Charlie's contributions
November 28, 2005
Some people (myself included) have wondered exactly how Charlie and his mathematics were helpful in last Friday's show. Here are the items that Charlie comments on.
(1) Information Theory. Charlie suggests that there was a common thread to the poisonings; this thread turns out to be the pharmaceutical company. Many of us would say that this was, perhaps, just common sense; however, remember that Charlie expresses his common sense in mathematical terms: it is his life and his vocabulary. Certainly the F.B.I. would have found this connection eventually, since they have the resources to find all connections, given enough time. Charlie perhaps saves them time by pointing them in the fruitful direction early on. I doubt, though, that information theory is necesssary here. (Information theory is a real, very useful, and fairly recent branch of mathematics, pretty much founded by the 20th century mathematician Claude Shannon. You can Google it.)
(2) Graph Theory and Euler's solution to the "Seven Bridges of Koenigsberg" problem. This is a very important bit of mathematics by Leonhard Euler, who was possibly the most prolific mathematician ever -- he wrote over 700 articles and papers, a whole bunch of which when he was an old man and totally blind. Euler's paper on the "Seven Bridges" is considered by many to mark the birth of what is now called "graph theory." It has also been argued that Euler was one of the first people to study "topology." In any case, Charlie's description of Euler's solution is not quite right.
What Euler proved was that if you had a bunch of regions separated from each other by rivers but connected by bridges, you could traverse all the bridges without repetition only if one of the following cases hold:
(A) Each region has an even number of bridges to it, OR
(B) There are no more than two regions with an odd number of bridges to each, in which case you must start in one of these regions and end in the other.
He also proves that it is impossible for there to be an odd number of regions each with an odd number of bridges.
Probably the best account of Euler's solution is by Euler himself. You can find a translation in "The World of Mathematics" edited by J.R. Newman --- available in most libraries, large bookstores or on-line.
However, as interesting as the Seven Bridges problem is, it is basically irrelevent to the episode. Charlie uses it to conclude that the fugitive, McHugh, must pass over some trail more than once. First of all, this probably does not follow from Euler's criteria since Charlie doesn't really know at that time how many campsites (regions) and trails (bridges) there actually are. Secondly, we don't need anything as complicated as Euler's analysis. There are only a finite number of trails (or campsites). If McHugh travels enough, he has to repeat one of them. For example, if there are 50 trails, then as soon as he makes 51 trips he must pass over some trail twice. This is sometimes called "The Pigeonhole Principle": If there are more pigeons than pigeonholes, there must be at least two pigeons in the same pigeonhole. Charlie's original idea of arranging the trails in order of likelihood (by studying the terrain and proximity to sources of food and water) was a better idea, though he never seemed to have implemented it.
(3) (Mathematical) Negotiation Theory. I have no idea what this is, but I'm generally suspicious of any supposed combination of mathematics and analysis of human behavior. In any case, to say that in a negotiation each side must offer the other something, which is how this part of the episode plays out, is about as banal and obvious a statement as can be made. That Charlie scratches some sort of pseudo-math in the dirt with his stick doesn't make it any less silly. Well, you can't win them all...
(4) Steiner points and Steiner paths. I've already discussed the Steiner point of a triangle. However, in the context of the episode, this doesn't make much sense. Distance is measured along straight lines and the paths in the woods are curvy. Furthermore, there is no way a fugitive can make the elaborate construction (described in the last blog) necessary to find the Steiner point, and it might turn out to be in the middle of an impenetrable thorn thicket or a bog. Finally, anyone looking at the map Charlie shows would immediately see McHugh's ranch right smack in the middle of the three campgrounds forming the triangle, Steiner point or no Steiner point.
We have to cut Charlie and the writers of Numb3rs some slack, and the episode was fun. Nevertheless, sometimes simple and straightforward logic and elementary mathematics are what's called for, not fancy postgraduate complications.