Message #612

From: David Smith <djs314djs314@yahoo.com>
Subject: A Few Updates
Date: Tue, 06 Jan 2009 22:18:04 -0000

Hello everyone,

First of all I would like to be another person to congratulate
Noel on solving Magic120Cell! It must have taken a lot of patience
and dedication. I must admit I considered taking a shot at it,
but then rechecked the number of pieces and decided to leave it
to more capable hands! :) Good job! Also, my thanks go to Roice
for making that great video of Noel’s solution. It clearly shows
how much time and thought were required on your part, Noel.

Also, I would like to welcome Chris to the group. Thanks a lot
for sharing so much about yourself; you sound very friendly and
I am sure you will be a great addition to the group. I am
fascinated by your physics and math background, which is similar
to mine except that I am just starting school. :) I would love to
talk to you more about these and other things, would you mind if
I send you an email?

I have a few corrections in my work to announce. In my last post
regarding which cubes are theoretically constructable in various
dimensions, Roice discovered that my second equation is incorrect.
At the time, I realized my mistake and obtained a correct formula,
but I have since forgotten it. If it is of importance to anyone
I will re-derive it, but I didn’t really focus on it too much.

Also, about a month or two ago I was quite embarrassed to discover
a mistake in my formula for the number of permutations of an
n^4 Rubik’s Cube, which I had thought to be correct for a long
time. The correct version is here
<http://www.gravitation3d.com/david/n%5E4_Cube.pdf> , the mistake was
in the
denominator of the fifth term, the first one with a numerator
of 192!. Another mistake was to be found in the permutation
counts of MagicCube5D, which can be found on Roice’s website
here <http://www.gravitation3d.com/magiccube5d/permutations.html> .
These were corrected shortly after they were put up.
Thanks again to Roice for hosting my papers and results! :)

About a month ago I spent a week on a general formula for
the number of permutations of an n^d Rubik’s Cube. I made
great progress, but the formula was highly convoluted, consisting
of over 20 equations and riddled with absolute value and floor
functions, and other special cases. I took a break, and have not
had the interest to return to it. During this time, I have had a
few ideas which could simplify things, but I have been busy with
other projects. I have a feeling there might be a simple group-
theoretical way to describe such a formula, but I do not have
the required knowledge or familiarity in group theory to try to
find it. I may return to it eventually, but I have been considering
that if I can only come up with such an extremely complex set of
equations, which take into account many special cases and are
themselves recursive in nature, then perhaps I should leave this
problem to others who can do it more justice. If I do continue
work on higher-dimensional Rubik’s Cubes, it would probably
be on a general formula for 5D cubes.

I hope everyone had a great holiday, and best wishes to all for
the new year.

David