2009-07-06

Marble problem 1

Bengt so enjoyed having a themed month, that he insists on doing it again. This time it's Marble Month, so he invites Sture over to play with marbles. Sture says that he has lost his marbles, but that he'd be willing to make trite jokes about balls instead. But Bengt has made up his mind, and we all know how stubborn he can be.

Bengt brings out his bag of marbles. The marbles are physically identical; they have the diameter 1 cm, and mass 1 g. They start building pyramids, by placing three marbles next to each other in a triangle, and one on top.

  1. With four marbles you can build a pyramid of 2 levels; how many marbles do you need for n levels?
  2. And how high is such a pyramid?
  3. Another shape that one can imagine, altho it's hard to build with marbles, is a shape where a central marble is surrounded by as many other marbles as possible, each touching the central one. How many marbles can you fit in?
  4. In how many shapes can it be done?
  5. That arrangement can also be extended. You can add another layer, such that each marble in the new layer has to touch one of the last layer. If you keep going, the shape will look more and more like a regular polyhedron. Which one?
  6. How many marbles will you have in n layers?
  7. Imagine that the pyramid is enclosed in a tetrahedron. It is just big enough to contain the whole pyramid. The walls are water- and airtight, but have negligible mass and thickness. Would the contraption float on water?
  8. They build a number of pyramids - the small kind, just four marbles. If a pyramid "goes off", the middle marble drops down and the other three roll away at the same speed. What speed will they move at?
  9. In the end, they will have n pyramids. They are randomly distributed in a roughly circular area a of the floor (a really big area), and oriented in random directions. If a marble hits a pyramid, it will go off. It should therefore be possible to set up a chain reaction of the pyramids, with marbles from each busted pyramid setting off other pyramids. How big must n be to achieve "critical mass", so that setting off a few of the pyramids is likely to set off a large part of the others? Assume that there is no friction, and no energy loss in collisions, and that the marbles disappear when they leave the area.