Cycling algorithms

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

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  1. 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 S

    Cale S Member

    Jan 18, 2014
    Iowa, USA
    I remember someone had a list of all possible orders of algorithms
  3. xyzzy

    xyzzy Member

    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. 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. Chree

    Chree Member

    Jun 7, 2013
    Portland, OR, USA
    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. 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. xyzzy

    xyzzy Member

    Dec 24, 2015

    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)

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