Marble Problem 3

Sture has finally got tired of playing with marbles, but Bengt continues with the fervor of a true scientist. He is still building pyramids, of four marbles each. But the floor can be a little slippery sometimes, and then the pyramids won't stick together. Bengt intends to do something about that. One of his ideas is to give the marbles electrical charge, in order to make them stick to each other.
The marbles have diameter 1 cm and mass 1 g. They are not conductive, so they do not lose their charge when touching each other or the floor.

1. An easier way to get marbles to stick together is to use glue. Suppose each marble exerts a glue-force on each marble that it touches. How big must the glue-force be to hold the pyramids up, if there is no friction at all? (That is, neither between a marble and the floor, nor between a marble and another marble.)
   
2. Forget the glue, and let's investigate the friction a little more. What happens if there is a large friction between a marble and the floor, but no friction between a marble and another marble?
   
3. And what if it's the other way around - large friction between a marble and another marble, but no friction between a marble and the floor?
   
4. Suppose the friction against the floor is very large. How big must the the marble-marble friction coefficient be to keep the pyramid up?
   
5. Suppose both the frictions are the same. How big must the friction coefficient be to keep the pyramid up?
   
6. Bengt also wants to build the ball-shape mentioned in Marble Problem 1: One central marble, surrounded by as many other marbles as possible, all touching the central marble. With no friction, what glue-force is needed to keep this arrangement standing?
   
7. Without glue, could enough friction keep it up? Assume very large friction against the floor. How big must the the marble-marble friction coefficient be?
   
8. Now suppose we want to do something even more extreme: Let there be a force between two marbles, which is just enough to overcome gravity, so that Bengt can hold one marble and let the other hang from it. Clearly this is possible with a glue force. But could this force be described by a friction coefficient, if it is extremely large?
   
9. Let's try with those electric charges. Two marbles have equal and opposite charge, enough to hang one marble from the other. How big must the charge be?
   
10. What if you have three marbles hanging like a chain, the one in the middle being positive, and the others having equal but negative charge?
    
11. What if you have n marbles in a chain, with equal and alternating charges?
    
12. Returning to the case with three marbles - is there a better way to distribute the charges? Find the setup such that the sum of the absolute values of the charges is as small as possible.
    
13. And for n.
    
14. We can also use charges to get the pyramid to stay up. If the top marble has positive charge, and the bottom ones have negative charge of the same magnitude, will that help the pyramid stay together, or will it actually force it apart?
    
15. Suppose instead that the charge of the top one can be larger than the others. What should the charges be to get the pyramid to stay up? Assume there is large floor friction but no marble-marble friction, and minimise the largest charge magnitude.
    
16. Even more freedom: Arrange the charges, magnitude and polarity, any way you like, and again minimise the largest magnitude.
    
17. And what if there is no friction at all?
    
18. What about the ball shape? If there is more than one possible shape, which one is better? Assume that there is large floor friction but no marble-marble friction. Also assume for convenience that the middle marble is positively charged. To get the best result (that is, minimise the largest magnitude), which marbles should be positive and which should be negative?
    
19. And what would those charges be?
    
20. And what if there is no friction at all?
    
21. There is yet another, even more unrealistic way to get the marbles to stick together. What if they were really heavy, so that gravity kept them together? Consider the case of one marble hanging from another. If they have the same mass, how heavy must they be?
    
22. And what about the pyramid? Assuming large floor friction but no marble-marble friction, and same mass for all marbles.
    
23. And what if there is no friction at all?
    
24. For completeness: The ball shape. Large floor friction but no marble-marble friction.
    
25. And what if there is no friction at all?
    
26. Bengt decides to finish off the alphabet by putting u and i together. Take one of the marbles from question u, charge it up, and hang it from a normal 1 g marble like in question i. How big would the charge have to be? And how big, roughly, is the theoretically biggest positive charge 1 g of normal material can have (that is, with every electron removed)?
    

Marble problem 2

When we left them last week, Bengt and Sture were building pyramids out of marbles. Since time in problem-space is independent of time in the real world (and also, I wrote this problem a while ago) they are still working on that, but they are nearly finished. Oh boy, they have built lots of pyramids.

There are twelve marbles left in the bag. Each marble is either black or white. The marbles are otherwise identical. Sture takes four marbles, and then gives the bag to Bengt. Bengt takes three marbles and places them on the ground. They are all white. He reaches in to take one more.

"I wonder what the probability is of the last one being white." says Bengt.
"Well, if those three were white, then there are most likely fewer white left."
"But on the other hand, the fact that these three were white suggests that there were more white ones than black ones when you gave me the bag."
"Hm. Sounds like a good problem."
"I don't know. I think it might be a bad problem, actually."

1. Is it a good problem? Or, more specifically, is it a problem that can be solved? Is it possible to say anything about the probability with the given information, and in that case, what?
   
2. Assume that half of the marbles are white when Bengt gets the bag. Same question.
   
3. Assume instead that half of the marbles are white to begin with, and that Sture takes four at random.
   
4. Assume instead that half of them are white to begin with, but Sture takes four of the same colour.
   
5. Assume instead that half of them are white to begin with, but Sture starts by taking two of the same colour, and then picks the other two at random.
   
6. Assume instead that we don't know the original distribution, but that each marble is originally equally likely to be black or white, and that Sture takes four at random.
   
7. The same, but Sture takes two of each colour.
   
8. The same, but Sture takes four of the most prevalent colour. (If they are equally common he picks a colour at random.)
   

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.