Quicktime,Webm,Ogg | Over the winter break 2013, while spending some days at the cape with the family, we visited the Province town puzzle shop on New Years Eve. I bought there the Meffert's great challenge as well as Smooth operator, a maze puzzle. More about the second later. The Meffert challence consists of a cube in which the corners stay stationary but turn. Four of the vertices correspond gears with 8 teeth attached, while the other four vertices have 5 teeth attached. There are two modes. In the closed mode, everything is linked. In the open mode, two halfs can move independently. Let G denote the group of one side, then G x G is the group when the cube is open. It is easy to see that the group G is Abelian and cyclic and equal to C_{15}. So, in both cases, it appears to be a very simple puzzle, because the group just consists of two loops. But wait. Being confident and arrogant, I started to pay more attention to the food channel competitions while playing with it a bit more. Surprise: if the closing and separating of the cube halfs were allowed in more general situation, the group became bigger and appears now to have (8^{4} * 54)/2 = 1280000 elements. Still nothing comparable with the Rubik cube but still challenging. | Quicktime,Webm,Ogg |
Fortune cookie in Mews. |
Apropos randomness: while waiting for the food on the brink of New year
in the restaurant "Muws"
in Province town, we had fortune cookies
on the plates which included little toys. One was a classic metal puzzle
in the form of a bent nail which must be thousands of years old
at least. These puzzles are not groups but group theory is involved also if $G$ is the group generated by affine translations and rotations in space and the puzzle has $d$ pieces, then all the possible configurations form a maze bound by a hypersurface of situations, where the two objects touch. For two pieces, this is a 11 dimensional surface in a 12 dimensional space. While mazes are trivial in two dimensions (just have your hand on the left wall and walk), in higher dimensions, puzzles are difficult. By the way, I could not resist also to buy an other such toy at the toy shop the Smooth operator. It is labeled to be difficult and I can not yet solve it yet. P.S. In this paper an other story is told |
Smooth operator |
Back to the simple maze with two simple loops: in the restaurant we started to wonder how long it would take to solve this Monte Carlo type. Just shake the connected puzzle long enough until it falls apart. To our surprise it was possible. It worked twice with a couple of dozen shakes. If you make the experiment, the puzzle must be small so that you can shake it in the hand. Also, the solution path should not be too tight. If you have to slide along, it is too tight. What happens when shaking is that we do a random walk in an open subset of the group $G \times G$ intersected with a large sphere. If two points are connected, the random walker will reach from one point to the other, but it might take a long time. This depends on the geometry and on the smallness of the connection path. Shaking it and getting the nails hooked together did never work. It is obvious that the probability of achieving this is much smaller because the volume of the 12 dimensional region which needs to be transversed by the random walk is much smaller. A wild guess would be that one could put the two separate nails in a shaking box, wait a few years until it is solved. Thats how the first improbably combinations of complicated organic molecules could have occurred on a time scale of billion of years. | Quicktime,Webm,Ogg |