Date: Sat, 28 Mar 81 12:59:00 -0500 (EST)
From: David C. Plummer <DCP@MIT-MC >
~~~ ~~~ Subject: New toy (long message, but read it anyway!!)

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.