Date: 28 MAR 1981 1259-EST From: DCP at MIT-MC (David C. Plummer) Subject: New toy (long message, but read it anyway!!) To: CUBE-LOVERS at MIT-MC 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.