Message #508

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] On positions of the non-full puzzles
Date: Fri, 09 May 2008 19:51:31 -0500

I changed the name of the email thread for this because my gmail
conversation view was getting out of control with "Magic120Cell Realized"
replies ;)

Anyway, along these lines, it is also interesting to think about the number
of possible puzzles having a given number of colors. There is only one
puzzle with 120 colors and one puzzle with a single color, but how many
different puzzles with 9 colors are there? An upper bound is 120P9 =
3.79E18, but that has multiple counts of visually identical ones (equivalent
after 4D rotations like you described). Since understanding the 4D view
transforms will be key for what you are looking at too, maybe some of your
investigations will help be able to give the final answer here. It does
sound quite difficult.

Naturally once the calc for that is done, we’ll have to wonder what the
total number of possible puzzles is (given the freedom to set any number and
pattern of repeat colors desired), but that will just the be the sum of the
answers for the 1…120 cases. Btw, I felt justified in having these extra
puzzles because there is a common version of Megaminx that has 6 colors
instead of 12 (opposite colors are the same). Hmmm, that just made me
realize I guess I didn’t include the most relevant variant. It’d be nice to
add at least one more puzzle then, a 60 colored puzzle where antipodal cells
are the same color.

Also, somewhat related and worthy of note is that the full version of
Hyperminx is unique because it has more colors (120) than stickers-per-cell
(63). Contrast this with all the other puzzles (I’ve watched many
frustratingly try to scramble my Rubik’s cube so well that no colors are
repeated on a face, which is of course impossible and enjoyable to see
someone discover). I wonder if it is possible to scramble the Hyperminx so
that every sticker on any given cell is a different color? I’m not sure.

Roice

On Thu, May 8, 2008 at 8:29 PM, David Smith <djs314djs314@yahoo.com> wrote:

> Roice, that was a great article! Some of those numbers make
> the number I found look like nothing! Thank you again for
> putting my result on your website.
>
> spel_werdz_rite, thank you for verifying this result! I had
> no idea anyone else had calculated this number.
>
> I recently had another idea for Magic120Cell before I go
> back to the n^4 cube. It seems like it will be very difficult,
> but I am going to try to find the number of visually different
> positions of each of the other variations of puzzles (the
> 2-colored, both 9-colored, and the 12-colored versions) of
> Magic120Cell. This will involve accounting for the similarly
> colored pieces (4-colored pieces with the same colors may not be
> visually identical due to their orientation, and counting the pieces
> will require the use of the Magic120Cell program), and the similarly
> colored centers (accounting for apparently different positions
> acctually being visually identical due to rotations of the entire
> puzzle in 4-space; the corner orientation logic would also apply
> to the centers for counting how many ways the they can be visually
> identical when rotated. This would be made eaiser by imagining the
> 0-colored piece that Roice mentioned.) These are just a few quick
> observations, there may be more complications I am not yet aware of.
>
> All the best,
> David
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