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## Mysteries of Money, Pizza and Rockets

Published on Sunday, December 13, 2009 in , , , ,

Some problem remain unsolved because they're simply so challenging. Other mysteries remain unsolved simply because they haven't bugged enough people that someone finally sits down and solves the problem. Today, we'll take a look at some interesting mysteries that have finally been solved.

Want some money? I've got a game that will guarantee you money! I have 2 envelopes that both contain money, but 1 of the envelopes contains twice as much money as the other one. You only get to keep the contents of 1 envelope, but you do get to make a choice. Just to make in more interesting, I'll let you choose either envelope, check to see how much money is inside, and then decide if you want to switch or not. If you don't switch, you keep the money in that first chosen enevelope. If you do switch, you keep the money in that 2nd envelope.

At first, this sounds like a simple variation of the Monty Hall Dilemma, but there's an important difference. In the Monty Hall Dilemma, you know the hidden contents (2 goats, 1 car). In this two-envelope challenge, you start with no knowledge of the exact contents, and then gain a partial knowledge.

The long run of this challenge has always been how to maximize your return. You could just always switch, and you'd get the larger envelope half of the time. The same thing would happen if you never switched. Yet the calculation of expected return suggests you should always switch. Earlier this year, a group of researchers from Australia actually worked out a surprisingly simple strategy for maximizing your returns in this two-envelope paradox.

Either way, you've got your money, let's go get some pizza with it! On Grey Matters, I've actually discussed pizza quite a bit, from the Pizza Theorem to using geometry in a pizza parlor to save money. Here's another challenge: How do you fairly divide a pizza?

If pizza places always divided their pizzas into exactly equal slices and always cut exactly through the center, this would be a relatively simple problem. However, whenever you get a pizza in the real world, some pieces are larger, some are smaller, and the cut isn't always through the exact center. As Yogi Berra once said, "In theory, there's no difference between theory and practice. In practice, there is."

Under these conditions, how do you divide up the pizza fairly? It would be especially handy if there were no calculations, so that the pizza doesn't get cold while you work out the answer.

The answer to this problem was found, not surprisingly, by a group of mathematics students who enjoyed hanging out in their favorite pizzeria. Study the answer well, as many pizzas were left cold in pursuit of this important information.

One mystery that got plenty of attention this week was a mysterious spiral in the sky over Norway. It appeared as a white spiral with a blue streak shooting out of the middle. The mystery was solved when it was discovered that it was a misfiring rocket launched from Russia. The video here shows how the spiral and streak occured, along with some stunning photographs of how it actually appeared.

There are still plenty of unsolved math-related problems, but have you ever noticed that the closer a problem is to our immediate interests, the more quickly it gets solved? Even the backers of the Millennium Prize realize this, which is why they attach a \$1 million prize to these unsolved problems.