# Message #509

From: Jay Berkenbilt <ejb@ql.org>

Subject: Re: [MC4D] Magic120Cell Realized

Date: Sat, 10 May 2008 10:57:26 -0400

I have to add my voice to the rest in expression of awe at this

puzzle. It’s been years since I’ve even done mc4d – my life has

gotten busier. One day maybe I’ll try it, and I’m sure I’ll

eventually play around with it just to see what it feels like. As

with many of the other participants on this list, I have always had a

special affinity for the 120-cell. It always seemed to me that it

sort of snuck in to the regular polyhedron list, just barely fitting,

kind of like the pentagon just barely being able to be the face shape

of one of the platonic solids. :-) Do I recall correctly that this

polyhedron is its own dual?

> Honestly, the reason I wasn’t planning on working through a solution

> was that I am a bit scared of the sheer number of pieces! I just

> finished up the final parts that I felt were needed for it to be

> solvable today, and I actually haven’t even figured out a single

> sequence yet. So as of this evening, I only have the thoughts about

> it we’ve discussed in the past, which is that it will be easier in

> some ways than MC4D because of the larger space to sequester pieces,

> but that it will be a big effort in time. Also, I think I am ready

> for a bit of a rest and was too excited to share to let it sit on a

> shelf. Sarah will be happy to get my attention back now too since

> I’ve been spending a lot of time on it lately :)

My recollection of solving the megaminx is that you can do all but the

last few steps as localized solutions. Each twist affects such a

small number of pieces that the constraints don’t play a big role

until the end. It seems that each twist would necessarily alter

pieces on the 12 adjacent cells.

I don’t find it surprising that five random twists would result in

some interacting pieces. The first twist affects pieces on 12 of the

120 cells, not including the cell twisted. In order for the second

twist to not interact with any pieces, it must be on a cell that is

neither any of the 12 affected faces nor adjacent to any of them

(except that it could be another twist of the first face). I’m not

sure how many cells that is. If you managed to get one, there are

even fewer places for the third twist. It seems to me that the number

of twists after which there is some guaranteed interaction must be

very small….maybe three or four? I could probably work it out, but

I imagine others on this list could do it faster. My "math chops" may

be good compared to the general population, but not compared to many

of the readers of this list. :-)

Anyway, the 120 cell puzzle is a work of beauty!

–Jay