Timeline for How to tell if a Rubik's cube is solvable?
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
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| Jul 10, 2022 at 13:03 | comment | added | Paul Sinclair | For Edge Parity, I think a better way of defining the "correct orientation" is that a cubie is correctly oriented if either both of its sides agree with the adjacent centers, or neither side agrees with the adjacent face. It is wrongly oriented when only one side agrees with the adjacent center. The advantage of this definition is that in all cases, a quarter turn of a face will flip all four edges, making it easy to see that the count of correctly oriented faces must always be even. | |
| Jul 10, 2022 at 12:58 | comment | added | Paul Sinclair | @VoidStar - Henning never said anything about counting "out-of-place" cubies for Permutation parity. The "standard method" of computing the sign of a permutation referenced is to count how many swaps of two items in the permutation it would take to completely undo it. If that count is odd, it is an odd permutation, if it is even it is an even permutation. | |
| Nov 2, 2021 at 2:24 | comment | added | VoidStar | There may be a problem with your "Permutation parity" method. To demonstrate, swap YR and WR such that edge parity is maintained. As this is 1 swap (odd # of swaps), it should fail the permutation parity test. Yet using your hints, since the distance is 2 for 2 out-of-place cubies, it cannot fail. Your hint of "we temporarily forget the difference between opposing colors" implies similar confusion as opposing face swaps DO matter. I tried this on several online cube solver and not one of them has detected this properly, so it seems messing this up is common. Any tips? | |
| Nov 13, 2017 at 9:29 | history | edited | hmakholm left over Monica | CC BY-SA 3.0 | edited body |
| S Nov 13, 2017 at 9:27 | history | rollback | hmakholm left over Monica | Rollback to Revision 4 - Edit approval overridden by post owner or moderator | |
| Nov 13, 2017 at 7:03 | history | suggested | CommunityBot | CC BY-SA 3.0 | fixed syntax and spelling |
| Nov 13, 2017 at 6:43 | review | Suggested edits | |||
| S Nov 13, 2017 at 9:27 | |||||
| Mar 5, 2017 at 16:14 | history | edited | hmakholm left over Monica | CC BY-SA 3.0 | deleted 121 characters in body |
| S Mar 5, 2017 at 14:43 | history | suggested | CommunityBot | CC BY-SA 3.0 | wrong figure |
| Mar 5, 2017 at 14:19 | review | Suggested edits | |||
| S Mar 5, 2017 at 14:43 | |||||
| Jan 16, 2016 at 15:00 | comment | added | Soham | @HenningMakholm Sorry to disturb.....I had not read the word dismantled cube.... | |
| Apr 5, 2012 at 9:45 | vote | accept | Mel | ||
| Apr 4, 2012 at 16:30 | comment | added | hmakholm left over Monica | @Mel: When speaking about permutations, "even" is a technical term that is not equivalent to "divisible by 2" -- see parity of a permutation on Wikipedia. Swapping just two pieces is simplest example of an odd permutation. You're supposed (in one of the many equivalent ways to think about it) to be counting the number of swaps, not the number of pieces affected. The permutation is even if it can be created using an even number of swaps. | |
| Apr 4, 2012 at 7:24 | comment | added | Mel | Thank you for explaining in plain english! I'm just a little confused with the permutation/positional parity. I am supposed to compute for the sum and it should be divisible by four right? (not just even). If I swap 2 adjacent edges or corners, that would be an even permutation right? but the cube wont be solvable. | |
| Apr 3, 2012 at 18:05 | comment | added | hmakholm left over Monica | @Jyriki: Agree about parity but couldn't think of a better word. Triplity? The 4×4×4 and 5×5×5 properties follow because I can transpose two edge cubies (outer edges on the 5×5×5) without disturbing centers or corners, which together with conjugation generates the entire $S_{24}$ of edge pieces. The centers themselves are easily solved from any starting point because the pieces are interchangeable in groups of 4. | |
| Apr 3, 2012 at 17:49 | comment | added | Jyrki Lahtonen | +1 Well done, Henning. AFAICT I used exactly the same method to describe corner parity on my Rubik's cube page (in Finnish only, so I'm not gonna bother with the link - the page is old anyway :-). The word parity suggests to me that the check is two-valued, whereas here it is 3-valued. Doesn't matter! Interesting stuff about 4x4x4 and higher!!! I didn't know that these were the only checks (though I once had a working 4x4x4 and a 5x5x5, but they were not very durable). | |
| Apr 3, 2012 at 15:18 | history | edited | hmakholm left over Monica | CC BY-SA 3.0 | huh? where did this half of the sentence go? |
| Apr 3, 2012 at 15:07 | comment | added | hmakholm left over Monica | I didn't know that, but did notice it in your answer (somehow I didn't get a new-answers alert while typing, or I would have been considerably less verbose here). | |
| Apr 3, 2012 at 15:04 | comment | added | joriki | The regularity in the assignments you describe is useful for a computer evaluation but not required mathematically; any assignments from $\mathbb Z_2$ to edge pieces and edge slots and any consistently oriented assignments from $\mathbb Z_3$ to corner pieces and corner slots will do. | |
| Apr 3, 2012 at 14:53 | history | answered | hmakholm left over Monica | CC BY-SA 3.0 |