请教一道题~~~~Shaking Marbles

yvonneyang

Shaking Marbles

Suppose one has a glass cylinder jar of radius rand height h which contains, for simplicity, two types of spherical marbles ofthe same radius s. Initially, the marbles fill the jar to the height l. Themarbles can be easily told apart, say, some are red and some are green. Thinkof the red marbles as "dangerous" while the green marbles are"safe." It is desirable that the red marbles be separated from thegreen ones. Imagine that if one can see some part of a red marble sitting on onthe top it can be removed without disturbing any of the remaining marbles. Atany given time a top can be placed on the jar and the jar shaken for a periodof time t, and any red marbles that have risen to the top can be removed.

Your job is to design a system to remove all of the red marbles by asequence of "cycles" where one shakes the closed jar for some periodt and then removes those red marbles that can be removed from the top. Theshaking cycles are assumed to all last equally long.

You should consider two cases:a. The jar is opaque so that after some amount of shaking you are notsure where the the red marbles are located.
b. The jar is transparent and you can get a certain amount of informationabout where the red marbles are.


Various variants of this problem are:
a. Initially all of the exactly r red marbles areon the bottom of the jar in a layer.
b. Initially all of the exactly r red marbles are randomly mixed with thegreen marbles.
c. Initially, the value of l, the height to which the jar is full isclose to the height h of the jar.

In a general way the purpose of the modeling is to see how long it takesto remove the dangerous red marbles in terms of the initial height to which thejar is filled and the size of the marbles. Furthermore, how is this separationtime dependent on whether or not one can observe what is happening during theshaking process? Perhaps a good place to get started is to assume there exactly one red marble. In the process of carrying out your model state anyadditional assumptions you decide are necessary.

Clearly, there are many ways to generalize this problem and you shouldcomment on some of these, and whether or not the methods that you used to workon this problem will carry over to the generalizations.
不明白题目中的透明与不透明的区别该如何处理请大家帮帮忙~~~

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