![]() I was not lucky enough to > think up a sequence in 4D or 5D that would do the same but I'm > confident it could be done and it would relieve much of the > frustrations of solving the 4x4 puzzles if they were discovered. > In 3D, there exist complex sequences that will correct parities and > leave the rest of the cube unscrambled. ![]() In 4D, switch two 2-colour cubies is the same parity error as switching two 3-colour cubies, as a quarter-turn of a cell transfers one parity error on the other. In "noel.chalmers" wrote: > The possible parities are as follows: > In 3D: > Single edge flipped > Two edges switched > In 4D: > Single 2-colour cubie flipped > Two 2-colour cubies switched > Single 3-colour cubie flipped > Two 3-colour cubies switched In fact, in 3D, we can also switch two corners, but this is the same as switching to edges, since a quarter-turn of a face transfers one parity error on the other. I feel like I could write a book on this but, then again, who would read it? :p=20įrom: "thibaut.kirchner" Date: Wed, 13:47:48 -0000 Subject: Re: On Higher-Dimensional Parity That being said, I would be happy to answer questions and hear people's thoughts as to the exact causes of parity (and ways to avoid it) ) Cheers, Noel P.S. I hope this answers everyone's questions about higher dimensional parities. I was not lucky enough to think up a sequence in 4D or 5D that would do the same but I'm confident it could be done and it would relieve much of the frustrations of solving the 4x4 puzzles if they were discovered. This made the solving process very frustrating at times.=20 In 3D, there exist complex sequences that will correct parities and leave the rest of the cube unscrambled. While they were effective, they were inefficient and left much of the already placed pieces scrambled. When I encountered parities in 5D I was forced to make up a parity-correcting sequence on the spot. You will only be able to recognize that your solving has encountered an X-colour parity when almost all the X-coloured pieces have already been placed. The possible parities are as follows: In 3D: Single edge flipped Two edges switched In 4D: Single 2-colour cubie flipped Two 2-colour cubies switched Single 3-colour cubie flipped Two 3-colour cubies switched And so on for 5D. ![]() Meaning, in 3D the 3-colour pieces will not be affected by parity, in 4D the 4-colour cubies will not be affected, and similarly in 5D for the 5-colour cubies. What is surprising is that parity does not affect corner pieces of any puzzle. This is but one possible example of parity. However, a 4x4, when reduced to a 3x3, can have a single edge flipped. For example, in 3D, it is impossible to flip a single edge and leave the rest of the cube unchanged. The simplest definition of parity is: When the puzzle is simplified to a 3x3, it will have configurations that are normally impossible in a standard 3x3. Because of their movable centers, a phenomenon called parity can occur. This however is not the end of the story for puzzles like the 4x4 and 6圆 and other even-sized puzzles. Doing the same with 2-, 3-, and 4-colour cubies you can simplify the puzzle down to a familiar 3x3 puzzle. When all of these are positioned together they will behave exactly like a single 1-colour blue cubie as in a 3x3 when only the outside layers are turned. For example, a 4^4 has 4 1-colour, blue cubies. To do this you must match cubies of the same colours together into groups that will act as single piece when only the outer layers are turned. To understand what is meant by parity I must first describe how it is encountered.=20 When solving the puzzles larger than the 3x3 variety I first seek to align the cube in such a way as to make it into a 3x3 puzzle. ![]() Thread: "On Higher-Dimensional Parity" From: "noel.chalmers" Date: Fri, 16:55:22 -0000 Subject: On Higher-Dimensional ParityĮxplaining parity is no simple task but here goes.
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