Message #544

From: lucas_awad <lucasawad@gmail.com>
Subject: Something interesting and strange about permutations
Date: Wed, 06 Aug 2008 18:19:02 -0000

After solving the MC5D, I have discovered something a bit strange
about permutations.

As everyone who read the solution for MC4D know, we can permutate the
4-color hypercubies by doing the 3-color series two times (one of them
the reverse).

But, why we cannot permutate the 3-color pieces with doing two times a
2-color permutation with 2 moves on MC2D?

Because the face rotation is different.

When rotating a "face" in MC2D, the move is like this:

1 2 3 –> 3 2 1

In 3D, the same movement should be:

1 2 3 –> 3 2 1
4 5 6 –> 6 5 4
7 8 9 –> 9 8 7

But that’s not what we really do with a rubik’s cube, it is this (it
would we for example U2, if it is "U" face):

1 2 3 –> 9 8 7
4 5 6 –> 6 5 4
7 8 9 –> 3 2 1

If you see, this algorythm (2-color permutation in MC2D) doesn’t only
do 4-6 permutation, also 2-8, which don’t happen in MC4D with 3-color
series.

By doing the previous movement (the unreal one) we only affect two
faces which change their stickers (the same as MC2D), but with a
rubik’s cube (and also MC4D and 5D) we are affecting 4 adjacent faces
(the other keep still the same stickers). So with the unreal movement
we would be able 3-color pieces by doing the sequence: ( F - R ) U ( R

• F ) U

However, in MC4D we do movements that only affect 4 faces, and that
allows us to easily permutate the 4-color hypercubies by doing the
3-color series algorythm. The fact I’m thinking now is if in MC4D and
MC5D all adjacent faces should be affected to make the rotation real,
and we are understanding higher dimensional puzzles wrongly.

I hope that you understand what I have said.

Greetings
Lucas