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Tanya Sienko is visiting me, and she says that the cube is the

craze of Japan. She also presented me with a new toy, given to

her by some Japanese. (I don't know if is in this counrty --

yet.)

The thing is shaped like a barrel mounted on a supporting

structure. The barrel can move one UNIT up or down in the

structure. Around the circumference of the barrel there are five

equally distributed columns. Two of the columns have four rows,

and three of them have five. The ones with five have a plunger on

the associated part of both the top and bottom (or left and

right) parts of the supporting structure. Two plungers are next

to each other, and the third is opposite their midpoint. There

are 23 balls in the device: four each of green, yellow, blue,

red, orange (one for each column) and three black balls. (in a

minute you will see where these black balls go). The barrel is

divided into four parts. The left- and right-most parts are fixed

with respect to the supporting structure. Each has three cavities

either to hold a ball or one of the plungers. The barrel moves,

so either the left has balls in the cavity and the right has the

plungers, or vice versa. The middle two sections of the barrel

have two cavities in each row, and these rotate around the

circumference, taking balls with them.

I have been trying to say left and right, because I think the

corect way to thing of this devices is as follows: Hold it

horizontally, with the barrel centered in the supporting

structure. This means that each plunger is half way into its

cavity. A MOVE consists of moving the barrel one half unit right

or left, then moving one of the rotating middle sections forward

or backward one unit, and then returning the barrel to center

position. This creates four generators: move barrel [left,right],

then move middle section-[left,right] forward (or backward, which

is the inverse). Visually:

| | | | A A A A A B B B B B \ \ / / A A A A B B B B / / \ \ A A A A A B B B B B \ \ / / A A A A B B B B / / \ \ A A A A A B B B B B | | | | | | | | C C C C C D D D D D \ \ / / C C C C D D D D / / \ \ C C C C C D D D D D \ \ / / C C C C D D D D / / \ \ C C C C C D D D D D | | | |

Where A is move barrel left , move left section

B is move barrel right, move left section

C left , right

D right, right

The top and bottom of these drawings are connected, cavities

(filled with the balls) move along the lines. All balls move in

the same direction the same number of units (i.e., the middle

sections are rigid). I hope this is a good enough description, if

not send me mail and I will send an addendum.

The object, so I hear, is to get each column (row in these

pictures) a single color, and if there are five slots (of which

there are three), the fifth has a black ball in it, when the

barrel is pushed all the way to one side, the plungers take up

three of the outside-barrel-sections, and the black balls take up

the opposite three. from a symmetric point of view, I think it

would be more general to SOLVE it so that the black ball is in

the middle of the five balls (this may not be solvable though)..

If we ignore the obvoius left-right symmetry of the above

pictures, the first assumption of the combinatorics of this beast

is simply P(23;4,4,4,4,4,3)=numbers of ways to permute 4 balls of

each of 5 colors and 3 balls of another color=

23! ------------------- = 541111756185000 = 541 trillion 4! 4! 4! 4! 4! 3!

Until I have played with it for a while, I can't even guess on

how many orbits there are. Perhaps only one -- I don't know.

Super-groups come in a few classes: (1) Each non-black ball gets a second label (1-4) giving size 23!/3! = 4.3*10^21 (2) Each black gets a second label (1-3) giving size 23!/(4!)^5 = 3.25*10^15 (3) (1) and (2), all balls distinct giving size 23! = 25.8*10^21

If anybody sees one in this county, please let me know. Tanya

believes they are only in Japan at the moment. She has donated

the one I have seen to me/SIPB, so people at MIT and area are

free to come to 39-200. PLEASE BE CAREFUL with it. It is plastic

and it looks breakable -- especially the outer part of the

supporting structure looks like it dould break. I think a better

construction would be to have them be plates which are attached

to the axis with screws. This might lead to a temptation to

disassemble, which may be epsilon below breakage.