Wanted: 11-periodic optimal algorithm

Alsamoshelan

Member
Hello everyone,

I study the period of algorithms.

Period basics
Let A be an algorithm. A is "k-periodic" if A^k (A repetead k times) is the equivalent of doing nothing.
For instance, (RUR'U') is 12-periodic, 36-periodic, and its smallest period is 6.

What I'm looking for
I'm searching algorithms whose smallest period is 11 and whose height (HTM) is as small as possible.
Currently I found this one (with a program), its height is 10 :

D' L R' F U' R U' D F' L

But maybe there are smaller algorithms. That's why I need some help. If anybody find such an algorithm with a height <= 10, please let me know ! Important fact for the search
An algorithm whose smallest period is 11 must be a 11-edges cycle.

Thank you.

Alsamoshelan

Member
Ok, thank you, indeed intersting that he studied also 2^2 and 4^4.
So he seems to use STM (while I'm studying in HTM).
Interesting that he find "R L2 U' F' d" as an alg with a minimal period of 2520. In HTM the highest minimal period is 1260 (reached by R' B R' U L2 for instance).

Tao Yu

Member
Ben Whitmore found these just now using ksolve++:

U L U F D2 U2 F' R' B' D' (10 HTM)
U L U R D U' F' L' B' D' (10 HTM)

He said that these are the shortest algorithms that have a order of 11. He also said that there are probably thousands of algs like these (10 HTM with an order of 11).

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Alsamoshelan

Member
Thank you. Good to see my conjecture confirmed. Who is he ? I'd like to know how to proove that none of them has a height < 10.

Tao Yu

Member
Thank you. Good to see my conjecture confirmed. Who is he ? I'd like to know how to proove that none of them has a height < 10.
He's the writer of ksolve++ and told me that he used it to find those algorithms. You can probably reach him on reddit (he's not on this forum).

rokicki

Member
There are none of length less than 10. There are 17,760 of length 10 (canonical sequences; commuting moves have a
prescribed order). There are 194,496 of length 11 (canonical sequences). There are 2,355,600 of length 12.

The numbers increase geometrically from there.

These are pretty easy to find; corners must be solved, so you can use corners as a pruning table, and just find all
solutions to corners and check each for periodicity of 11.

-tom

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Alsamoshelan

Member
Thank you very much, very precise data!
so you can use corners as a pruning table, and just find all
solutions to corners and check each for periodicity of 11.
That's what I did to find my alg of length 10. However my program is not efficient, I think there are ways to easily filter algs preserving corners.

He's the writer of ksolve++ and told me that he used it to find those algorithms.
I didn't know this software. It seems to be very efficient.