as you can see, it didn't even generate a single scramble with order 2 or 3, not to mention other possible orders. That's expected, but the wide range of values means simulations can vary quite bit.

Just saw this thread bumped, and realized I have more definitive answers since last year!

The exact average order for a 3x3x3 position with fixed centers is 116061927260324415253 / 961155628321996800 (≈120.75248)

For3x3x3 with moving centers, it's 47395084787297179857763 / 346016026195918848000 (≈136.973669)

https://github.com/lgarron/permutation-order-tallies has data for a few other puzzles.

 

[k-punkt]

The exact average order for a 3x3x3 position

What do you mean by order for position?

Isn't this thread about orders of algorithms? An algorithm may be of any length / number of rotations of cube faces. As I understand it the order tells how often you can repeat this algorithm until you're back at the starting position. This should be completely independent of the actual state of the cube.
Perhaps it's even independent of the length of the algorithm. But I'm not sure about that.

 

[kubesolver]

What do you mean by order for position?

Isn't this thread about orders of algorithms?

For the purposes of this thread algorithm is equivalent to a position generated by this algorithm from solved state.

 

[k-punkt]

Ah, ok!

But be aware such a position may be reached by using a lot of different algorithms.

There's no 1 to 1 relation between position and algorithm.

 

[ruffleduck]

Ah, ok!

But be aware such a position may be reached by using a lot of different algorithms.

There's no 1 to 1 relation between position and algorithm.

The difference does not matter: an algorithm and its corresponding position application are the same bijection on the permutation group of the cube.