Here’s a cool spin on a game you probably haven’t played since you solved it in elementary school. You and another player take turns naming numbers one through nine. Each number may only be used once. The player who collects any three numbers that add up to 15 wins. If all numbers are used up without someone getting 15, the game is a draw.

Once two players have figured out all the strategies to winning the game, it becomes easy, and imperative, to prevent the opponent from winning. Just like Tic-Tac-Toe, two experienced players will have games that always end in a draw, which is important for teaching an advanced Artificial Intelligence why nobody wins playing Global Thermo-Nuclear Warfare–sorry, had a flashback to the 1983 movie War Games there.

In the research paper The Game of JAM: An Isomorph of Tic-Tac-Toe, John A. Michon notes how this game promotes a different way of thinking about a classic problem (he refers to this game as “Number Scrabble”):

Although the games are mathematically equivalent, they are likely to differ psychologically, because they require different sorts of information. Number Scrabble is a numerical game requiring addition and subtraction of numbers, whereas Tic-Tac-Toe requires a spatial representation, which can only be disposed of with some difficulty, even by fairly experienced players.

Tic-Tac-Toe and the Numerical version are Mathematically equivalent, and we can plot this out visually. In the table below, all rows, columns, and diagonals add up to 15.

4	3	8
9	5	1
2	7	6

Plaing Tic-Tac-Toe on this grid is the same as playing the numerical version. You can find a shockwave demonstration of this principle to play for yourself. This website calls the game Add Fast.

Taking the number five opens up four paths to victory. The numbers {1, 3, 7, 9} each open another path. {2, 4, 6, 8} are the least valuable strategically.

This seems like a cool game to play with your kids. When they get good at it, show them the relationship to Tic-Tac-Toe. Or you can learn the number-grid, and amaze your friends by kicking their butts at this game.

In 1964 Lyndon Johnson’s campaign ran the following ad scaring Americans in to voting for him with the idea that Barry Goldwater would start a nuclear war if elected:

[youtube=http://www.youtube.com/v/OKs-bTL-pRg]

Hillary Clinton’s campaign is spinning their recent primary wins as attributable to their “3a.m. Phone Call” ad, which uses a similar tactic against Obama:

[youtube=http://www.youtube.com/v/M70emIFxETs]

Fear is a powerful motivator. The 9/11 terrorist attacks took President Bush from a 50 percent approval rating all the way up to 90 percent in a month. They convinced us to invade a country that had nothing to do with the attacks, and without a plan for securing the country once occupied.

In the world of rhetoric, this is known as the Politics of Fear, but geeks aren’t fooled for a moment, because we have the awesome power of mathematics to embolden us!

Behold! Putting things in perspective, 2,998 people died as a direct result of the 2001 terrorist attacks in America. Comparing that number to a 2002 list of causes of death by rate chart, we can determine how worried we should be about terrorism in comparison to all the other things in the world trying to kill us, and decide whether we are getting our $2 Trillion Worth out of the Iraq war.

We can quickly see that we are 19 times more likely to die in an automobile accident than from terrorism, a figure that will surely go up as our roads crumble as our all our infrastructure money goes to the “War on Terror.” We are 263 times more likely to die of Cardiovascular disease, which, for the price of the Iraq War, we could buy a whole lot of research, education, and prevention.

Admittedly, I’m fudging things a bit by comparing 2001’s terrorism deaths to 2002’s causes of death, but there were zero deaths in America from terrorism from 2002 to 2007 (unless you count Americans in Iraq, another preventable tragedy), but my calculator keeps giving me an error when I divide by zero, so I’m biasing these numbers heavily towards terrorism’s favor as a cause of death, and it still looks miniscule.

Without that bias, 2,998 deaths spread over seven years would make us 11 times more likely to die of Hepatitis B, a threat that requires a well-funded CDC to protect us, and 131 times more likely to die in a car accident. Remember that when you see a new pot-hole or a delayed road project. We know that education, more than any other factor, extends life spans, too bad we can’t quantify the lives were loosing on that front.

This chart from Wired best illustrates the disproportionate nature of our fears, and it’s not just terrorism that we are disproportionately afraid of shark attacks, airplane crashes, and other unlikely causes top our lists also.

As for 3AM phone calls, we all know first hand what an experienced leader is capable of when informed of a national crisis:

 

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Note: I am well aware of the “You don’t know how many terrorist acts have been prevented since 2001” argument. I don’t believe a Department of Homeland Security, looking to justify its funding, would keep a thwarted terrorist act a secret for a second.

One Gigabyte 20 Years Ago (left) One Gigabyte Today (Right) source

Bill Gates is often misquoted as having said, “no one will ever need more than 640K of memory,” in the 1980s. 24 years ago, my Commodore 64 personal computer ran games like “Mail-Order Monsters” and “Archon” on a mere 64 kilobytes of memory. This was a huge advance over my 1977 Atari 2600 game console, which ran “Pong” and “Space Invaders” on a scant 128 bytes of memory. Today my dual-core Pentium uses a gigabyte of RAM, about 7.8 million times as much memory as the Atari, and, after upgrading to Windows Vista, even that doesn’t cut it anymore.

From bits to bytes, kilobytes, megabytes, gigabytes, and, with impending DVD technological advances, terabytes, our computing power grows exponentially. This empirically observed fact is known as Moore’s Law, named after Intel co-founder Gordon E. Moore, who observed in 1965 that the number of transistors on an integrated component doubles every 18 months. In other words, computers double in power every year and a half. This Law of Computing has held true now for over 40 years in an explosion of processing power that allows for what history will record as the Information Age, the times in which we are currently living.

Now it’s time to familiarize ourselves with a new measurement, the exabyte. We can thank research firm IDC’s white paper The Expanding Digital Universe for introducing us to this latest milestone, which estimates the human race collectively produced 161 exabytes of data in 2006.

So what’s an exabyte? To visualize this number, it’s helpful to begin at the smallest measurement of data, the bit. A bit is a 1 or 0, “on” or “off,” “true” or “false.” Up one level from this binary state we have the byte, which is 8 bits. If you open Notepad on your computer, type any one letter and save the file, you have generated one byte of data, which you can verify by right-clicking on the file and selecting “Properties.”

Every additional character typed and saved will add another byte to the file’s size. Every 1,000 characters is a kilobyte, and every 1,000 kilobytes a megabyte. A 90,000-word novel translates into about 0.5 megabytes¹. An exabyte is 1,000,000,000,000,000,000 bytes of data, or 500 billion novels. That’s 77 novels written for every person on Earth², and we are producing 161 times that much data, 230 billion CDs worth³, or nearly 12,400 novels for every person on Earth every year.

We produced more data last year than has been produced in the last 5,000 years of human history. That’s just for 2006, and that’s only the beginning. “In 2010, the amount of digital information created and copied worldwide will rise six fold to a staggering 988 exabytes,” that’s 12 Petabytes short of having to adopt yet another term of measurement, the Zettabyte.

The search engine Google is named after the largest number the nephew of mathematician Edward Kasner could think of, the googol. It is the number one followed by 100 zeros. By one recent estimate, it takes 450,000 computers networked on server farms to run the Google search engine, indexing 8 billion Web pages every year. I wonder when we’ll be talking about our hard drives (or maybe they’ll be flash drives by then) in terms of googlebytes?

And then we still have the googolplex waiting for us in the distant future, the number one followed by a googol of zeroes.

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¹500,000 characters in Novel based on a Microsoft Word Count and Character count of one of my novels, which came out to 450,000 characters for a 82,000 world novel. So this is a very conservative estimate.

²1,000,000,000,000,000,000 bytes translates to
1,000,000,000,000 megabytes which translates to
500,000,000,000 novels divided by 6.5 billion human beings

³CDs hold 700MB of Data
700,000,000
161,000,000,000,000,000,000
230,000,000,000

A New Kind of Science

Many books I like to read with a yellow highlighter, reading Stephen Wolfram’s ANKOS I was compelled to whip out a red pen. While his 1,000-plus page field-guide to cellular automata and complexity theory is brimming with fantastic examples of all shapes, sizes, and dimensions, Wolfram’s writing and failure to acknowledge accomplishments in the field beyond his own research make this book a difficult read.

Wolfram violates the rule of science writing that you must disassociate yourself from your research. I was skeptical of the importance of this principle, until I saw what happens when you don’t follow it:

Just over twenty years ago I made what at first seemed like a small discovery: a computer experiment of mine showed something I did not expect. But the more I investigated, the more I realized that what I had seen was the beginning of a crack in the very foundations of existing science, and a first clue towards a whole new kind of science.

This book is the culmination of nearly twenty years of work that I have done to develop that new kind of science. I had never expected it would take anything like that long, but I have discovered vastly more than I ever thought possible, and in fact what I have done now touches almost every existing area of science, and quite a bit besides.

Wow! Stephen Wolfram considers his book an Earth-shattering iconoclasm that will revolutionize science, and it’s all on Wolfram himself and his 20 years of research; however, despite his repeated use of “I” and casual dismissal of all the research preceding him, Wolfram is not publishing in a vacuum, and that hurts his efforts profoundly.

Put simply, Wolfram believes he has discovered Emergence, the idea that complex systems and patterns can arise out of simple processes or rules. Wolfram mentions searching for patterns in primes, but never mentions Ulam’s spiral. Mentions seeking patterns in pi, but never mentions Carl Sagan’s Contact, which entertained the idea first. Chaos/Complexity Theory gets mentioned in a footnote. A footnote!!! Wolfram never acknowledges that he is standing on the shoulders of giants like Alan Turing, John Von Neumann, or Edward Lorenz.

Lines of Prime Numbers in Ulam’s Spiral

Maybe Wolfram isn’t ignoring all the history behind his subject, maybe in the 15 years of writing his book, he simply never noticed that it’s all been discovered without him, before he even started writing. If we were to lose Einstein’s Theory of Relativity today, someone else would uncover it within a few years. That’s the nature of truth, everyone can arrive at it independently.

The problem is that Wolfram’s failure to explore the near century’s worth of work by his peers on this subject cripples his presentation. Instead of a broad, eclectic overview of ideas from across the field of research shedding light on each of his examples, we are forced to look at them with Wolfram’s blinders on, and given only his insights alone. This is a frustrating treatment, teasing at enlightenment, but never yielding any depth.

Wolfram hasn’t invented anything. Speculation isn’t invention. In the end nothing has been discovered. There is only more wonder. People speculated on these patterns before Wolfram, and they will speculate after him.

Cellular automata, emergence, chaos theory, and other incredibly complex mathematical wonders produced by basic rules allowed to play out over time are absolutely fascinating concepts. You can lose yourself for hours staring at fractals. You can wonder at the increasing wave function of unpredictability produced on a system by something as seemingly mathematically insignificant as a butterfly flapping its wings. You can ponder infinitely complex numbers like pi and phi impossibly running away forever, while appreciating the way they somehow manifest in nature. It defies logic.

Luckily, Wolfram’s book repeatedly appeals to his readers to take up this subject, to explore the phenomena of which he provides so many wonderful examples. Anyone experiencing an Ionian Enchantment from Wolfram’s book will continue his train of thought and discover Turing, Neumann, and myriad of mathematicians and computer scientists immersed in this field. They will discover the whole realm of mighty minds who have also immersed themselves in these puzzles.

Then they will return to A New Kind of Science, and appreciate that Stephen Wolfram has put together a very good coffee table book on cellular automata, just not a revolutionary one.

Flatland the Movie VS Flatland the Film

I really enjoyed and appreciated Edwin Abott’s 1884 classic book Flatland, A Romance of Many Dimensions, which tells the story of Square, a lawyer living in Flatland, a two-dimensional world that has height and width, but not length. It’s in the public domain and free to download at a variety of places if you’re interested in checking out a book that will change the way you look at the world.

In 2007, two animated adaptations of Abott’s book arrived in DVD format Flatland the Film and Flatland the Movie. While neither was wholly satisfying, they each had their good points.

FtF was definitely the more hard-core of the two films. We can see its Flatlander’s internal organs, the clockwork of their brains and hearts, just as we should being Spacelanders looking down on them, and just as four-dimensional beings would see our insides. The social dynamics of Abott’s world are preserved here, in all its male-chauvinist, authoritarian glory. The Flatlanders in this representation are covered with wiggling hairs, which we may assume aid their locomotion and interacts with the world. Unfortunately, the film is filled with intertitles that don’t add anything to understanding Flatland, but do everything to let you know the writer thinks you’re too stupid to get it. I definitely didn’t appreciate having my film interrupted so I could be insulted every few minutes with statements like, “Did you get that important plot point?” and “SuchandSuch should be obvious to you.”

FtM side-steps many of Abott’s more controversial social issues, or rather dumbs them down into a substantially less controversial form. Women and Men are both Squares, unlike Abott’s world, where women are intellectually inferior, however physically superior lines. FtM’s Flatlanders have fractals for their insides, and they carry suitcases with them by magical means. When they turn upside down, the eye and mouth of these Flatlanders magically switch places so as not to upset the viewer. The movie does present a disclaimer that it is not a true representation of Flatland, so as to make it more palatable to Spacelanders like ourselves.

FtM was 100% kid-safe, its concepts presented in an easily digestible format, and was filled with characters resembling those we have here in Spaceland.

FtF was most definitely not something you could watch with your kids. In fact, one scene, where an asymmetrically-shaped senator with revolutionary ideas is assassinated in the public forum, drags on forever as isosceles triangles hack him to pieces, and then into smaller pieces, and then even smaller pieces. Not cool. I was looking for enlightenment and got gross juvenile indulgence.

At 30 minutes in length, FtM barely skimmed the multitude of fascinating aspects to Abott’s world and left me wanting for more mathematical goodies. Luckily the special features on the DVD included a talk with a mathematician who walked through a thought experiment of going through our Spaceland’s three-dimensions into Hyper-Spaceland’s four-dimensions.

At an hour and a half, FtF had me checking my watch about halfway through, trying to figure out how much longer they could draw it out, and then was left gawking as the credits rolled, “That’s how they ended it??? Nooooooo!!!”

FtM has a vastly superior website with flash animations and sound effects. FtF has a flat brochure website with black text on a white background. FtM runs $30, FtF runs $22. These factoids had no affect on my impression of either movie, I mention them because there they are.

I have to go with Flatland the Movie, despite what I think is the flaw of not being alien enough in its presentation of the two-dimensional world, the film is accessible and it focuses on the intellectual, enlightenment principles I admire. The Movie’s website does make the dishonest claim that you need to buy the Special Educational Edition of the DVD if you want to show it in the classroom.

However Section 110(1) of the Copyright Act qualifies showing any film in a classroom for education as Fair Use; and, therefore, not a violation of copyright law. So share this film with your students, follow up with the extras, and have an enlightening discussion about life in dimensions one through four and beyond. You can supplement this discussion with the book, and maybe provide a few screenshots of Flatland the Film to explore the hard-mathematical realities of these worlds.

All Hail Hypnotoad!

I finally got to pick up a copy of the First Carbon Neutral DVD from FOX, Futurama: Bender’s Big Score, this weekend, and I’m very much enjoying my geek-humor fix.

Some big brains go into making Futurama. Executive Producer David X. Cohen has a master’s degree in computer science from UC Berkeley. Writer and executive producer Ken Keeler has a Ph.D. in applied mathematics from Harvard University. When asked if all those years of education ever paid off, Keeler replied in an interview:

Well, sure. For example, Bender’s serial number is 1729, a historically significant integer to mathematicians everywhere; that “joke” alone is worth six years of grad school, I’d say.

My favorite extra on the DVD is a lecture by Dr. Sarah J. Greenwald explaining many of the mathematical jokes appearing on the show. She also has a website of Futurama’s Math and Science references.

My all time favorite reference from the show’s history comes when the winner of a horse race is announced, and Professor Farnsworth angrily tears up his losing ticket and exclaims, “No fair! You changed the outcome by measuring it!” This references the Heisenberg Uncertainty Principle, a fact of quantum physics that drove Einstein crazy, and oh boy you should hear the crickets chirping when I quote this joke while watching sports with my friends.

There’s also a half-hour long pilot episode of “Everybody Loves Hypnotoad,” which is a half-hour of Hypnotoad doing it’s thing with a commercial break and laugh track. I sat through all of it, regularly breaking out into laughter because I… just… love… Hypnotoad. Excuse me, I have to go watch that episode again.