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Hi, I'm new to the group, but I have read the entire archive. I noticed

rather little work done on Square 1. It seems to me that this puzzle

deserves a closer look for finding God's algorithm. Mike Reid's

calculations notwithstanding (archive 17), I have found that the problem

can be reduced by at least a factor of 400 if we just get rid of

combinations that result from trivial face turns, and if we note that

the Start position has a degeneracy of 16. (One center slice is assumed

fixed - another factor of 2 is tempting but not possible)

Mike's calcualtion for the number of states would reduce to: (2*(1/6)*(9/2)+2*(28/3)*3+(35/4)*(35/4))*2*8!*8!=435891456000 combinations. Divide by the start degeneracy, multiply by 2 storage bits per state, and you get a storage requirement of 6.81GB. This seems very close to being doable. Maybe in another 10 years, I can do this project on my PC, if no one has done it yet.

On another note, when I signed up, I mentioned to Alan that I must be

crazy enough to join this group since I have a five foot mockup of a

rubik's type puzzle as my coffee table. He thought its description

might be of general interest. Skip the rest of this paragraph if you

couldn't care less about its origins. I built it for Caltech's ditch day event

Maybe you have heard of it. That's where all the seniors leave for the

day with their room locked only with a puzzle of some sort, and the

object is for the undergraduates to get into the room by solving it

(with a couple of clues, of course). Anyway, being as it was that I

had a mechanical engineer roommate... The rest is history, and I now

have a five foot diameter puzzle coffee table.

OK, a description. The puzzle is a three centered version of the

Puzzler, widely available in the last few years in puzzle/game specialty

stores. The differences being that it is colored so that the maximum

number of combinatins are possible (including the supergroup of distinguishing

face centers). For those of you who have not seen the Puzzler, and thus

have no frame of reference, consider one vertex of a cube and it's

surrounding faces. 7 vertices, 9 edges. Faces can undergo 4 quarter

turns. Extrapolate to the Megaminx and you again get one central vertex

for a total of 10 vertices and 12 edges. Faces can undergo 1/5 turns.

Extrapolate again to six sided faces, and you get a flat puzzle with one

central vertex for a total of 13 vertices, and 15 edges. Faces can

undergo 1/6 turns.

So it is basically the <URF> group for a' cube' with

hexagonal faces. The extra face over the Puzzler also serves to remove

the significant parity constraints on the edge pieces. (compare <UR>

group to <URF> group of regular cube).

You, too, can make a smaller version of the Ditch Day puzzle at home.

The advantage of the flat puzzle is that it is easily constructed. I

built the 6 inch diameter prototype with poster board, lamination, magic

markers, and an easily machined smooth pressboard frame. You only need

to drill three 3 inch holes. The rest is trimming. Oh yeah, you will

need a plexiglass faceplate to keep the pieces in too. Cutting out and

gluing together the poster board to make sufficiently thick pieces was

the hardest part.

Number of combinations = (13!*15!*3^13*2^15*6^3)/24 = 3.83E33 Difficulty is comparable with Megaminx.

Happy cubing. This is already too long.

mikem.

Mike Masonjones.