Suppose we have a perfectly flat and infinite table. On this table are a bunch of marbles. At a moment we will call "the big bang", each of the marbles begins moving at some constant speed and in a constant direction. From marble to marble, these speeds and directions differ. Someone has thoughtfully chosen these speeds and directions in such a way that no two marbles will ever collide.
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.
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. Is there a pattern to this size?
In particular, is this size "random" or chaotic? Can you show that the size of this ring must grow with time? If not, can you show that it must "eventually" grow with time? Making some modest assumptions, can you show that, over time, the ring size function 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. (It's OK, we can assume there are at least three!) A marble might be part of the enclosing ring at one time, and then later not be. 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.
Back to the puzzle page.