The Monty Hall Problem

Posted on 9 October 2010 by Xander

Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say 1, and the host, who knows what’s behind the doors, opens another door, say 3, which has a goat. He says to you, “Do you want to pick door 2?” Is it to your advantage to switch your choice of doors?1http://www.marilynvossavant.com/articles/gameshow.html

This problem, called the Monty Hall Problem, is one of my favorite mathematical puzzles. Because it has been discussed to death (a quick Google search brings up hundreds of thousands of results), there is probably not much that I can add to the discussion. That being said, it is an interesting problem, and I think that it is worth sharing.

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Monday Mandelbrot Madness I

Posted on 4 October 2010 by Xander

Fractal Fire-Detail.

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Spherical Cows

Posted on 2 October 2010 by Xander

There is an old mathematical joke that goes like this:

A dairy farmer is interested in increasing his milk production. He goes to the local university, and seeks the advice of three experts: a biologist, a pharmacist, and a mathematician.

The biologist comes back with her report first. “Well,” she says, “I’ve gone over your numbers, and I am optimistic. If you you begin a selective breeding program now, in 10 years you should be able to increase your milk production by around 15%.”

The pharmacist was next to submit his report. “Well,” he says, “I’ve gone over your numbers, and I am optimistic. If you start administering bovine growth hormones and antibiotics now, in one year you should be able to increase your milk production by around 50%.”

The mathematician was the last to submit his report. “I’ve got great news for you!” he says. “I can increase your milk production by 500% before the end of the week!”

The farmer is incredulous. “How are you going to do that?” he asks.

“Quite simply,” the mathematician replies. “First, we must assume a spherical cow…”

At the risk of ruining the joke by over analysis, I present a question: why is this joke funny?

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Mathemagic

Posted on 30 September 2010 by Xander

I love magic.

A friend of mine, who works as a professional magician, introduces one of her tricks by explaining that she was trained as a dancer. In dance, she says, the performers want the audience to see everything that they are doing. Dance is an expressive art form where movements are often exaggerated so that the audience can see every detail. Magic, by contrast, is secretive. Magicians use movement to conceal what they are doing, so that the audience is surprised by what happens, and leave a performance thinking, “How did they do that?”

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Mathematics and Beauty

Posted on 28 September 2010 by Xander

Why are numbers beautiful? It’s like asking why Beethoven’s Ninth Symphony is beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.

—Paul Erdös

I think that most people can agree that beauty can be found in music. People may have different taste, and find different pieces of music to be beautiful for different reasons, but I doubt that there are many people that could not name a piece of music that they think is beautiful.

The same could be said for other forms of artistic expression. There are beautiful paintings; beautiful sculptures; beautiful novels, poems, and stories; even beautiful buildings and foods. People are used to, and comfortable with, the idea that art and artistic expression can be beautiful.

On the other hand, for many people mathematics is a nothing more than a tool. They learn math in school, and are told that it is useful; that it is a tool for finding answers. They may have heard at some point in their lives that math is beautiful, but they probably dismissed that idea—how can a bunch of black and white scribbles on a page be compared to beautiful things like a flower or symphony?

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