One famous achievement concerns that misunderstood branch of mathematics – statistics. The God of Whizz once fell asleep mid-sentence whilst taking notes at a statistics lecture. For those who did pay attention in class statistics would become a vital tool in the defeat of Axis forces.

The problem was this: Traditional military intelligence was failing to determine the true numbers of tanks produced by Germany’s factories. This was important; the Allies needed reliable information on the numbers of new Mark IV and V tanks being produced. To be forewarned is to be forearmed, after all…

Mathematicians showed that it wasn’t necessary to steal factory logs, bribe officials, or photograph high-security facilities. All you needed was a serial number.

The German Army gratefully obliged by stamping every tank that left their factories with a unique value. Every now and then one such tank would be captured or destroyed, yielding its serial number. The mathematicians realised that you could infer the total number of tanks produced by logging the numbers of all captured German tanks, provided the tanks were numbered in order of production.

Here’s the explanation, courtesy of the Guardian:

The basic idea was that the highest serial number among the captured tanks could be used to calculate the overall total. The German tanks were numbered as follows: 1, 2, 3 … N, where N was the desired total number of tanks produced. Imagine that they had captured five tanks, with serial numbers 20, 31, 43, 78 and 92. They now had a sample of five, with a maximum serial number of 92. Call the sample size S and the maximum serial number M. After some experimentation with other series, the statisticians reckoned that a good estimator of the number of tanks would probably be provided by the simple equation (M-1)(S+1)/S. In the example given, this translates to (92-1)(5+1)/5, which is equal to 109.2. Therefore the estimate of tanks produced at that time would be 109.

The statisticians used this method to estimate German tank production at 246 per month between 1940 and 1942 – a fraction of the estimates produced by traditional intelligence. This gave the Allies greater confidence to attack the Western Front in 1944 and the rest, as they always say, is history.

[via Wired Autopia & The Guardian]

So it was disappointing to read a recent too-smart-by-half ’10 worst…’ blog post at nymag.com. The post identifies ‘Ridiculous-Sounding Math Classes Currently Offered at Liberal-Arts Colleges’. [A 'liberal arts college', for those unused to the term, has nothing to do with liberal politics, and tends to teach science alongside the arts.]

This post is a classic illustration of the maxim that “a little knowledge is a dangerous thing.” In their attempt to demonstrate their disdain for wishy-washy pseudo-science courses for maths dunces the authors have managed to highlight their own ignorance of maths and science. A warning about people in glass houses springs to mind.

Click through for the offending top ten, but it’s clear that the authors have misunderstood some of these course descriptions entirely:

10. Topology: The Nature of Shape and Space: “In geometry we ask: How big is it? How long is it? But in topology we ask: Is it connected? Is it compact? Does it have holes?”…

…6. Models of Life: “In particular, we will ask such questions as: How do you model the growth of a population of animals? How can you model the growth of a tree? How do sunflowers and seashells grow?

Topology is hard maths. Modelling nature (such as the hidden mathematics in natural structures) is fascinating, mathematically robust, and not for slouches. Other mocked subjects include probability, folding, patterns of natural numbers, and so forth.

The authors have got a bit of a pasting from some of the comments, many pointing out that their selection of courses says more about the authors’ ignorance than the courses themselves, and we’d be inclined to agree.

The NYMag post ends:

This is why Asia is winning, by the way.

…A reference to the fact that China and India are producing vastly more pure and applied-science graduates than the USA or Europe, and how their numbers may be giving them an advantage in the global race.

We absolutely want more maths and sciences taught in colleges and schools. In fact, the GoW would be inclined to force students to take some basic scientific courses at all levels of secondary and tertiary education.

But the courses mocked in this list sound like just the sort of things arts undergraduates should be studying (alongside the fundamentals of mathematics, of course) to give them an appreciation for how this beautiful subject is completely interwoven in the creative and and natural worlds.

Not all of the courses listed seem essential to maths studies, but those courses that deal with creatively exploring the world through maths (never mind those that deal with the ‘proper’ maths of topology, probability, modelling and the like) are hugely valuable. GH Hardy who, in the early C20, was one of the most famous mathematicians alive, said:

“I am interested in mathematics only as a creative art.”

And from the man who introduced the world to Ramanujan, that sounds like confirmation that students should have a greater appreciation for the creative elements of maths, not less.

[via NYMag]

For proof of this, you need only look at the multi-billion-pound gambling industry. Bookmakers and casinos profit because we find it hard to evaluate odds and probabilities. I am reminded of an intelligent friend who couldn’t understand that buying two different tickets doubled your chances of winning the lottery with one ticket, but that you had to buy 14 million different tickets to be pretty much guaranteed a win.

In this vein, TV-hungry uber-mathematician Marcus du Sautoy explains a cunning probability problem to the wonderful Alan Davies (a man paid to act the dunce on BBC’s QI but who, I suspect, knows almost as much as the Fryster himself…).

The Monty Hall problem is the general name for a puzzle based on a common game-show scenario. I’ll let you watch Marcus and Alan explain all, or you can watch Kevin Spacey and an unnaturally attentive maths class discuss it, from the maths-and-gambling movie ’21′, below.

Of course, Maths-Whizzers know full well the importance of probability. We start to introduce the concept in Key Stage 2 maths and students are evaluating complex probablity scenarios by year 8.