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12/17/17, 6(02 PM4x4 Cube Twisty Puzzle - Parity Cases
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Parity on the4x4 Rubik’sCubeParity is something that mostpuzzle solvers despise. Extraalgorithms that you have tolearn in order to solve the cube100% of the time. It is somethingthat can slow speedsolvers
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12/17/17, 6(02 PM4x4 Cube Twisty Puzzle - Parity Cases
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a a
b b
Edge parity
down immensely in officialsolves (https://ruwix.com/the-rubiks-cube/rubiks-cube-competitions-wca/), and isgenerally seen as a pain to dealwith. Today I’m going to take alook at parity in all its forms andsee why it happens and thereasons why it’s impossible toprevent completely.
Where Parity Occursand how to solve it.One type ofpseudo-parity isedgeparity. Forevenlayeredcubes, this isn’t technically a parityat all, but it still requires an extra“algorithm” to solve, even though itis very short and just involvesusing the same moves you do tosolve the rest of the edges, but ina different order.
It happens when all but two edgesare solved. These two edges lookthe same, but inverted. In thepicture, the blue-red edge on the
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12/17/17, 6(02 PM4x4 Cube Twisty Puzzle - Parity Cases
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left needs to be paired with thered-blue edge on the right, and thesame for the blue-orange edges.
If you know how to solve a 4x4(https://ruwix.com/twisty-puzzles/4x4x4-rubiks-cube-rubiks-revenge/), you will know theflipping algorithm. This is used toflip an edge during edge solving(this algorithm can’t be used forOLL (Orient Last Layer) paritybecause it affects other parts ofthe cube that aren’t necessary topreserve during edge pairing suchas corner orientation and thepositions of the edges on top. Theflipping algorithm is as follows – RU R’ F R’ F’ R. If you do this to asolved cube, you can see how thealgorithm affects the rest of thepuzzle, but this is not noticeableduring edge pairing.
Now solving this parity should besimple. If you think about what thealgorithm is doing, it’s actuallyquite simple – Uw’ R U R’ F R’ F’R Uw.
All this algorithm(https://ruwix.com/the-rubiks-cube/algorithm/) does is slicealong so that the other red-blue
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12/17/17, 6(02 PM4x4 Cube Twisty Puzzle - Parity Cases
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piece is above the orange-bluepiece, flips the edge, then slicesback, solving both edges.
The main types of Parity are OLLand PLL Parity for NxN twistypuzzles (https://ruwix.com/twisty-puzzles/big-cubes-nxnxn-solution/). It is important to notethat these parities can ONLY occuron even layered cubes (4x4, 6x6etc.) and never on odd layeredcubes (3x3, 5x5(https://ruwix.com/twisty-puzzles/5x5x5-rubiks-cube-professors-cube/) etc.). This will beexplained later.
OLL ParityOLL Parityis a
speedsolver’s(https://ruwix.com/the-rubiks-cube/speedcubing-speedsolving/)worst nightmare. PLL Parity canalso be a pain, but nothing incomparison to OLL parity. Thisimage shows OLL parity in itspurest form, but any state where
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12/17/17, 6(02 PM4x4 Cube Twisty Puzzle - Parity Cases
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there is an odd number of yellowedges facing upwards is the same.This is the indication that you haveparity. It is called so because it isfirst noticed during the OLL stageof a solve.
It is caused when solving theedges. You see, on a 3x3 puzzle(https://ruwix.com/the-rubiks-cube/how-to-solve-the-rubiks-cube-beginners-method/) each ofthe edges are basically “made” foryou. They can’t be split in two,therefore they can’t be “flipped”.However, when solving edges on a4x4, you have no way of knowingwhether an edge is “solved” or“flipped”. This means when youhave reduced the cube to a 3x3with the reduction method,although it may just look like ascrambled 3x3, picture it as ascrambled 3x3 after one edge hasbeen removed and put back in thewrong way around.
Because there is no way of tellingwhich way round an edge shouldgo, it’s really a 50-50 chance as towhether or not the number of“flipped” edges you will end upwith after edge solving is odd oreven. An even number of flippededges means there is no OLL
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parity. If you want to see foryourself, take out two edges froma 3x3 and put them back in theirplaces but flipped. Now scramblethe cube, and you’ll find that youcan solve it. An odd number offlipped edges means that there isOLL parity. Do the same thing:take a 3x3 cube and this time flipthree edges in their places. Nomatter what you do, there willalways be one flipped edgeremaining when you try to solve it.
This is combined with the fact thatyou have to solve the centres,meaning they can all be in differentplaces. When solving the first twocentres, you can solve them onany side, as long as they areopposite one another. Thereforethere is absolutely no way of tellingwhether an edge is solvedcorrectly or not, meaning OLLparity is completely random.
There are many algorithms thatare all unique apart from one thing– They are all very long anddifficult to memorise. Manyspeedcubers try a few differentalgorithms before they find onethey’re able to solve decently fast.However, many speedcubers alsouse muscle memory for most of
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the algorithm memorisation,meaning they won’t be able toperform the same algorithm slowlywithout making a mistake. Evenwith the best memory and the bestalgorithm, most speedcubers willstruggle to solve OLL (/the-rubiks-cube/advanced-cfop-fridrich/orient-the-last-layer-oll/) parity in under 3-4 seconds, meaning that it can bethe cause of a ruined average.OLL parity is also known for thelarge risk of pieces popping fromthe puzzle during execution.Because it involves almostexclusively slice turns (turns thatwould be impossible on a 3x3) andis executed as fast as possible, apop during this stage can be muchlarger than a normal pop, andcombined with the strong relianceon muscle memory that mostspeedcubers have, this canprevent the solver from completingthe rest of the algorithm withouterror when the cube is put backtogether (https://ruwix.com/the-rubiks-cube/how-take-apart-disassemble-the-rubiks-cube-and-put-back-together/).
PLL Parity
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PLL (Permute Last Layers) Parityis like the final nail in the coffin fora speedcuber who has alreadyhad to deal with OLL parity.Although it is much easier toexecute and the algorithms areeasier to remember, the time ittakes to execute combined withthe time of OLL parity can add 6-7seconds total to an averagespeedcuber’s solve time(https://ruwix.com/online-rubiks-stopwatch-timer/).
These images show PLL parity intheir pure forms. However, mostspeedcubers solve PLL(https://ruwix.com/the-rubiks-cube/advanced-cfop-fridrich/permutate-the-last-layer-pll/) parity as soon as theyrecognise it, which is when OLLhas been completed and any OLLparity has been dealt with. Thebest way to spot PLL parity iswhen you see a PLL case that isimpossible to get on a 3x3 cube.One of the best examples for thisis the pseudo-T permutation. Afterfinishing OLL, you will notice thetwo bars that indicate a Tpermutation. But instead of having
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headlights on the side that thebars point towards, you will have asolved bar.
The main algorithm people learnfor PLL parity swaps the two edgesshown in the first picture. This iscalled the Half-H perm case. Theother case is the Half-Z permcase, and the same algorithm isused but with a setup beforehand.The setup moves the edge on theright so that it is opposite the edgeon the front, by doing R2 D’ andthen rotating so that the F face isnow the U face, then performingthe same algorithm.
It occurs for a similar reason toOLL parity. During edge pairing itis impossible to tell whether edges
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are swapped, therefore having twoswapped edges results in someform of PLL parity.
These parities can happen on anyeven NxN layered cube. For a 4x4(/twisty-puzzles/4x4x4-rubiks-cube-rubiks-revenge/) there aretwo edges that need pairing, butwith a 6x6 there are 4. Paritiesoccur, but they are exactly thesame as 4x4 parity, they just lookbigger.
I won’t provide specific algorithmsfor OLL and PLL parity herebecause there are so many thatare widely used. If you want to finda list of these, check out this page(https://www.speedsolving.com/wiki/index.php/4x4x4_Parity_Algorithms)for a full list of all 4x4 Parityalgorithms.
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