Numb3rs has just wrapped up its third season with some big surprises and puzzling character questions (WARNING: This link contains spoilers!).

Numb3rs' season 3 DVD is tentatively set for a US release on September 18, 2007, and CBS has renewed the show for a 4th season, so now seems like a good time to go over the series.

There are a few things that I haven't covered in my previous Numb3rs posts.

Popular Science Magazine covers one thing I've puzzled over in the past few years, and that is just when prime-time TV went from bathing suits to lab coats.

Eric Weisstein, whose Mathworld site I've written about numerous times, and covers The Math(ematica)Behind Numb3rs. This is an especially good article, bothe because of the links that allow you to dig further into the math used in the show, and because Eric himself is part of the mathematical research team for the show!

Until the Season 3 DVD is released, I suggest the summer re-runs, and perhaps the upcoming August release of The Numbers Behind NUMB3RS: Solving Crime with Mathematics.

Imagine you and another person are arrested by the police. The police don't have sufficient evidence for a conviction, so the police split you and this other person up into different rooms. They offer you and the other person the same deal. If both you and the other person stay silent, both of you will wind up spending 6 months in jail (the police have enough evidence for a lesser charge). If both you and the other person agree to testify against each other, you will both spend the next 2 years in jail. If one agrees to testify for the prosecution, and the other person remains silent, the one who testifies will go free, while the one who remains silent will go to jail for 10 years.

Considering that you can't know what choice the other person is going to make, what is the best choice you can make to minimize your jail time?

This is a classic puzzle known as the Prisoner's Dilemma. The version stated as above has a very simple solution. If you stay silent, you'll either serve 6 months or 10 years. If you testify against the other person, you'll either serve 2 years or go free. Obviously, the better choice is to testify against the other person.

This seems fairly obvious, and doesn't seem like much of a puzzle. Perhaps it needs one more aspect to make it more interesting.

Imagine this is done not just once, but repeated numerous times (perhaps both you and the other person have been charged with numerous crimes). Only after you have made your choice each time, and before the next round, are you told the other person's choice. Now, what is your best strategy?

This makes it a bit more difficult, doesn't it? This variation is known as the iterated prisoner's dilemma. Before we discuss the best strategy for this version, try some online simulations to see if you can work it out for yourself. You can try out a version using the reward of gold coins. If this doesn't help, and you have Java installed, try another version that allows you to control the other person's strategy.

If the Monte Carlo method gives you trouble, perhaps applying the Nash equilibrium might help. True, the article might be a bit long, but here is the basic idea of the Nash equilibrium, from the movie A Beautiful Mind.

Have you figured it out yet? If not, I'll let Charlie Eppes (David Krumholtz) explain the optimum strategy, from the The Art of Reckoning episode of Numb3rs, with a great rock-climbing analogy:

Yes, Tit For Tat is the best overall strategy. However, there is one strategy that beat it out in the Prisoner's Dilemma competition. A team from Southampton University created the programs that did just that.

How is this possible? The Southampton strategy was to submit 60 programs. Each of the programs was able to recognize the others by the sequence of the first 5-10 choices made. If a Southampton program recognized another Southampton program, one of the programs would always choose to cooperate with the police, while the other one would always choose to remain silent. If a Southampton program realized it was playing a non-Southampton program, it would always choose to cooperate with the police, thereby minimizing the score of the competing program. This strategy took the top three places in the competition when it was applied, not to mention numerous places at the bottom for the programs that would consistently stay silent.

So why do people still claim Tit For Tat is the best? In the classic iterated problem, it is assumed that you only have control of your own actions, and have not pre-arranged a code with the other person (as you were separated before the offer was first made).

To wind up this post, I'd like to point you to Bill Whittle's site, in which he discusses the philosophical implications of the prisoner's dilemma. If you haven't visited this site before, I also suggest you explore the previous entries, too. His essays and even his simple posts are always clear and thought-provoking.

More and more, I've been running across updates to stories I've previously posted here on Grey Matters, so here's the latest:

* Back in December, I posted Verizon: Dollars and Sense, about their difficulty telling the difference between .002 dollars and .002 cents. While Verizon claims to have straightened this matter out and trained their staff, as of last month, Verizon is still having trouble with this same issue! The original complaint was eventually refunded, but the trouble continues for other customers.

* Richard Wiseman, who can be found performing a magic square on Grey Matters Videos, has a new YouTube video, concerning an amazing color change. See if you can catch it before the secret is revealed:

* In March's Mathematical Humor post, I limited myself to humor from the Mathworld site. However, I should add Numericana's great mathematical humor section. If you haven't checked out Numericana after I mentioned it in my Monty Hall Dilemma and Mind-Boggling Amount of Mind-Boggling posts, now is a great time to check it out!

* Finally, as a fun update to all my Pi posts, check out Mazda's new car model (image opens in new window)!

Back in my lightning calculator post, I mentioned plus magazine, a free online magazine showing the fun side of math.

Now, the Pacific Institute for the Mathematical Sciences has jumped into the free mathematical magazine arena with their Pi In The Sky Magazine. It's geared towards high school math students, but is a great and fun read for just about anyone interested in math.

All of the issues (10, at this writing) are available online as PDF files. Issue 10 caught my attention because of its focus on mathematical games and puzzles. They have great articles on games and puzzes like Sudoku, Nim and the Monty Hall Dilemma.

If you enjoy this issue, you'll find plenty to enjoy in the past issues, too!

Over at Geomblog, they've posted the 8th Carnival of Mathematics!

JD2718 will be hosting the next Carnival of Mathematics on June 1st. June will wind up, on the 29th, with the Carnival of Mathematics being hosted right here at Grey Matters!

Attention, Brain Gang! Over at the various Gawker Media sites there have been numerous posts which should pique your interest.

* From Lifehacker: You can put your short term memory to the test. If you're worried about how you did, you can find 10 ways to keep your memory strong.

They also recently discussed a study suggesting that you can improve your memory with your eyes.

Would you like to learn a new language? Lifehacker shows you how to find podcasts that teach you a new language!

If you just want to learn anything, including a language, you can find 77 ways to learn faster, better and deeper. You could also try collaborative mind mapping with MindMeister.

If you like Sudoku puzzles, learn how to solve them!

* From Kotaku: The European release of More Brain Training in June is discussed. This is the sequel to the popular Brain Age game.

They also have screenshots from Big Brain Academy for the Wii.

* From Gizmodo: Learn while you're on the toilet with Brain Training Toilet Paper.

Here's some quick snippets from around the web that regular readers may enjoy:

* After a pizza parlor made an error, they made another offer to make it up to their patrons. Unfortunately for the restaurant, their patrons used geometry to figure out it was a bad deal. Who knew geometry and pizza could work so well together? Oh, yes, I did!

* Grey Matters favorite Arthur Benjamin has a 4-DVD (or 2-Transcript) set out called Joy of Mathematics. It's a set of 24 lectures focusing on making mathematics less frightening, and it's a bargain at $69.95, especially when it's normally $254.95!

* Richard Wiseman, who has been on here performing a magic square, has a new video featuring an amazing color change. Try and catch it if you can!

* In Apple's newest batch of I'm a Mac, and I'm a PC ads, the Apple Genius shows how much of a genius she truly is. This is my kind of woman!

* Speaking of genius women, here are some real-life celebrity women with high IQs. This list neglects several notable women, such as Danica McKellar (Wonder Years' Winnie Cooper) and MENSA member Geena Davis, but it's still a great read nonetheless.

Just how does our society think of mathematicians? One of the best ways to find out is to look at their portrayal in movies and TV.

The question is, just where do you find such portrayals? Oliver Knill's Mathematics and Movies page (Flash required) is an excellent place to start. It can be surprising how much math has appeared in your favorite movies and TV shows.

Once you realize the quantity of math out in popular media, how is the quality? Just because you hear the Pythagorean theorem stated doesn't mean that time was put into researching it. Thankfully, Alex Kasman comes to the rescue with his Mathematical Fiction page, in which the stories are rated not only for the quality of the story, but the accuracy of the mathematical principles, as well.

Of course, some shows have already established their reputation for mathematical accuracy among their fans. Numb3rs quickly comes to mind. Futurama and the Simpsons (surprised?) are also great sources of high quality mathematical principles, and even mathematical humor! As noted on Bender Bending Rodriguez' Mathematical Curiosities page, even jokes in the background of Futurama are well researched.

If you like math humor like this, you'll be see more like this when new Futurama episodes arrive in 2008!

Yes, the 7th Carnival of Mathematics is up over at nOnoscience. This edition of the Carnival of Mathematics has plenty of recreational mathematics posts that are perfect for Grey Matters readers!

The Geomblog will be hosting the next Carnival on May 18th. Also, don't forget that the Carnival will be hosted right here on June 29th!

While I've discussed methods of calculations numerous times, I haven't talked much about the means.

The oldest calculating device would probably be the abacus.

In the west, however, the most popular calculating device, for the early part of the 20th century, was the slide rule. If you're not familiar with them, Eric's Slide Rule Site features a wonderful introduction to them. Once you get used to the idea of logarithms, they're easier to understand. This YouTube explanation of slide rules may help you understand further.

Once you've read and understand about slide rules, you may want to try them out yourself. You can try out the UniVirtual Slide Rule (Java required) or even try out different slide rule models (Javascript required). If you prefer to run a slide rule application on your system, you can download Caveman's Calculator for Mac OS X or Slide Rule 1.1 for any system.

Once you get the basics down, you can even learn a few slide rule tricks to make your work easier.

The death knell for the slide rule, of course, was the modern electronic calculator. Most operating systems today even come with a virtual calculator. The modern calculator is worth exploring further. You may not even realize that the calculator with your operating system can be switched into other modes of which you may not be aware (See here and here).

You can download numerous other calculators, depending on your goals. For example, let's say you wanted to show off your ability to multiply two 9-digit numbers in your head. In this case, you would need a high-precision calculator, such as the Magic Number Machine for Mac users or the Precise Calculator for Windows users. The variety of specialized calculators is truly mind-boggling.

That covers the past and present of calculators, but what about the future? The most amazing version of a future calculator that I've seen is the one in this video:

You may be thinking that this free-form calculator is simply a pipe dream, but it is available as a free download (Java required) from Weapons of Math Construction.

That's enough for now. I've given you plenty of calculators to play with, so I'll let you go play.