Mr Honner

Proofs Without Words

Here are two of my favorite Proofs Without Words. I’ve been thinking about infinite geometric series a lot lately, and these are two lovely, well-known, visualizations of two amazing infinite sums:

In a square of side length 1 (and therefore, area 1), cut the square in half; then cut one half in half (that’s a quarter); now cut one of the quarters in half (that’s an eighth); and so on and so on and so on (this puts the infinite in infinite sum). Eventually you’ll fill up the whole square! So this is a demonstration of the following amazing, and somewhat counterintuitive, fact that

Similarly, this diagram

is a visual representation of the following sum:

As any good, lazy mathematician would say, the details are left to the reader.

Application Sports

Football Calculator

In an ESPN blog-post, Bill Belichick’s decision to go for it on 4th down against the San Diego Chargers was analyzed, and the ultimate conclusion was that, by a few percentage points, this was wrong decision. Last season, Belichick was second-guessed after a similar decision led to a Patriot loss to the Colts. Interestingly, a similar analysis deemed that particular decision to be correct.

The idea of applying serious risk-reward analyses to football seems to increasing in popularity, even though a rigorous study by a world-class economist was conducted nearly 10 years ago .

The author at ESPN utilized a Win Calculator at the Advanced NFL Stats website. It’s a pretty cool idea–input the current score, time remaining, quarter, field position, down and distance, and the calculator returns Win Probability, Expected Score, and some other projected data.

Of course, the devil’s in the details–that is, the algorithms–but it’s cool to see the quantitative analysis of sports continue to spread.

By patrick honner, 16 yearsOctober 28, 2010 ago

Appreciation Probability

Paul the Octopus, 2008-2010

Paul the Octopus, whose prognosticating skills captured the imaginations of World Cup viewers everywhere, died this week at the age of 2.5. He died of natural causes.

Paul defied probability by correctly predicting the results of all of Germany’s seven World Cup matches. After making it through the tournament with a perfect record–during which he received death threats and had a stamp printed in his honor–Paul retired from predicting. Rediscovering his British roots, Paul was appointed an official ambassador for England’s 2018 World Cup bid, a post he held until his untimely demise.

Apparently there have been many copy-cats, so to speak, including “a saltwater crocodile named Dirty Harry, who predicted Spain’s World Cup final win and called the result of Australia’s general election by snatching a chicken carcass dangling beneath a caricature of Prime Minister Julia Gillard”. But Paul will always have a special place in our hearts.

By patrick honner, 16 yearsOctober 27, 2010 ago

Appreciation Art Geometry

Wireframe Torus

This is a cool example of wire-sculpture: a single piece of wire woven into the shape of the torus.

http://makezine.com/2010/10/24/math-monday-wire-torus-challenge/

(I doubt you can play pool on this one, though.)

The author of the blog post, George Hart, is the proprietor of the soon-to-be Math Museum, and it seems he is something of a sculptor himself. According to the article, this piece was on display at a conference held by the European Society for Mathematics and the Arts.

Poking around their website and admiring the the multitudinous mathart is a pleasant way to pass a little time.

By patrick honner, 16 yearsOctober 26, 2010 ago

Challenge

Coffee and Cream

I was recently reminded of an excellent math problem involving mixtures.

Imagine yourself sitting in front of a cup of coffee and a cup of cream.

Suppose you take a spoonful of cream, pour it into the coffee, and stir it up. Now once that’s thoroughly mixed, you take a spoonful of the mixture and pour it back into the cream. Then you mix that up. After all of this, is there more coffee in the cream, more cream in the coffee, or equal amounts in both?

I encourage you to think about the problem before perusing the several solutions below!

Proofs Without Words

Football Calculator

Paul the Octopus, 2008-2010

Wireframe Torus

Coffee and Cream

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