tag:blogger.com,1999:blog-362699732021-02-28T16:08:49.605-05:00The Mathematical TouristMath Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.comBlogger1846125tag:blogger.com,1999:blog-36269973.post-52260466840492532522021-02-25T17:00:00.000-05:002021-02-25T17:00:28.151-05:00Dartboard EstimatesThrowing darts at a target may sound like a curiously haphazard way to solve a mathematical problem. Properly applied as a kind of intelligent guessing, however, it can become a highly effective technique for obtaining answers to certain problems in Ramsey theory (see "Playing Fields of Logic") and in other areas of mathematics.Suppose that instead of the usual rings and diagonals, a dartboard Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-9064047423424434272021-02-24T17:11:00.001-05:002021-02-24T17:11:47.173-05:00Puzzling Groups and Ramsey NumbersThroughout his long, itinerant life, Paul Erdős (1913-1996) spent most of his waking hours and, apparently, all his sleeping hours doing mathematics. He was a superb problem solver, and his phenomenal memory allowed him to cite exact references to thousands of mathematical papers, including their page numbers."If you don't know how to attack a particular problem, ask Erdős" was the constant Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-44018464973181561622021-02-22T16:46:00.001-05:002021-02-24T17:12:56.958-05:00Planes of BudapestNearly every Sunday during the winter of 1933 in Budapest, a small group of students would meet somewhere in the city at a park or cafe to discuss mathematics. The gathering typically included Paul Erdős (1913-1996), George Szekeres (1911-2005), and Esther Klein (1910-2005).In addition to feeding their passion for mathematics, the students enjoyed exchanging personal gossip and talking politics. Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-68654458989738488712021-02-21T13:38:00.001-05:002021-02-21T23:53:49.660-05:00Playing Fields of LogicRamsey theory owes it name to Frank Plumpton Ramsey, an English mathematician, philosopher, and economist. His father, Arthur Stanley Ramsey, was a professor of mathematics and the president of Magdalene College at the University of Cambridge.Frank Ramsey (1903-1930). MAA Convergence Portrait Gallery.Frank Ramsey was born in 1903 and spent nearly his entire life in Cambridge. After he graduated Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-35939085301487429532021-02-18T14:13:00.000-05:002021-02-18T14:13:46.026-05:00Pigeonhole CongestionSorting the mail that comes into an office generally requires that each piece be slipped into the appropriate slot of an array of pigeonholes—one for each employee. Suppose that a small business needs ten such slots. When 11 pieces of mail arrive, one or more of the slots will have to contain at least two items.So, if there are more pigeons than holes, some of the pigeons have to double up. Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-56335232185614888332021-02-11T17:25:00.001-05:002021-02-11T17:38:46.109-05:00The Long RunQuite often, in a race game governed strictly by chance, the player who starts out ahead stays ahead for most, if not all, of the race. This striking feature is worth examining more closely.In a two-player coin-flipping game, heads and tails will each win half the time, on average (see "Rolls and Flips"). But in a game of a thousand flips, when the total number of heads versus the total number ofMath Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-69256124278160340942021-02-09T13:26:00.000-05:002021-02-09T13:26:49.723-05:00Change of Face The serious gamblers in casinos hang out at the craps tables. The basic rules of this two-dice game are simple, but the bewildering array of options for betting on various outcomes creates a fast-paced, insidiously seductive pastime, in which a heady brew of chance, intuition, experience, calculation, and superstition come into play.The shooter tosses two dice. If a total of seven or eleven Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-21721613048327263272021-02-08T13:36:00.000-05:002021-02-08T13:36:21.251-05:00Climbing and SlidingThe origins of the game of Chutes and Ladders (or its Snakes and Ladders counterpart) go back many centuries to India and a board game called moksha patam (a Hindu concept that is akin to heaven and hell). Designed as a way of instructing children in religious values, the game graphically depicts the coexistence of good and evil and illustrates the impact of chance on human affairs.In the game, Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-73682623969260698792021-02-07T13:26:00.001-05:002021-02-08T14:02:00.807-05:00Tumbling DiceLike coins (see "Coin Toss Randomness"), cubic dice are subject to physical laws. An unscrupulous player can take advantage of this physics to manipulate chance. A cheat, for instance, can control a throw by spinning a die so that a particular face remains uppermost or by rolling it so that two faces stay vertical. In each case, the maneuver reduces the number of possible outcomes.A grossly Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-76707925420784813362021-02-06T10:46:00.001-05:002021-02-06T17:21:46.896-05:00Dice BiasDice represent an intriguing example of the interplay between randomness, chance, and physical law.The simplest mathematical model of a standard die is based on the assumption that the die is a perfect cube. A cube has six faces, and each face has an equal probability of coming up—one in six (1/6).A store-bought die, however, isn't really a perfect cube. Such dice, like those used in children's Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-77045708244170012842021-02-05T10:24:00.002-05:002021-02-07T13:28:06.024-05:00Rolls and Flips"Iacta alea est."—Attributed by Suetonius to Julius Caesar (100-44 B.C.)Dice are among the oldest known randomizers used in games of chance. In 49 B.C., when Julius Caesar ordered his troops across the river Rubicon to wage civil war in Italy, the alea of the well-known proverb he quoted already had the standard form of the die we use today: a cube engraved or painted with one to six dots, Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-22150075944127301692021-02-04T13:59:00.002-05:002021-02-04T13:59:35.240-05:00Coin Toss RandomnessFlipping a coin in the air, catching it, then determining whether it has come up heads or tails is a common way to decide who starts off a two-person game or resolve a binary question. Because you expect that heads is as likely to come up as tails, it sounds like a fair way to make a choice.But coin tossing isn't really random at all. A mechanical gadget can flip a properly positioned coin so Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-60903435570628661142021-02-02T14:04:00.002-05:002021-02-05T10:25:04.376-05:00The Die Is Cast"Iacta alea est."—Attributed by Suetonius to Julius Caesar (100-44 B.C.)A die tumbles out of a cupped hand, bounces on the carpet a few times, rolls a short distance, then teeters to a stop. The uppermost face of the white cube shows four black dots arranged in a square.Grinning, a child briskly moves a red token four squares to the right along the bottom row of a large checkerboard grid. The Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-5113416792911368922021-02-01T18:05:00.002-05:002021-02-02T14:05:57.785-05:00Infinite PossibilityAll mimsy were the borogoves,And the mome raths outgrabe.—Lewis Carroll (1832-1898), "Jabberwocky"An ape sits hunched over a keyboard. A long hairy finger bangs a key, and the letter a appears on the computer screen. Another random stab produces n, then a space, then a, p, and e.That an ape would generate this particular sequence of characters is, of course, highly improbable. In the realm of Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-57528789742905449352021-01-27T17:40:00.005-05:002021-02-22T17:52:50.510-05:00Party GamesLike driftwood spars, which meet and passUpon the boundless ocean-plain,So on the sea of life, alas!Man meets man,—meets and quits again.—Matthew Arnold (1822-1881), "Switzerland"In work and play. from hour to hour and year to year, we pass from one gathering to another. Sometimes we are surrounded by people we know; sometimes we find ourselves in the presence of strangers. The restless ebb and Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-8921798395215013382021-01-24T16:20:00.000-05:002021-01-24T16:20:06.741-05:00Playing with Ruth-Aaron PairsOn April 8, 1974, Henry (Hank) Aaron hit his 715th major league home run, surpassing the previous mark of 714 career home runs long held by baseball great Babe Ruth. Understandably, the event received considerable coverage in newspapers and magazines and on television.However, those reports invariably overlooked the mathematical aspects of the achievement, particularly the curious properties of Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-14940704398083298932021-01-20T08:00:00.075-05:002021-01-20T08:00:02.065-05:00Poe's SecretsWriter Edgar Allan Poe (1809-1849) is famous for his short stories of the mysterious and the macabre. Poe's tale "The Gold-Bug," published in 1843, is sometimes cited as one of the best works of fiction that turn upon a secret message."The Gold-Bug" is about a secret message written in invisible ink on a scrap of parchment. The deciphered message leads to a buried chest filled with fabulous Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-15831583168339922392021-01-18T12:59:00.000-05:002021-01-18T12:59:23.144-05:00Cracking a Medieval CodeThe first printed book on cryptology was written by Johannes Trithemius (1462-1516), an abbot in Germany who was one of the leading intellectuals of his day. Bearing the title Polygraphia, it was published in 1518 after Trithemius's death.Polygraphia title page.The first of the six books of Polygraphia contains 384 columns of Latin words, two columns per page. Each word stands for a letter of theMath Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-27147759371834775092021-01-12T16:27:00.000-05:002021-01-12T16:27:16.171-05:00The Limits of MathematicsAt the beginning of the 20th century, the German mathematician David Hilbert (1862–1943) advocated an ambitious program to formulate a system of axioms and rules of inference that would encompass all mathematics, from basic arithmetic to advanced calculus. His dream was to codify the methods of mathematical reasoning and put them within a single framework.David Hilbert. MAA Convergence Portrait Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-13587590942227116502021-01-11T08:00:00.001-05:002021-01-11T08:00:02.395-05:00Coyote Fence Passageway Dale Ball Trails South (Wilderness Gate), Santa Fe, New Mexico, 2021.Photo by I. PetersonMath Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-50790742090375912302021-01-10T08:00:00.001-05:002021-01-10T08:00:05.664-05:00Winter Grass Feather reed grass (Karl Foerster), Santa Fe, New Mexico, 2021.Photo by I. PetersonMath Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-67060253687427529452021-01-09T08:00:00.013-05:002021-01-09T08:00:03.754-05:00Mountain Clouds Sierra Del Norte (Dale Ball Trails North), Santa Fe, New Mexico, 2020.Photo by I. PetersonMath Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-78720087101668675942021-01-08T08:00:00.001-05:002021-01-08T08:00:04.787-05:00Santa Fe Cholla Santa Fe cholla (Cylindropuntia viridiflora). Santa Fe Botanical Garden, Santa Fe, New Mexico, 2020.Photo by I. PetersonMath Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-17385680864188142292021-01-07T08:00:00.134-05:002021-01-07T08:00:00.569-05:00Dancing Musical ChaosPoint by glowing point, the image swirls into view. As it builds up on the computer screen, it begins to resemble a delicate, stylized butterfly with translucent wings held lazily askew.It's called the Lorenz attractor, named for Edward N. Lorenz, who in 1963 discovered this curious form encoded in a set of equations describing air flows in the atmosphere. The computer image arises out of a Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-83348956304271314592021-01-05T08:00:00.114-05:002021-01-05T08:00:00.842-05:00Calendar QuirksCalendars represent our efforts to create frameworks that allow us to reckon time over extended periods.We normally count the day—the time it takes Earth to rotate once on its axis—as the smallest unit of calendrical time. The measurement of fractions of a day fits, by convention, into the category of timekeeping.Of the dozens of calendars presently used in the world, the most common ones group Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0