In the arms and legs race it's all about the numbers

By Burkard Polster and Marty Ross

The Age, 27 October 2008

With a cunning smile on his face, your mathematician friend Isaac shows you a photo taken at one of Melbourne’s many fun runs. He asks you how many people took part in the run. You have absolutely no idea. But you pause, and you notice there are a few entry numbers visible on the athletes: 4329, 2081, 16081, 18256 and 9264.

Assuming the participants are numbered consecutively, from 1 to whatever, you can reasonably answer “At least 18256”. And there are probably more. But what is a good guess as to how many more?

Think of all the runners’ numbers written out in sequence. Then the five visible numbers chop this string of numbers into six parts: the first section goes from 1 to 2081, the second section from 2082 to 4329, and so on up to the sixth and last section, which consists of the numbers beyond 18256. So, we know exactly the extent of each of the first five sections, but not the last section.

Combined, the first five sections go up to 18256. So, on average each section corresponds to 18256/5 runners. So, a plausible estimate is that this last section is of the same size. This amounts to a guess of the total number of runners being 18256 + 18256/5 = 21907.

Your answer wipes the smug smile from Isaac’s face, but are you right? Probably not. But there is a sense in which your estimate is close to the best possible guess. If you play this game over and over, this averaging strategy will work better than any other. And, if you base your estimate on a larger sample of visible numbers, then your estimate will almost certainly be very close to the true figure.

Does this all seem a silly diversion? In WWII, mathematicians ended up playing these very games. Each side in the conflict wanted to know the number of tanks, and other major equipment, possessed by the other side. One source of evidence was from traditional spying. But another source was the serial numbers on the few captured pieces of equipment. The data available and the mathematical approach used were not as simple as for our fun run, but it was basically the same game.

Usually, mathematical estimates based upon the serial numbers were much more accurate than from spying. This is clearly illustrated by the table, showing the actual and estimated numbers of German Panther tanks manufactured in different months. So, the image of a cunning James Bond may be much more exciting than mathematical detective work, but the latter is likely to be much more reliable.