Mining for the ‘Prime’ Jewels of Numbers

Prime numbers, numbers which are only divisible by 1 and themselves, are probably familiar to everybody from grade school as they’ve been studied intensely going back to the ancient Greeks. Back in the 17th century a monk named Marin Mersenne thought he had a formula that would produce primes every time. He didn’t, but his formula still seems to produce them a significant part of the time, and in particular it’s currently the best bet to produce hitherto-unknown large prime numbers. And by large we mean LARGE: the largest known prime is nearly 13 million digits long. If you wrote 10 digits per inch, that prime would still stretch over 20 miles long!

This report from NPR describes the Great Internet Mersenne Prime Search (GIMPS for short), which is how many of these large Mersenne primes have been discovered. GIMPS distributes a free, downloadable program that runs on any computer and helps in the search by using portions of your computer’s downtime. Instead of being in screen saver mode your computer, together with about 30,000 other computers worldwide, takes part in some computations. There’s prizes involved, too: if your computer is the one that finds the next Mersenne prime, you could win $3000 dollars. (If and when a prime with 100,000,000 digits is found, then a prize of $150,000 is up for grabs!)

World Series of Poker: Attack of the Math Brats


The time we live in has been called The Information Age by some because of the mountains of data that have become available. Often, analyzing that data can end up revealing better ways to do things, or even completely overturn traditional wisdom in favor of new techniques, techniques that are actually backed up by data evidence. (Super Crunchers was a recent book about this very trend.) As this Time magazine article relates, this phenomenon is happening very quickly and very publicly in the high-stakes world of Texas hold’ em poker. The old guard, whose play is based in part on “reading” their opponents, is being overwhelmed by young players armed with probability-based strategies, strategies divined by analyzing reams of data obtained from the millions of online poker games played on the internet. The article leads off with a quote from old-guarder Phil Hellmuth, who’s won a record 11 World Series of Poker championship bracelets:

“The reason I won 11 bracelets is my ability to read opponents,” he explains. “These new guys are focused on the math. And they are changing everything.”

The old guard is not going out without a fight, of course. Many of them are picking up the new techniques and trying to meld them with their own expertise. But the new ‘math brats’ are setting the pace: 21-year-old Joe Cada won last year’s Poker Main event, netting $9 million and becoming its youngest winner ever. The previous youngest winner was 22-year-old Peter Eastgate, who won in 2008. The youngest winner before that was Hellmuth, and he’d held that record for nearly 20 years.

Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present

This new book by George Szpiro details the various systems people have used through the centuries to elect popes, presidents, mayors, and even #1 ranked sports teams. Voting theory has made an appearance on this blog previously, when the Academy of Motion Picture Arts and Sciences made a switch in their voting tabulation system recently. Szpiro’s book received an excellent review in the New Yorker magazine, a review that itself does a very nice job of sketching the various voting systems that have been tried, and the problems and advantages of each.

20 Moves or Less Will Solve Rubik’s Cube


People have been solving Rubik’s Cube since it was invented in 1974, and some have gotten quite fast at it. But just how fast is possible? In other words, if you were an omniscient solver, what’s the most number of moves it would take you to solve any starting position of Rubik’s cube? Since there are 43,252,003,274,489,856,000 possible starting positions, many of which require different strategies to solve, answering that question seemed outside the realm of possibility: a computer dealing with each starting position in less than 20 seconds would still require 35 years of analysis. But by using some high-powered mathematics to chop the problem down to size, a team consisting of a mathematician, a Google engineer, a math teacher, and a programmer (and yes, a powerful computer) have indeed answered the question, determining that all starting positions are solvable in 20 moves or less. The story was picked up by a number of news outlets, including NPR, Discover, BBC News, and others.

Short-Circuiting Malaria

Malaria affects hundreds of millions of people throughout the world annually, killing over one million of them. Almost all of these victims live in third-world countries, and many of them are children—one estimate is that a child dies of malaria every 30 seconds. This Newsweek article, by Daniel Lyons, describes Intellectual Ventures, an American start-up that is trying to use first-world know-how to combat the disease. The company got its start when its founder, Nathan Myhrvold, was told by Bill Gates, “Come up with some good ideas and I’ll come up with some money to pursue them.” Some of the inventors mentioned are mathematicians, and one of those ideas is a massive mathematical model of the disease that “lets researchers see the effect of potential vaccines that don’t exist, so they can choose which one to develop.” Other ideas are also mentioned and many of them, as Myhrvold admits, sound farfetched. But really, as Lyons concludes, how can you argue against trying?

Addendum: In another development on the malaria front, researchers at Case Western Reserve University recently developed techniques that can quickly identify drug resistance in strains of malaria. The new technique is expected to “enable the medical community to react quickly to inevitable resistance and thereby save lives while increasing the lifespan of drugs used against the disease,” according to the article “New Methods, New Math Speed Detection of Drug-Resistant Malaria” from ScienceDaily. The key—developed by mathematics Prof. Peter Thomas and a student, Drew Kouri—involved “using a nontraditional mathematical analysis that’s proved more accurate than traditional methods.” (That ‘nontraditional analysis’ mentioned, by the way, was simply switching to polar coordinates.)

Finally, 140-year-old Boltzmann Equation solved

During the late 1800’s, James Clerk Maxwell and Ludwig Boltzmann developed a differential equation that predicted how gaseous material distributes itself in space and how it reacts to changes in things like temperature, pressure or velocity. The only problem: for 140 years, solutions to the equation could only be found for gases that were perfect equilibrium. Enter Philip Gressman and Robert Strain of the University of Pennsylvania. Using recently developed mathematical techniques, they were able to describe solutions to the equation under any conditions. As this article in the Time of India notes, now it’s possible to “describe the location of gas molecules probabilistically and predict the likelihood that a molecule will reside at any particular location and have a particular momentum at any given time in the future.”

Monkey Brain ‘Hardwired’ for Simple Math

There seem to have been a number of articles recently that detail evidence that humans are “hard-wired” for math; in fact, one such article appeared on this blog already. Now comes further evidence that not just humanity, but primates in general are innately mathematical. An article appearing on Yahoo describes findings by German neurobiologists that “rhesus macaques can engage in abstract mathematical reasoning using specific brain cells dedicated to the comprehension of math rules and relationships.” In fact, the article notes that this recent observation is just the latest piece of evidence for innate math abilities: apparently a 2007 study found that young chimpanzees actually out-performed some human adults in tracking numbers and remembering sequencing!

Addendum: And here comes more evidence. A new article entitled “Preschoolers Use Statistics to Understand Others” details the result of an experiment where babies intuited a character’s feelings about a kind of toy by observing how likely the character was to select the toy. I’ll let you read the article to see exactly how the experiment worked. I’m guessing that many more articles with findings like this one are on the way.

Packing Tetrahedrons, and Closing In on a Perfect Fit


Another NY Times story, this time taking its starting point with Aristotle: “More than 2,300 years ago, Aristotle was wrong.” What he was wrong about was regular tetrahedrons, four-sided figures where each side is an equilateral triangle. (Dungeons & Dragons players will recognize the tetrahedron as the game’s 4-sided die.) Aristotle thought that tetrahedrons could be packed together perfectly, so that, for instance, you could fill a box with tetrahedrons with no unfilled space appearing anywhere in the box. 1800 years passed before somebody realized he was wrong, but even then nobody had a clue as to just how ‘well’ tetrahedrons could pack.

Fast-forward to the modern era, when tetrahedron packing underwent something of an arms race. John Conway and Salvatore Torquato from Princeton found a packing that they could mathematically prove would fill at least 72% of the space. Paul Chaikin at NYU then had high school students fill aquariums and other things with hundreds of the D&D dice, discovering that 72% was easy! You could always do better. But how much better? And was there a systematic way to do it (rather than dumping dice into an aquarium)?

This past year saw the arms race accelerate, with competing groups coming out with successively better packings: first Elizabeth Chen and Jeff Lagarias, from the mathematics department at Michigan, published a packing pattern that guaranteed 78% coverage. Then two distinct patterns emerged that got to 85%: an extremely complicated one from Sharon Glotzer, also from Michigan, but in the chemistry department (and coming at things from the point of view of developing new materials for the Air Force), and an extremely simple one by a group from Cornell University. Just weeks ago, Dr. Torquato and Yang Jiao, his graduate student, tweaked the Cornell pattern to get coverage of 85.55%. And finally, just this past Monday, Elizabeth Chen posted a pattern that reached 85.63%. Can you do better?

The Ninth Annual Year in Ideas

2009 Year In Ideas

The NY Times Magazine annually publishes its The Year in Ideas issue, devoted entirely to “the most clever, important, silly and just plain weird innovations … from all corners of the thinking world.” A surprising number of these ideas are based on a study or research article or something similar that employs some bit of mathematical and/or statistical analysis. The ones I’ve listed below are chosen as being the ones that most prominently feature mathematics ideas, or feature mathematics and/or mathematicians centrally. Listed alphabetically:

  • Black Quarterbacks Are Underpaid by Jason Zengerle describes the statistical analysis of two economists, David J. Berri and Rob Simmons, who discovered that in the NFL black quarterbacks are typically paid much less than white quarterbacks. Their analysis goes farther, however, and notes that the apparent cause is not necessarily racism. Instead, the NFL quarterback rating statistic is the culprit. NFL contracts are often based on hitting certain statistical levels, and for quarterbacks the statistic used is often the QB rating. Since QB rating fails to count rushing yards at all–something that black quarterbacks typically excel at–black quarterbacks are typically ‘discounted’, QB rating-wise.
  • Forensic Polling Analysis visits a topic seen already here in this blog: the suspicious polling numbers of the polling firm Strategic Vision LLC. You can visit that entry or the Times article for more info.
  • In a blow to meritocracy-lovers everywhere, another entry notes that Random Promotions, rather than merit-based ones, can actually produce better businesses (and typically do, at least in simulations). The article by Clive Thompson describes a study done by a trio of Italian scientists in which the researchers created a virtual 160-person company and then tried out various different promotion schemes within the company, with the aim of seeing which scheme improved the company’s productivity the most. Promoting on merit turned out to be a lousy idea (at least for the company as a whole) while promoting at random turned out to be the top strategy. In the middle was the curious idea of alternately promoting the best and then the worst employees. The fact that the mixed best/worst strategy outperformed the merit strategy is yet another example of Parrando’s Paradox, a phenomenon first identified by game theory.
  • Massively Collaborative Mathematics features the first mathematical theorem proved by a ‘collective mind’, if you will. In January 2009, Timothy Gowers, one of the top mathematicians in the field, proposed on his blog that the mathematical community, as a whole–or at least that portion that knew and read his blog–attack a long-standing unsolved problem in mathematics known as the Density Hales-Jewett Theorem. Contributors ranged from eminent mathematicians to high school teachers, and hundreds of thousands of words worth of ideas were eventually proposed, developed, discarded, combined, and so forth. Gowers had initially set the bar low, hoping this ‘Polymath’ project would result in “anything that could count as genuine progress toward an understanding of the problem.” Instead, six weeks later the problem was completely solved. A paper detailing the result, authored by D.H.J. Polymath, has been submitted to a leading journal.
  • Finally, the (alphabetically) last idea listed, “Zombie-Attack Science,” features a story that appeared on this blog previously. See that entry, or the Times article, of course, for details.

Computer Program Cracks Cipher That Stumped Thomas Jefferson


Nunmber 68 of the Top 100 Science Stories of 2009 (courtesy of Discover magazine) is the story of Robert Patterson, an American mathematician from 200 years ago. Patterson and our third president, Thomas Jefferson, shared an interest in cryptography, and in 1801 Patterson sent Jefferson a letter containing a message encrypted with a code that Patterson claimed would stump humanity “until the end of time.”

Well, it did stump humanity for over two centuries, but Lawren Smithline of Princeton, New Jersey cracked it last year. You can read how in the article. (The encrypted message? The Preamble to the Declaration of Independence.)