Suppose we have a perfectly flat and infinite table. Scattered on this table are a (finite number of) marbles. At a moment we will call "the big bang", each of the marbles, individually, begins moving at some constant speed and in a constant direction on the table. From marble to marble, these speeds and directions may differ.
To avoid trivialities, we assume that there are at least 3 marbles, that all marbles have nonzero speeds, and that at least two marbles have different speeds.
To avoid the complication of dealing with collisions, we can assume that someone has thoughtfully chosen the speeds and directions so that no two marbles will ever collide. Or else we may assume that marbles don't interact, and that if their paths cross, each continues with its original speed and direction. This is a stronger assumption, but it turns out not to matter which one we choose.
Now at any particular time, we could always draw a ring that would enclose all the marbles. While there are many rings we could draw, there is always a smallest such ring, whose diameter we might denote by D.
Now if we were very nosy, we could check on the marbles at every instant of time, and determine the size of the smallest ring that would enclose the marbles. In other words, the diameter is a function of time, and we should really denote it by D(t).
Now it is time to ask whether there are any rules that D(t) must obey? In particular, we wonder whether the behavior of D(t) is completely unpredictable, or if it is somewhat regular, with sudden unpredictable jumps? Is it continuous? Differentiable? Can you show that D(t) must grow with time? If not, can you show that it must "eventually" grow with time? (That is, can you show that no matter what the initial configuration, speeds and directions, there is always some time after which D(t) can only increase?).
Making some modest assumptions, can you show that, over time, D(t) may have no more than one minimum value, and must tend to infinity?
Can you demonstrate the last statement above using a handful of a common household item?
If you are mathematically inclined, you know that at any time, the smallest enclosing ring is uniquely defined, and must touch at least three of the marbles. A marble might be part of the enclosing ring at one time, and then later not be.
Now, ask yourself, can a marble be part of the enclosing ring two times, with a time in between where it is not? Is the enclosing ring a continuous function of the marble positions? Is it continuous with respect to time? What about differentiability? At some point, can we show that the asymptotic behavior of the enclosing ring is linear growth about some fixed center? Perhaps about a center moving with constant velocity?
I give up, show me the solution.