Games and Toys: Math games and toys are the focus of these summaries as we embark on the summer months. An interesting TED talk concerning the value of logic games and puzzles was referenced on a recent post from Math for Love. If you agree with the speaker, who is a high school math teacher, that math as a required subject should be omitted from the high school curriculum, then there's even more reason to spend some time exploring games and toys.
"Statistical Mechanices of Magnet Balls," by Brian Hayes. Bit-Player, 4 May 2012.
Science writer Brian Hayes documents his search for the "lowest energy" configurations of the magnetic buckyballs that one can buy at many toy stores and science gift shops. The post includes the results of his own exploration in which he dropped different configurations of balls down the side of a large bowl to see how they collided and combined to create one formation. Mathematically, one of the most interesting parts of the post was his observation that a collection of balls arranged in a rectangle or in a parallelogram were both stable and wrapped up into sturdy cylinders, but the rectangle could not be easily sheared to create the parallelogram without its bouncing back. It made me want to experiment myself! Interesting connections here between math (sphere packing), physics, and ideas from chemistry come to the surface in this blog entry.
"How To Win at Battleship," by Alisha Harris. Browbeat--Slate's Culture Blog, 16 May 2012.
As the game-inspired movie Battleship cruises into theaters, we may see a surge in sale of the classic toy. And while classic board games like Battleship might be easy to play, after a while most players abandon random guessing and develop their own strategies. This blog reminds readers of the rules of Battleship (and its variations) and explores a variety of progressively more effective strategies. The Slate post is actually a summary of a much more in-depth analysis given in the December 2011 blog post from Data Genetics. One typical strategy is "hunt and target" in which players "shoot" at random until they hit a "ship" and then target the adjacent squares to "sink" the ship. Just by observing that the shortest ship has length two, one can use parity to cut in half the number of squares that are searched in the "hunt" phase. Then by using the fact that the players are required to call out when their ship has been sunk, another improvement can be made. To analyze the effectiveness of each strategy, the number of of games completed after a given number of shots is plotted. The Data Genetics blog also includes mathematical analyses of Candyland, RISK, and most recently Hangman. The Hangman post is my favorite as it takes the reader through a variety of reasonable but less-than-ideal strategies. There's even an old entry entitled "Markov Chains, Monte-Carlo, and Chutes and Ladders."
"Engaging Math Activities for the Summer Break," by Alex Bogomolny. CTK Insights, 21 May 2012.
Learning through play is, by some accounts, the best way for children to learn. This post highlights a variety of simple mathematical games for children, including some that encourage kids to come up with informal proofs. One game that encourages a sense of "proof" involves figuring out all the combinations of ways that a number of matches or "pins" can be arranged to form a polygon. Aside from merely explaining the rules of these games, many of the entries include an interactive applet, so that kids who want to play on the computer can do it on their own. For instance, in the polygon game the user can choose the number of matches, and then click on a match to change its orientation. The game keeps a tally of the number of head-head, tail-tail, and head-tail vertices so that the player can see emerging patterns. Another game that encourages explanation is the "breaking chocolate bars" game that asks the player to find the minimum number of breaks that is required to completely break up a 5 x 6 chocolate bar into the smallest square pieces (that are normally found scored into the bar). Try it!
--- Brie Finegold"The Ancient Sexual Revolution that May Have Spurred Human Monogamy," by Maia Szalavitz. Time Magazine, 29 May 2012.
"Researchers have long wondered why---unlike our sexually promiscuous chimpanzee-like ancestors---humans developed strong pair bonds with individual partners," writes Szalavitz. Several scenarios have been proposed. Now, in a recently published study in the Proceedings of the National Academy of Sciences, Sergey Gavrilets, a professor of ecology, evolutionary biology and mathematics, has "shown mathematically that some of these scenarios are more likely than others." In particular, "Gavrilets' study suggests that a sexual revolution occurred, led by low-ranked males and faithful females. Low-ranked males, who had no hopes of physically overcoming the dominant members of their groups, instead began providing extra food to certain females, to curry sexual favor," Szalavitz explains. "These females responded by remaining faithful to their breadwinning males. That change in behavior favored the reproductive success of these monogamous couples."
Leading evolutionary theorist, Sarah Hrdy, "commended Gavrilets' research, though she took issue with some of the assumptions made in his mathematical model." These include the fact that Gavrilets' model depends upon research that doesn't account for women's cooperative rearing of children. In addition, Hrdy says, "I don't think human mothers in the past could count on the long-term survival and fidelity of provisioning mates any more than mothers today can. It looks to me like males are responding to a wider range of factors than can be represented in such a model." Gavrilets is working on adding variables to make the model more complete.
--- Claudia Clark
"Space-filling solution could boost Wi-Fi security," by Jacob Aron. New Scientist, 17 May 2012.
Mathematicians have long investigated packing problems, such as the famous Kepler Problem, which seeks the most efficient way to pack spheres in three-dimensional space, as well as covering problems, which ask questions such as, How many square tiles are needed to cover a circular floor? Physicist Carolyn Phillips of the University of Michigan and her colleagues work on what are called filling problems. An example of a two-dimensional filling problem is, How do you cover a triangle with disks of varying sizes, without the disks extending beyond the boundary of the triangle? "It is essentially a mathematical version of coloring in a shape using differently sized blobs of paint," the article says. Phillips and her colleagues have explained how to solve the filling problem for two-dimensional shapes. Solving the problem in three-dimensions is conceptually similar, she told New Scientist, but designing an algorithm to produce an optimal solution is more challenging. The work grew out of studies of nanoparticles and could eventually have applications to cancer treatments and Wi-Fi networks. The work appeared in the article "Optimal Filling of Shapes" in May 2012 in Physical Review Letters.
--- Allyn Jackson
"A Mathematical Challenge to Obesity," by Claudia Dreifus. The New York Times, 14 May 2012;
Using a mathematical model of how a person's body weight changes in relation to food intake and exercise, mathematician Carson Chow challenged conventional wisdom on dieting and probed the cause of America's obesity epidemic. Chow, a researcher at the National Institute of Diabetes and Digestive and Kidney Diseases, worked with another mathematician, Kevin Hall, to develop a simple model for the myriad factors affecting body weight, and found that reaching a maintainable new, lower weight by lowering food intake takes about three years--much longer than previously thought. As for the bigger question of the cause for the obesity epidemic in America, Chow's research indicated a simple solution: a large increase in food intake. Chow explained during an interview that the amount of food available to a person has increased by 1,000 calories per day since the 1970s, due to more efficient farming processes and technology, and his model shows the increase in calories more than accounts for America's weight gain. (Image: Mathematical Moment on obesity. Click on the image to hear Chow's colleague, Kevin Hall, talk about math and obesity.)
--- Lisa DeKeukeleare
"The Case For A Presidential Science Debate." NPR's Talk of the Nation--Science Friday, 11 May 2012.
Ira Flatow, the host of Science Friday, leads a discussion about the case for a presidential science debate with Dr. John Allen Paulos, Professor of Mathematics at Temple University; Shawn Otto, CEO and Co-Founder of ScienceDebate.org; and former Republican congressman Vernon Ehlers, who is also a physicist. They discuss how public policy decisions need to be based on evidence instead of opinion or belief and how scientific and mathematical knowledge is crucial for a president to be able to deal with all the intricacies of government. Whether it is understanding the financial structure of the United States government, tackling the budget deficit or dealing with the decline in education standards, scientific understanding is essential.
--- Baldur Hedinsson
"'Invisible' planet found by 150-year old gravity measuring technique," by Rob Waugh. Mail Online, 11 May 2012.
The Science section of the Mail Online has a story about a 150-year-old math technique being put to a new use. More than 150 years ago French mathematician Urbain Le Verrier predicted the existence of Neptune based on small deviations in the motions of Uranus. Le Verrier had no way of seeing Neptune, but using a mathematical method to assess the gravitational effects that another planet would have on Uranus, Le Verrier predicted the position of the unseen planet. Now Dr. David Nesvorny of Southwest Research Institute and his fellow researchers have used this technique to find an unseen planet orbiting a star in a far away solar system. The team’s claim will be put to the test by new observations from NASA’s Kepler Telescope. (Image: Southwest Research Institute.)
--- Baldur Hedinsson
"Roulette beater spills physics behind victory," by Michael Slezak. New Scientist, 10 May 2012.
Physicist Doyne Farmer used math and a wearable computer to beat Nevada roulette tables in the 1970s, but never revealed how he did it--until now. Farmer recently circulated a draft paper on his method after researchers from Australia and Hong Kong published a similar method in the journal Chaos, revealing the moneymaking secret Farmer kept from the casinos. Both methods use a counting device and mathematical calculations to predict where the ball will land based on when the ball and a specific part of the wheel pass a certain point. Farmer's prediction relied on the amount of air resistance the ball encounters, while the newly published method focuses on frictional forces to determine where the ball will ultimately fall after the chaotic bouncing that occurs when it first drops from the wheel rim. The Chaos paper indicates the method correctly predicted the half of the wheel in which the ball would land in 13 of 22 trials, turning the odds of winning from 2.7% in favor of the casino to 18% in favor of the player. Farmer, who received his PhD in math from the University of California, Santa Cruz, is currently the director of economic modeling at the University of Oxford's Martin School.
--- Lisa DeKeukeleare
"Letter from Germany: There's billions and Billions," by Burkard Polster and Marty Ross. The Age, 7 May 2012.
Burkard Polster and Marty Ross write about two different customs of referring to large numbers. Most European languages use the so-called "long scale" for large numbers ending in -illion, such as million and trillion (in which a billion is 1012). Whereas the standard in English-speaking countries is the "short scale" (in which a billion is 109).The article explains the origin and logic behind the two scales. It also has a great picture (reproduced at left) illustrating for what numbers the scales match and where they disagree. See a video discussion of the subject.
--- Baldur Hedinsson
"Physicists go totally random," by Alexandra Witze. Science News, 6 May 2012.
Physicists at ETH Zurich in Switzerland have developed a new method for achieving perfect randomness in selecting numbers, a finding with implications ranging from casino floors to encryption. The new method is based on quantum theory, in particular the principle that two particles separated by vast distances can be linked if that measurement of a specific property of one particle reveals the value of the same measurement of the other. By using a stream of partially random information to select the specific properties of two linked particles to measure, the physicists were able to produce an outcome completely independent of any other variables. A fellow Swiss physicist notes that the possibility of improving randomness is "new and surprising," and the article explains that such improvements could benefit casinos and code-makers by eliminating any patterns that could allow for exploitation.
--- Lisa DeKeukeleare
"First Spinoff of African Math Institute Takes Root in Senegal," by Martin Enserink. Science, 4 May 2012, page 533.
Four years ago, Science reported on the African Institute for Mathematical Sciences (AIMS), an institute founded in Muizenberg, South Africa in 2003 to provide rigorous training to promising young African mathematicians. The institute was founded by South-African born mathematician Neil Turok, who "believes excellence in math is one of the keys to development in Africa." His dream: "to create a network of 15 AIMS institutes" around the continent. Now, due to the support of several large donors in the last two years, the second AIMS institute--AIMS Senegal--opened its doors in M'bour, Senegal last fall, and a third and fourth institute--one in Ghana and the other in Ethiopia--will open within the year. Despite some differences between AIMS South Africa and AIMS Senegal, the idea behind the institutes is the same: students participate in several months of "day-and-night immersion in high-level mathematics." Their lecturers are "internationally renowned mathematicians, who each come to spend three full weeks. They are extremely accessible; discussions often continue during the communal meals and into the night." The result is an experience that Odumodu Nneka Chigozie, one of the 31 students currently enrolled in the Senegal institute, describes as "amazing and very inspiring." At the end of this 10-month term, she will become one of 450 alumni of the two AIMS institutes.
--- Claudia Clark
"A Math-Free Guide to the Math of Alice in Wonderland," by Esther Inglis-Arkell. io9.com, 3 May 2012.
You may know that Alice In Wonderland was penned by Charles Dodgson, an Oxford logician who wrote under the name Lewis Carroll. But you may not know how much math - or rather, how much criticism of what was at the time contemporary mathematics - Dodgson packed into his book. The "new math" of the mid-1800s represented the first flowering of modern mathematics - from Augustus De Morgan's development of symbolic algebra and coining of its name (derived from the Arabic phrase for "restoration and reduction"), to increasing acceptance of complex numbers and William Rowan Hamilton's development of quaternions, to the beginnings of topology and projective geometry through Jean-Victor Poncelet's "principle of continuity." As math developed into a specialized language for the description of abstract structures, it increasingly departed from direct correspondences with physical reality, a fact that bothered Dodgson, a fan of Euclid's Elements and a dedicated tutor.
A "conservative ... cautious mathematician who produced little original work", Dodgson ground his axe in fiction when he found himself outmanned in the mathematical literature. This post on the science and sci-fi blog io9, in which the analysis has more to do with literature than real-valued functions, explains some of his most famous gibes. For instance, the scene with the caterpillar can be interpreted as a jab at algebra, which Dodgson suggests sprouted from nowhere, like a mushroom, and clouded the heads of mathematicians, like opium. Alice came to the mushroom to "restore" her size, and then was "reduced" in size when she ate it, shrinking so rapidly that her chin hit her foot. The tea party is all about Hamilton's quaternions. Hamilton introduced a fourth dimension (time) in order to devise a number system describing three-dimensional rotations; without time, he could only describe planar rotations. At the tea party, four guests are stuck rotating around the table, forever, after Time leaves. When Alice herself departs, the Hare and the Hatter are trying to stuff the Dormouse into the tea kettle, returning them to a two-parameter number system (the complex numbers), and supposedly their freedom. In another scene, the Duchess' baby turns part of the way into a pig, while maintaining many of the features of a baby, lampooning Poncelet's continuous transformations of one geometrical object into another. As the informative links to NPR, New Scientist, and The New York Times at the bottom of the post indicate, much of this mathematical satire has been only recently unearthed. All of it only crept into the story after Dodgson first recounted the backbone of the tale to Alice Liddell and her two sisters, and it accounts for much of the dark absurdity that makes Alice unique and timeless. So, paradoxically enough, if it wasn't for the modern mathematics that so confounded him, Dodgson's name might not have followed Alice down into history at all. (Which it didn't, actually, if you think about it.) (Photo by Annette Emerson, who, like Alice, has often seen a cat without a grin but never a grin without a cat.)
--- Ben Polletta
"Is Origami the Future of Tech?" by Drake Bennett. Bloomberg Business Week, 3 May 2012.
Consider the processes--stamping, casting, carving, punching, molding, and stitching--by which so many of the products we use every day are fabricated. And yet, writer Drake Bennet notes, "the natural world doesn't use any of them. One of its favorite methods is to take something flat and fold it into a three-dimensional form. Flowers, leaves, wings, proteins, mountain ranges, eyelids, ears, DNA—all are created by folding." Today, researchers in robotics, biology, math, and computer science have found that "fabrication by folding has the potential to be far faster, cheaper, and less energy-intensive than traditional methods and to work at very, very small scales, where even the most precise mills and lathes have all the accuracy of an earthquake." In this article, Bennet describes some of the ways these researchers are using folding: to design "programmable matter" (materials that can change their own physical properties), to build tiny aerial robots that could gather intelligence or explore hazardous environments, to create new proteins which could lead to the design of useful compounds, and to identify and kill cancerous cells. (Image: Harvard Microrobotics Lab.)
--- Claudia Clark
"Goldbach's Prime Numbers," by Davide Castelvecchi. Scientific American, May 2012, page 23.
The Goldbach conjecture states that every even prime number bigger than 2 can be written as the sum of two prime numbers (ex. 30 = 13 + 17). The statement is still a conjecture even after almost 300 years after its first statement. The weak Goldbach conjecture states that every odd number greater than 7 can be written as the sum of three primes (ex. 15 = 3 + 5 + 7). This statement follows from the Goldbach conjecture. Each conjecture has been verified for numbers up to 19 digits. Now Terry Tao (UCLA) has shown that you can write any odd number as the sum of at most 5 primes. He hopes to extend his proof technique to prove the weak Goldbach conjecture, but does not think that the approach will extend to the original conjecture.
In the same issue of the magazine (but at the other end of the mathematical spectrum) is "Math rules," in which Steve Mirksy (author of the monthly Anti Gravity column), in response to Ian Stewart's book In Pursuit of the Unknown: 17 Equations That Changed the World, looks at some additional equations not in the book, such as 20x + 10y + 5z = 0: "when attempting to use a vending machine that takes singles, you will have in your possession some integer numbers of 20s, 10s and fives but no ones--and, therefore, no candy."
--- Mike Breen
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