# 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