Why does parity occur

You cannot get parity on a 2x2 or 3x3 cube because it is designed like that. And I didn't mean that if 3x3 is designed in some other way it will get parity. So, 3x3 has 1 center while 2x2 has 0 and because the 3x3 has 1 center, the center has no other center spots to go to and create a parity, while on 2x2, there are no centers at all!

There are some blind memorization tricks to know the odd and even pieces. You can do many other things with that time. There are some memorization techniques which can be used to identify odd and evenness of the centre slices.

And 4x4 has many more pieces to solve: 24 centers, 8 corners, 24 edges, in total So, if the 4 x4 could be done equally efficiently as a 3x3, it would only take It's now about 18 seconds, so one could also say that the 4x4 is 1.

I went through and searched the term "parity" and found the threads that directly related to the parity on the 4x4x4. This parity is the same concept as for 5x5x5 and on the larger cubes. There is also a different kind of parity that you get when trying to solve as a 3x3x3 cube at the end of the reduction step on any even sized cube.

AJelsma Member. Joined Jan 20, Messages Parity can be explained very fancily but here is your questions 1. Parity is basically something on the cube or puzzle it just is the way the puzzle has been made all the same it occurs on a 3x3x3 in times like bld solving 2.

I don't really believe that nobody understands what they really do so why couldn't somebody just explain it to me?! I found some scattered information about parity on some sites and from what I understood its all about the center pieces which can move on the 4x4x4 and not on the 3x3x3. So I got the theory that these parity algs let some of the center pieces switch positions while solving the incorrect pieces.

So you do get the even number of swaps but you just can't see them because the centers look identical. Is this correct? Lux Aeterna Member. Joined Nov 25, Messages The term is borrowed from algebra abstract algebra, not the silly solve-for-x high school algebra sense of the word. Put very simply, the Rubik's cube is a permutation group. Elements of a permutation group i. So let's say you have a scrambled cube.

That position has a "unique" decomposition into various cycles, and if you apply all those cycles in the right order, the cube becomes solved.

Well, technically you're applying the inverse of those cycles, for obvious reasons, but that doesn't matter. Now, in that cyclic decomposition there's going to be some number of two element cycles, or transpositions, where two pieces just have to swap places with each other. Here's where parity comes in. If there's an even number of transpositions in the given state, we call that an even permutation, and if there's an odd number of transpositions in the given state, we call that an odd permutation.

That's all parity is, it's the sign of the permutation, even or odd. Now, on the 3x3 you can ONLY solve the cube if the parity is even. For example, if you swap two edge pieces in the last layer, that's unsolvable, but if you swap two pairs of edges, then you can solve it i. Z-perm or H-perm. In the picture, the blue-red edge on the left needs to be paired with the red-blue edge on the right, and the same for the blue-orange edges.

If you know how to solve a 4x4 , you will know the flipping algorithm. If you do this to a solved cube, you can see how the algorithm affects the rest of the puzzle, but this is not noticeable during edge pairing. Now solving this parity should be simple. All this algorithm does is slice along so that the other red-blue piece is above the orange-blue piece, flips the edge, then slices back, solving both edges. It is important to note that these parities can ONLY occur on even layered cubes 4x4, 6x6 etc.

This will be explained later. This image shows OLL parity in its purest form, but any state where there is an odd number of yellow edges facing upwards is the same. This is the indication that you have parity.

It is called so because it is first noticed during the OLL stage of a solve. When you get a two dedge swap or a 2 corner swap , you created a 3x3x3 edge formation whose parity doesn't match the parity of the corners. OLL parity occurring on the 4x4x4 strictly has to deal with how many inner slice quarter turns you used to solve the first three centers, and PLL parity occurring on the 4x4x4 strictly has to deal with how you paired up the last few dedges.

The key is that the four center cubules on each face of a Rubik's Revenge are indistinguishable. A picture cube wouldn't have this problem, and thus it would be much harder to solve. Although this is true for the single dedge flip OLL parity on the 5x5x5 and larger supercubes, it is not true for the 4x4x4 supercube: the parity of the wing edges on the 4x4x4 is independent of the parity of the X centers of the 4x4x4 supercube, and therefore you are ultimately incorrect.

There exists algorithms which flip an edge but do not distort any centers on the 4x4x4 supercube. For the two dedge swap PLL parity , there exists algorithms which do not affect centers on any size supercube, no exceptions, and therefore if your statement was referring to PLL parity regarding same color centers, it is clearly always wrong. Joe Z has it correct. Ultimately it's caused by the fact that some of the pieces are indistinguishable. I think tho he meant to say edge or center pieces not just centers only because of course in a 4x4 there are a bunch of edge pieces that look the same, not just the centers.

In order to create what cubers call a "parity error" you need to move around some of the indistinguishable pieces either the edges or the centers. Eric's answer is not totally correct. In other words, because the inner layer moves do an odd number of exchanges, it is possible on the 4x4 to exchange 2 edges and leave the rest intact.

As a side note, PLL parity is not really a parity issue at all. It only looks like it because we try to reduce the 4x4 to a 3x3.

Actually, PLL parity in the 4x4 is a simple exchange of 2 pairs, and we don't think twice about that in the 3x3. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why does the 4x4 Rubik's cube have parity cases, while the 3x3 does not? Ask Question. Asked 6 years, 11 months ago. Active 3 years, 7 months ago. Viewed 21k times. The first parity case that can occur is this one, where an edge is simply flipped upside down: The second one that can occur is this one, where two edges are swapped: Neither of these are valid positions on a 3x3 Rubik's cube.

If not, what makes this a legal position on a 4x4 cube, and not a 3x3 cube? Improve this question. Shokhet 1 1 silver badge 10 10 bronze badges. This specific question is in fact unique to the Rubik's Revenge.

I had a rubix cube question to but if I post I might start a rubix cube [Part ] riot lol! In the last part, we got to [this position]. How do we solve the sixth edge now? Remember to put your answer in spoilers! Show 3 more comments.