# Cycling algorithms

Discussion in 'Puzzle Theory' started by shadowslice e, Jan 6, 2017.

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1. ### shadowslice eMember

Are there any algorithms that cycle in a prime number greater than 12 that do not have rotations or wide moves in them? (eg R,y would be R B L F etc).

2. ### Cale SMember

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I remember someone had a list of all possible orders of algorithms

3. ### xyzzyMember

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Dec 24, 2015
It's not possible. The order of the cube group is 2^27 × 3^14 × 5^3 × 7^2 × 11, so the largest prime order of any element is 11. (It doesn't matter whether you allow rotations or not.)

On big cubes you can get order-23 elements but not any larger, for the same reason. (Larger as in larger primes; 29, 31, 37, 43, etc. are all illegal on big cubes, but you can get order 24, 26, 28, 30, 33, 34, etc.)

shadowslice e likes this.
4. ### shadowslice eMember

Well I didn't go into that really. My logic was that you could have a 2 swap of c/e, 3 swap of c/e... 8swap c/e,9 swap e... up to 12 and you could flip but that you only take 2 and you could have a 3 for twist but (but as said before 3<12) so the largest prime would be 11 and all else would be combinations of the cycles.

I guess it would be the same on big cubes but you could have up to a 24-cycle of outer corner and edge centres (i forgot what they're called though) so 23 would be the biggest.

I guess you could have up to a 29 (prime) swap for a megaminx then and a 59 (prime) swap for a gigaminx or bigger minx.

Last edited: Jan 6, 2017
5. ### ChreeMember

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Now I'm all curious... what are the known cases for prime cycles so far? Can we get example algs of lower primes?

# of Cycles - Alg
2 - T Perm
3 - U perm
5 - (R U R' U)
7 - ???
11 - ???
13 - ???

6. ### shadowslice eMember

Well, for a start they can only affect either only corners or only edges or the same number of both (so only 7 out of the algs you've listed would have both corners and edges at the same time).
R U R' U' cycles in 6. You could try a 5 cycle of corners or edges (I don't know any off the top of my head).
Same goes for 7 cycle of corner/edges
11 or 13 cycle of edges works too.

7. ### xyzzyMember

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Dec 24, 2015
From http://www.jaapsch.net/puzzles/subgroup.htm:

7 - (U R U' F)2
11 - (U R' L F D2)2

On 4x4x4:

13 - (U D 2L2 2R' U D 2R 2L U 2R2 U 2L 2R D2 2L D2 U 2L2 U 2L D 2R 2L D U2 2L' D2 U 2L D 2R 2L U 2L D2 2R D 2L 2R D2 U 2L U 2R D2 U')4
(randomly generated lol)