# Wanted: 11-periodic optimal algorithm



## Alsamoshelan (Aug 29, 2018)

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.


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## Tao Yu (Aug 29, 2018)

You might find this interesting: http://mzrg.com/rubik/orders.shtml

U B E F' R' B' E2 F y' is listed as an alg with an order of 11


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## Alsamoshelan (Aug 29, 2018)

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).


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## Tao Yu (Aug 29, 2018)

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 (Aug 29, 2018)

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.


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## Tao Yu (Aug 29, 2018)

Alsamoshelan said:


> 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).


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## rokicki (Aug 30, 2018)

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 (Aug 30, 2018)

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.


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