# 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

#### 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 and Keroma12

#### 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.