One-Answer Puzzle Theory Question Thread

Discussion in 'One-Answer Question Forum' started by shadowslice e, Jan 3, 2016.

Welcome to the Speedsolving.com. You are currently viewing our boards as a guest which gives you limited access to join discussions and access our other features. By joining our free community of over 30,000 people, you will have access to post topics, communicate privately with other members (PM), respond to polls, upload content and access many other special features. Registration is fast, simple and absolutely free so please, join our community today!

If you have any problems with the registration process or your account login, please contact us and we'll help you get started. We look forward to seeing you on the forums!

Already a member? Login to stop seeing this message.
  1. Niklas is 7 moves. And the fastest way to swap two adjacent corners, with the block at the back left, is probably the JB PLL
     
  2. xyzzy

    xyzzy Member

    720
    300
    Dec 24, 2015
    An alternative is R2 D' r U2 r' D R U2 R (or one of its reflections/inverses). It's just the standard headlights OCLL but with some wide moves, and it's also optimal for swapping two edges while preserving the block.
     
  3. MethodNeutral

    MethodNeutral Member

    31
    1
    Jan 4, 2017
    That's perfect, thanks. I did have to mirror it because I don't normally use that one, but that works perfectly.

    Yeah, for Niklas I just meant until the 2-gen part it's 5 moves, should've been more specific. I would use the Jb PLL but I was pretty sure it wasn't optimal.
     
  4. MethodNeutral

    MethodNeutral Member

    31
    1
    Jan 4, 2017
    I wanted to find some algs for TTLL that are more ergonomic, but I have no idea how I can do that. Can someone point me to an alg-generator/cube solver that I can make solve the cube using, say, <RUF>?


    Sent from my iPhone using Tapatalk
     
  5. AlphaSheep

    AlphaSheep Member

    1,005
    503
    Nov 11, 2014
    Gauteng, South Africa
    WCA:
    2014GRAY03
    Cube Explorer
     
  6. MethodNeutral

    MethodNeutral Member

    31
    1
    Jan 4, 2017
  7. LexCubing

    LexCubing Member

    77
    3
    Aug 16, 2015
    First of all I'm confused how PLL is 22 if the formula is

    (4! x 4!)/2(4) = 72
    Those factorial represent the EP/CP and EO/CO is in only 1 state. 2 is the Permutation Parity and 4 is the AUFs.

    Also there are 7 OCLLs?
    (1 x 3^3)/8 = 7?

    57 OLLs?
    (3^3 x 2^3)/4 = 54?

    How do you calculate the # of algs if CP solved(not really solved it's just they're 2 gen) If CP is solved there are only 24 TTLLs and 84 ZBLL and just 4 PLL? The PLL is easy to solve but how do you solve it when you're trying to find out how many LSLL cases? If CP is solved* and the edges is and EO is solved how many cases? I don't know how to take in CP solved* calculations.
     
  8. xyzzy

    xyzzy Member

    720
    300
    Dec 24, 2015
    Short answer: for PLL, you're overcounting cases. For OLL and OCLL, you can't just blindly divide by AUF symmetry because you would undercount some cases, and don't forget about the skip case.

    Of the six TTLL subsets (each subset being 12 cases and not counting the set of PLLs), exactly one of them has the corners in the solved coset of the <R2,U> subgroup, and exactly one of them has the corners in the solved coset of the <F2,U> subgroup. These subsets are disjoint (i.e. have nothing in common), which is why you have 24 "2-gen" TTLLs (plus the 4 EPLLs).

    As for the last part of your question, you have to carefully define what you mean by CP first. Does a H-perm + U2 have solved CP by the definition you want to use?

    Long answer: Burnside's lemma.
     
    AlphaSheep likes this.
  9. AlphaSheep

    AlphaSheep Member

    1,005
    503
    Nov 11, 2014
    Gauteng, South Africa
    WCA:
    2014GRAY03
    Let's look at the easy one. OCLL has four twisted corners, but the twist of the last one is determined by the other three, so there are 3^3 = 27 cases. Some cases can have rotational symmetry - there is only only one case that has no twisted corners, and that can't be changed to any other case with an AUF. The H case has a different type of symmetry. There are only two H cases - one with the arms pointing to the front and back, and one with the arms pointing left and right. A U2 of an H case doesn't change the case, but a U of U' does. For the remaining cases, there is no rotational symmetry. For example the headlights OCLL has four distinct forms - headlights facing forwards, left, back or right.

    So the number of cases is
    (6 cases * 4 symmetries) + (1 case * 2 symmetries) + (1 case * 1 symmetry) = 27
    If you ignore symmetry, it is 6+1+1 = 8, and ignoring the OCLL skip case, it is 7 cases.

    With PLL, there are 72 cases, but skip, H, Na and Nb can't be transformed into other cases by AUFing. Z and E have U2 symmetry so there are two Z and two E perms that can be transformed into each other with a U move. The rest all have no symmetry, so a U move cycles between 4 distinct cases.

    That gives (16 * 4) + (2 * 2) + (4 * 1) = 72
    But ignoring symmetry, it's (16 + 2 + 4) = 22, or 21 of you exclude the skip case.

    This method of listing out all the cases and then counting symmetries works regardless of the restrictions on the cases, so you can still use it to work out the number of cases if CP is solved*

    * assuming that by CP solved you actually mean that the corners are such that they can be solved with R and U moves only. If you actually mean all of the corners are literally in their correct positions then there can't be any rotational symmetry which actually makes things easier.
     
  10. LexCubing

    LexCubing Member

    77
    3
    Aug 16, 2015
    If you read again I do mean by CP solved is the cases are now 2-gen. So there's no formula for working out symmetries? I need to work it brutally? So even if CP is solved*(2-gen) the # of cases are the same if I brutally list it out every cases nothing would change?

    Since I'm working on a LSLL method I want to know how to calc it properly. For ex. for TTLL
    The EO/CO is solved so they're 1.
    I can AUF the F2L corner to stay always at UBL and there's no more symmetry right?
    So the cases are (4! x 4!)/8 = 72?

    So for when CO is not solved? Assuming the F2L corner is at the U(AUFed at UBL always) layer:
    4! x 4! x 3^4?

    How do yo take in CP solved(2-gen)? Also since we're talking about LSLL we consider these cases where you rotate only by one y or y' (D/D') and the case is now 2-gen* also 2-gen like how Right Opp TTLLs can be solved.
     
    Last edited: Apr 17, 2017
  11. FJT97

    FJT97 Member

    359
    43
    Apr 3, 2013
    WCA:
    2012THIE01
    Hi
    A zzf2l question: I figured, that its more efficient to leave the f2l slot open after ive done a pair. But how many moves do I actually save on average? Is there a way to calculate that? To make the calculation easier, lets say that the next pair is always the pair of the same block and lets assume that I solve that pair optimally using <RUL>.

    And is there a difference of saved moves whether its about the first or the second block?
     
  12. xyzzy

    xyzzy Member

    720
    300
    Dec 24, 2015
    Conditioned on EO being solved, how does the move count distribution differ from the usual one? Is the maximum distance still 20 moves, or is it lower? (Let's say I still allow F and B quarter turns in solutions, so something like F U F' U F U2 F' counts as seven moves, not nine.)
     
  13. AlphaSheep

    AlphaSheep Member

    1,005
    503
    Nov 11, 2014
    Gauteng, South Africa
    WCA:
    2014GRAY03
    Interesting question. The distribution would take some effort to work out, but you get a simple yes/no answer to whether the maximum is still 20 moves by checking whether any of the 20 move states have solved EO. Here's the list of 20 move states.
     
  14. xyzzy

    xyzzy Member

    720
    300
    Dec 24, 2015
    I was hoping someone else had already done the work of checking, because I'm kind of lazy… (I have most of the code needed to check this except for string parsing, and I dislike string parsing because it mostly involves messy state machines and bleaurgh.)
     
  15. Roman

    Roman Member

    557
    260
    Jan 10, 2013
    Moscow, Russia
    WCA:
    2012STRA02
    YouTube:
    MrWormster
    What's the god number for the 3x3 OLL stage? What OLL case requires the most moves to solve?
     
  16. xyzzy

    xyzzy Member

    720
    300
    Dec 24, 2015
    12 moves, and the only OLL case that needs 12 moves is dot OLL with corners oriented (OLL 20).
     
    Roman likes this.

Share This Page