Message #520

From: "noel.chalmers" <ltd.dv8r@gmail.com>
Subject: On Higher-Dimensional Parity
Date: Fri, 23 May 2008 16:55:22 -0000

Explaining parity is no simple task but here goes.

To understand what is meant by parity I must first describe how it is
encountered.

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. 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.
For example, a 4^4 has 4 1-colour, blue cubies. 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. Doing
the same with 2-, 3-, and 4-colour cubies you can simplify the puzzle
down to a familiar 3x3 puzzle.

This however is not the end of the story for puzzles like the 4x4 and
6x6 and other even-sized puzzles. Because of their movable centers, a
phenomenon called parity can occur. 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. For
example, in 3D, it is impossible to flip a single edge and leave the
rest of the cube unchanged. However, a 4x4, when reduced to a 3x3, can
have a single edge flipped. This is but one possible example of parity.

What is surprising is that parity does not affect corner pieces of any
puzzle. 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.

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.

You will only be able to recognize that your solving has encountered<br> an X-colour parity when almost all the X-coloured pieces have already<br> been placed. When I encountered parities in 5D I was forced to make up<br> a parity-correcting sequence on the spot. While they were effective,<br> they were inefficient and left much of the already placed pieces<br> scrambled. This made the solving process very frustrating at times. <br> In 3D, there exist complex sequences that will correct parities and<br> leave the rest of the cube unscrambled. I was not lucky enough to<br> think up a sequence in 4D or 5D that would do the same but I'm<br> confident it could be done and it would relieve much of the<br> frustrations of solving the 4x4 puzzles if they were discovered.

I hope this answers everyone’s questions about higher dimensional
parities. 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. Sometimes I think I write too much, lol. I feel like I could
write a book on this but, then again, who would read it? :p