#### Shaking Marbles

Joseph Malkevitch, Math Department, York College (CUNY)

Suppose one has a glass cylinder jar of radius $r$ and height $h$ which contains, for simplicity, two types of spherical marbles of the same radius $s$. Initially, the marbles fill the jar to the height $l$. The marbles can be easily told apart, say, some are red and some are green. Think of the red marbles as "dangerous" while the green marbles are "safe." It is desirable that the red marbles be separated from the green ones. Imagine that if one can see some part of a red marble sitting on on the top it can be removed without disturbing any of the remaining marbles. At any given time a top can be placed on the jar and the jar shaken for a period of 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 a sequence of "cycles" where one shakes the closed jar for some period $t$ and then removes those red marbles that can be removed from the top. The shaking 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 not sure where the the red marbles are located.

b. The jar is transparent and you can get a certain amount of information about where the red marbles are.

**Various variants of this problem are:**

a. Initially all of the exactly $r$ red marbles are on the bottom of the jar in a layer.

b. Initially all of the exactly $r$ red marbles are randomly mixed with the green marbles.

c. Initially, the value of $l$, the height to which the jar is full is close to the height $h$ of the jar.

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

Clearly, there are many ways to generalize this problem and you should comment on some of these, and whether or not the methods that you used to work on this problem will carry over to the generalizations.