tag:blogger.com,1999:blog-758931270865409762011-08-20T07:15:19.586-05:00Articles for kids about math in everyday life, written by Ivars Peterson for <i>Muse</i> magazine.Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.comBlogger881100tag:blogger.com,1999:blog-75893127086540976.post-65347907051292168292008-03-03T12:58:00.003-05:002008-03-03T13:24:13.839-05:00

You're done with your iPod, so you carefully coil the headphone cord around the player and stuff it in your pocket. The next time you take it out, however, you find that you have to unravel the cord and undo a knot before you can go back to listening to your music. In fact, it's pretty amazing how easily knots can form by themselves—not only in headphone cords, but also in necklaces, coiled ropes

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-69342519267062204442008-02-17T13:06:00.004-05:002008-02-17T13:33:00.615-05:00

Take a look at a map of the United States and locate Colorado and Wyoming. On many maps, these states look like perfect rectangles.The laws that created these two states specify that each one lies between certain lines of latitude and longitude. (Latitude lines are drawn side-to-side on a globe; longitude lines go top-to-bottom.) Wyoming stretches from 41°N to 45°N latitude and from 104°W to 111°

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-22245636630851746912008-01-02T15:25:00.000-05:002008-01-02T15:54:54.198-05:00

When you're a little kid, one of the joys of having a train set is assembling all the pieces into the longest possible loop, then operating a train that follows the railroad's twists and turns. And, as your collection of tracks—straight pieces, curves, bridges, tunnels, switches, and crossings—gets larger, your designs get more and more elaborate.Mathematician Mark Snavely of Carthage College in

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-91272762661792300472007-11-05T15:50:00.000-05:002007-11-05T16:54:03.980-05:00

It's easy to learn to play checkers, but it takes a lot of practice to become an expert player, and no one has ever played checkers flawlessly—until now. This spring, a computer program became the first perfect checkers player ever. Tic-tac-toe, connect four, and other simple games were "solved" long ago, but checkers is the first of the complex games to fall.Computer scientist Jonathan Schaeffer

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-10479706896581602962007-10-10T11:35:00.000-05:002007-10-10T11:58:12.489-05:00

Lots of computer games have you whacking monsters as you struggle to find better weapons and power up. Some games, however, rely on puzzles more than single combat. If your game is one of the puzzling kind, trial and error might get you by, but a bit of logic or mathematical knowledge will often speed you on your way.Near the beginning of the game Myst, for example, you encounter a contraption

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-64925032428659513252007-08-24T13:01:00.001-05:002007-08-24T13:19:43.733-05:00

If you've kicked around a soccer ball, you've probably noticed the distinctive pattern on the ball's surface. For a long time, soccer balls have been stitched or glued together from 32 pieces of material. Twelve of these patches are five-sided (pentagons), and twenty of them are six-sided (hexagons). These patches are arranged so that each pentagon is surrounded by hexagons. Traditionally, the

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-91143254787505685902007-07-06T11:42:00.001-05:002007-07-06T12:23:50.174-05:00

Suppose that you live in a town that sprawls across the banks of a branching river, with an island in the middle. Seven bridges link the different parts of town. One sunny summer day, you decide to take a bike ride. Looking at a map, you try to find a route that would take you across all the bridges and back to where you started. You also add the challenge never to cross the same bridge more than

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-64901038746938267472007-07-05T08:19:00.000-05:002007-07-05T08:38:00.261-05:00

Playing with triangles can lead to amazing patterns and three-dimensional structures. That's what Hungarian designer Dániel Erdély (below) found when he created an intriguing geometric form out of two spirals of triangles that get smaller and smaller.Daniel Erdély holds a complex polyhedron constructed from spidrons. Photo by Regina Márkus.He called the resulting S-shaped object a spidron. Each

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-12160192495770855822007-07-04T10:32:00.000-05:002007-07-04T10:40:40.323-05:00

Nearly everyone has 10 fingers and 10 toes, and it's been like that for a long, long time. So, it's probably natural that we count by tens—so natural that the decimal system is, by far, the most common way of expressing numbers in both spoken and written language around the world.In ancient times, Pythagoras and his followers taught that "everything is number." They considered 10 to be special

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-35095846034824490672007-07-03T07:28:00.000-05:002007-07-03T07:36:50.542-05:00

Many people watch The Simpsons for its zany characters, political jokes, and outrageous situations. But other viewers keep a sharp eye out for references to mathematics. Really.Several of the show's writers studied math or computer science in college. And, from time to time, they just can't resist sneaking in a mathematical bit or two (or three). But, unless you're looking carefully, these inside

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-78559590326939000312007-07-02T08:06:00.000-05:002007-07-02T08:11:55.599-05:00

So it's already been a pretty tough school year? Some of your grades are OK, but others are quite dismal (especially that test you took the day after staying up way too late playing Guild Wars online)? Luckily, your teacher says you can drop a single test. Alright! But wait, which one?The answer is easy if all of the tests are worth the same number of points. You simply drop the lowest score. But

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-14642021597646427082007-07-01T19:18:00.000-05:002007-07-01T19:29:53.214-05:00

Some insects can walk on water. They take advantage of water's high surface tension to skate across a pond or puddle. But at the edge of the pond, where wet meets dry, surface tension makes the water curve upward in a meniscus. For tiny, water-walking insects, scaling this slope isn't easy. If they try to stride up the slope, they simply slide back down.So how do these insects get out of the pond

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-74131073413593696512007-06-30T08:37:00.000-05:002007-06-30T08:46:38.810-05:00

You're probably familiar with board games such as Monopoly, Candy Land, or Clue. These games are old enough that your parents likely enjoyed them when they were young, too. There are other games, which people have enjoyed across generations, that have been around even longer—some for thousands of years. One such game is awari, which originated in Africa. If you haven't tried it, you might be

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-12478885352525325982007-06-29T09:28:00.000-05:002007-06-29T09:33:58.889-05:00

We learn to count at such an early age that we tend to take the notion of numbers for granted. We know that two can stand for two apples, two oranges, or two argyle socks. But abstract numbers are the product of a long cultural evolution. They may even have played a role in the invention of writing, or so archaeologist Denise Schmandt-Besserat argues.Schmandt-Besserat has studied mysterious clay

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-42281105454465152472007-06-28T15:10:00.000-05:002007-06-28T15:16:21.449-05:00

"We all use math every day." That's a line you might hear in math class or see at a science museum. You don't expect to find a TV crime series that celebrates it. Yet millions of people hear this line every week at the beginning of the popular CBS show Numb3rs, which features a young math professor as a crime fighter.Charlie Eppes, played by David Krumholtz, uses math to help his older brother,

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-63177054414917246802007-06-27T19:38:00.000-05:002007-06-27T19:44:49.143-05:00

One of the oldest collections of mathematical problems we know of is Problems to Sharpen the Young. No one knows who wrote the book, but some scholars say that the author might have been someone named Alcuin, who lived from about 732 to the year 804 (three-digit years!). Alcuin was born near the city of York in England and was a student, then a teacher, and then head of the Cathedral School at

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-40152484783827351542007-06-26T18:59:00.000-05:002007-06-26T19:07:58.194-05:00

What could you do for fun with some coins and a sheet of paper? That's all Bob Hearn, a graduate student at the Massachusetts Institute of Technology, needed to invent a new kind of puzzle. Here's an example.Place four coins on the bottom row of circles (covering G, D, E, and R). This leaves the letters MARTIN exposed. Your challenge is to slide the coins along the paths joining the circles until

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-26571556847454827482007-06-25T19:19:00.000-05:002007-06-25T19:35:12.733-05:00

The humble brown-paper bag that you use to carry groceries is actually a technological masterpiece that solves many practical problems. Unlike a plastic bag, it can stand upright by itself; you don't need an extra hand to hold it open while you fill it. Yet it folds flat for easy storage.A standard brown-paper bag can stand upright, as shown above.It took inventors years to come up with a design

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-10441160030035211252007-06-24T13:33:00.000-05:002007-06-24T13:57:22.075-05:00

These days, a lot of people recognize the name Harry Potter, the hero of J.K. Rowling's immensely popular books. But Harry isn't a very popular baby name. In the United States in 2004, it ranked only 531st among names for boys. It was much more popular about 100 years ago, when it ranked in the top 15.The popularity of the name Harry (above) has fallen since its peak more than 100 years ago. ©

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-51960662189631232102007-06-19T20:16:00.000-05:002007-06-19T20:30:00.938-05:00

Millions of people around the world can't get started every day without their sudoku fix. Have you joined them?A sudoku puzzle usually consists of a nine-by-nine grid. Some of the spaces contain numbers; the rest are blank. Your goal is to fill in the blanks with numbers from 1 to 9 so that each row, each column, and each of the nine three-by-three blocks making up the grid contains just one of

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-57706057556556786092007-06-17T10:12:00.000-05:002007-06-17T10:15:16.425-05:00

There are just a few seconds left in the football game. Your team is all set to kick a field goal to win. The opposing team, however, calls a timeout. Its players hope that the extra time will make your kicker think about his upcoming kick—and then miss the field goal. The strategy is called "icing the kicker."But does this time-honored trick really work? Does making a kicker wait an extra minute

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-32545032082190619512007-06-16T06:56:00.000-05:002007-06-16T08:42:38.170-05:00

You've probably split sunlight into a rainbow of colors with a prism. Maybe you've also split sunlight with a spray of water from a garden hose or even with a CD. But have you ever tried splitting light with your fingernail?That's right. Your fingernail. On a bright, sunny day, if you look at sunlight reflecting off your fingernail at just the right angle, you might see a dancing pattern of

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-33509548540148436652007-06-15T07:28:00.000-05:002007-06-15T07:37:01.108-05:00

In the right hands, mathematics can be a musical instrument, a spooky player piano that turns numbers, number sequences, or mathematical functions into mysterious and haunting melodies.Composers create music out of math by inventing special recipes, or formulas, that match numbers or mathematical patterns with musical notes. One example is Daniel Cummerow's recipe for a pi tune. (Pi is the number

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-60324251198053992592007-06-14T20:07:00.000-05:002007-06-14T20:19:01.949-05:00

Have you ever tried drawing something without lifting your pencil from the paper? You usually end up with a squiggly mess that sometimes looks a bit like the object you were trying to draw.Now computer scientists have written a program that does first-rate "continuous-line" drawings. Why are they wasting their time on such frivolous things? The answer is that the drawings are actually a version

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-20311909809508725632007-06-12T19:51:00.000-05:002007-06-12T19:59:24.845-05:00

Have you ever left a necklace or a piece of string lying around on a table in a jumbled heap? There's a good chance that it will have formed a knot when you pick it up again, especially if it has been jostled a little. The same thing can happen to a garden hose left in an untidy pile on the ground.Sailors and rock climbers know about this problem, so they take great care to store their ropes in

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-40223240801751190722007-06-10T07:00:00.000-05:002007-06-10T07:12:17.415-05:00

It's often tough to figure out how to win even a really simple game.Consider the two-player game called Chomp. A move consists of picking one checker anywhere in a rectangular array of checkers, then removing that checker along with all the checkers above and to the right of it. It's like taking a big, neat bite out of a chocolate bar divided into easy-to-break-off sections.You and your opponent

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-60510184775415905022007-06-09T09:32:00.000-05:002007-06-09T09:45:51.065-05:00

You've probably heard the one about Archimedes, the Greek mathematician who jumped out of the tub and ran naked through the streets shouting, "Eureka! Eureka!" Legend has it he suddenly realized how to tell a wreath made of pure gold from one made of a mixture of gold and silver. First, put a weight of pure gold equal to the wreath's weight into a bowl and fill the bowl with water. Then, take out

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-88026239017582695252007-06-07T19:47:00.000-05:002007-06-07T19:48:33.961-05:00

Forget about the heat and the pressure and the iron core. Imagine there's a hole that goes all the way through Earth, passing through its center. If you dropped a stone into this hole, what do you think would happen?Normally when you drop a stone, it falls to the ground, pulled by Earth's gravity. For a stone falling through Earth, the situation is a little different. At the surface, it's

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-75893127086540976.post-51709846937602348682007-06-06T19:31:00.001-05:002009-01-12T20:38:23.454-05:00

Quick! What are the first nine digits of π, the number known as pi?You probably know pi as the number you get when you divide a circle's circumference by its diameter. You might even have a calculator that gives you the value of pi to eight or more decimal places: 3.14159265. . . . Remarkably, mathematicians have proved that the digits of pi go on forever, although only about 1 trillion have been

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-75893127086540976.post-48635628217043513602007-06-05T19:21:00.000-05:002007-06-05T19:26:20.949-05:00

Have you ever heard the one about a farmer traveling with a fox, a goose, and a bag of grain?The farmer comes to a river and finds a small boat that will hold only two passengers. For obvious reasons, he can't leave the fox alone with the goose, or the goose with the grain. How does he get his cargo safely to the other side?Brainteasers that involve ferrying people and their belongings across a

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-78880318103489453642007-06-04T19:48:00.000-05:002007-06-04T19:59:06.075-05:00

You've probably heard that it's impossible to fold a sheet of paper in half more than seven or eight times. Usually you're also told that it doesn't matter how big or thin the sheet is.Try folding a sheet of notebook paper. You'll probably find that it is pretty tough to get beyond eight folds. However, just because people—even experts—say something's impossible doesn't mean it is. That's what

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-44435251498928828802007-06-03T07:39:00.000-05:002007-06-03T07:53:59.552-05:00

It's sink or be sunk in the game of Battleship, the cat-and-mouse game where players try to figure out where their opponent's fleet is deployed. You probably know the paper-and-pencil, pegboard, or electronic versions. And if you don't, you can play a version of the game online at javascript.internet.com/games/battleship.html.But Mogens Esrom Larsen of the University of Copenhagen has invented a

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-57761841909430269382007-06-02T08:48:00.000-05:002007-06-02T08:55:46.947-05:00

"I'm going to be a knight," said Ron. With that remark, Ron Weasley took charge of a deadly, larger-than-life game of wizard chess, played near the climax of Harry Potter and the Sorcerer's Stone.Deciding to be a knight was an interesting choice. Chess has six different kinds of pieces: king, queen, bishop, knight, rook, and pawn. Each piece moves in its own special way. The knight stands out

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-90908893271057646362007-06-01T20:11:00.000-05:002007-06-01T20:17:09.221-05:00

It's a tense moment in Die Hard with a Vengeance. A time bomb is about to go off. The frazzled heroes, detective John McClane and his sidekick Zeus, a dry cleaner from Harlem, have just seconds to defuse it. To their horror, they discover they must solve a mathematical puzzle to do so.Sitting on the lip of a fountain in a park are a 5-gallon jug and a 3-gallon jug. The timer will stop if they

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-47022596951712607332007-05-31T19:39:00.000-05:002007-05-31T19:41:46.910-05:00

What's the biggest number you can think of? A billion? A trillion? A googol? (That's 1 followed by a hundred zeroes.) Whatever number you come up with, there's always a larger one. You could write down 1 and keep adding zeroes after it until you hand gets tired, and you still wouldn't get to the "last" number. There's always another number right after whatever you've written down. Just add 1 and

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-74791340510550957102007-05-30T19:48:00.000-05:002007-05-30T19:56:01.344-05:00

Some dogs live to play fetch. Others do it only when bribed. At least one really gonzo dog, however, takes the game seriously enough to do a bit of math to figure out the best way to catch the ball.Or so it seems.The dog is a Welsh corgi named Elvis, who belongs to mathematician (no surprise there) Tim Pennings of Hope College in Holland, Michigan. When Elvis and Pennings go to the beach, they

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-4976851135285424322007-05-29T19:57:00.000-05:002007-05-29T20:21:12.585-05:00

Do you have a lucky number? In ancient China, people believed that a special arrangement of nine numbers in a square was especially lucky. They engraved this pattern on stones or medallions that were worn as charms to ward off evil or bring good fortune.Here's the pattern. Can you tell what's special about it?Notice it contains all the numbers from 1 to 9. Better yet, the numbers in each row,

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-6980057336858366062007-05-28T08:30:00.000-05:002007-05-28T08:48:28.895-05:00

If you're good at keeping track of cards, here's a fairly simple but nearly foolproof mind-reading trick you can try out on your friends.Ask a friend to shuffle a standard deck of 52 playing cards, then have her secretly pick a number between 1 and 10. Tell your friend to slowly and steadily deal out the cards, one by one and face up, to form a pile. As she does so, she is to count them silently,

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-45395212786415843292007-05-27T05:32:00.000-05:002007-05-27T05:51:21.920-05:00

Every time you look at an image on your TV set or computer screen, you're really looking at a whole bunch of tiny dots, some dark, some light. You usually don't see those dots unless you look very closely or use a magnifying glass. Observed from a comfortable distance, they blend together to give you a recognizable Homer Simpson or a scene from Star Wars.Illustrators and artists have taken this

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-48201808530225926572007-05-26T19:31:00.000-05:002007-05-26T19:41:37.738-05:00

You probably know how to make a lacy snowflake, a chain of identical spruce trees, or a line of paper people by folding paper and cutting some notches out of the folded wad. Ah, but do you know the one-cut angelfish? It's a paper cutout that will amaze and astound your friends.Intrigued by paper cutting, computer scientists Erik and Martin Demaine and Anna Lubiw wondered what sorts of shapes it

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-21308245472264803762007-05-25T19:30:00.000-05:002007-05-25T19:34:00.676-05:00

Have you ever been dealt a gin rummy hand and realized you already had gin? Or an incredible run of hearts, so you were very close even though you didn't quite have gin? Did you think you were lucky? Or did you think that the dealer should have shuffled more times?Card players sometimes get lazy and fail to shuffle decks of cards as fully as they should. That sloppiness leaves traces of patterns

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-82963084957589298072007-05-24T19:04:00.000-05:002007-05-24T19:11:52.208-05:00

Heads or tails?Flipping a coin is a common way to start off a game or settle a question. Because you expect that the coin is as likely to come up heads as tails, it sounds like a fair way to make a choice.But is it really? Here's something you can try with a U.S. penny. The results may surprise you.Stand a dozen or more pennies on edge on the surface of a table. It may take patience, but if the

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-68543650137095643662007-05-23T19:28:00.000-05:002007-05-23T19:37:51.676-05:00

Have you ever been stuck in a waiting room with nothing but your boredom and a Rush Hour puzzle? You know, the plastic tray of colorful cars and trucks that are stuck in a traffic jam. The goal is to clear a path for a car to the only exit on the grid. Chances are you tried the puzzle. Chances are so did the person before you. And the person after you. Rush Hour looks easy, but it is hard. We're

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-42141851207425108392007-05-22T18:45:00.000-05:002007-05-22T18:48:28.033-05:00

Nothing could be grayer, more predictable, or less surprising than the endless sequence of whole numbers. Right? That's why people count to calm down and count to put themselves to sleep. Whole numbers define booooooooring.Not so fast. Many mathematicians like playing with numbers, and sometimes they discover weird patterns that are hard to explain. Here's a mysterious one you can try on your

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-75893127086540976.post-860218853147369022007-05-21T19:43:00.000-05:002007-05-21T19:47:42.724-05:00

Have you ever been stuck in one of those never-ending Monopoly games,* the sort your mom makes you move to the basement because she doesn't want tiny hotels in the mashed potatoes? If you have, you probably think Monopoly is pretty much a game of endurance (who can stand it the longest) rather than of strategy. After all, you have no control of where you land when you throw the dice and every

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-58356225715845783622007-05-19T13:35:00.000-05:002007-05-20T07:27:07.320-05:00

How do you train a dog to sit?One way is to give your dog a treat—a cookie, toy, or meat-flavored byproduct—the instant it does the right thing. This seems obvious enough. But what should you do once the dog has learned the trick? Do you dole out a treat every time your dog sits? Surprisingly, the answer is no. The dog will be much more persistent if the reward is unpredictable than if it gets a

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.comtag:blogger.com,1999:blog-75893127086540976.post-13133092906829288692007-05-18T20:55:00.000-05:002007-05-18T21:03:20.229-05:00

Voting sounds so simple. Whether you're picking a class president or deciding which snack food you and your pals should get, you just make a choice, someone counts the votes, and the majority wins. Right?That's true if there were only two candidates or two choices. But as soon as you have more than two choices, things can get pretty complicated and crazy. The winner may depend on the voting

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-45033004717392143852007-05-13T11:35:00.000-05:002007-05-13T11:51:09.369-05:00

Imagine yourself inside a fishbowl, looking out. What might you see? Perhaps a fish darting around in the water, strands of seaweed, the table on which the fishbowl rests, a packet of fish food on the table, and a cat staring into the bowl, its paw touching the glass.The puzzle for artist Dick Termes, who lives in Spearfish, South Dakota, was how to paint such a scene, showing all that surrounds

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-58674886177742497122007-05-12T08:40:00.000-05:002007-05-12T08:51:23.460-05:00

A collapsible umbrella looks downright simple next to one of Chuck Hoberman's amazing unfolding toys. His most famous toy is the Hoberman sphere—a geometric ball of plastic struts and pivoting joints. At the touch of a finger, it magically unfurls from the size of a basketball to a latticework sphere large enough for a toddler to sit inside. Just as readily, it shrinks back to its compact form.As

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-81628822618663070622007-05-11T20:47:00.000-05:002007-05-11T20:54:48.031-05:00

For baseball fans, one of the highlights of the 2001 season was the home-run record set by Barry Bonds of the San Francisco Giants.All summer long, newspapers printed charts showing how many home runs Bonds had hit and how many he would have at the end of the season if he continued hitting them at the same rate. On average, Bonds was hitting a home run once every second game. So his total was

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-91450407884063427402007-05-10T19:35:00.000-05:002007-05-10T19:45:38.296-05:00

Want to impress your friends? Tell them you can do trigonometry just like that—without even thinking about it. You don't have to tell your friends your brain does the calculations without any help from you.Trigonometry is a branch of mathematics that has to do with using angles to figure out distances; it is the basis for all calculations a surveyor does to make a map.When a surveyor wants to

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-75893127086540976.post-32885644036825503572007-05-09T20:04:00.000-05:002007-05-09T20:12:59.694-05:00

What do you think the chances are that two or more kids in your class have the same birthday? A year has 365 days, so to have a 50-50 chance that two kids have the same birthday, you'd need to have 180 kids. Right?Wrong. It turns out that you'd need far fewer.Suppose there are only two kids in the class. Allen's birthday is July 4. One day is used up, so if Brandi's birthday is different, it can

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-44752205710605533232007-05-08T19:51:00.000-05:002007-05-08T19:52:09.967-05:00

Maps of the United States often show the states in different colors. In general, mapmakers use enough colors to make sure states that touch are never the same color.Suppose you have only three colors of pens. Is that enough to fill in all the states? Not quite. Nevada, for example, is surrounded by five other states: California, Oregon, Idaho, Utah, and Arizona. There's no way to color this group

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-75893127086540976.post-12885862641858308212007-05-07T20:09:00.000-05:002007-05-07T20:26:02.089-05:00

The writer Edgar Allan Poe is famous for his scary stories and poems, but he also loved secret messages. His mystery story "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 treasure.Here's the coded message that Poe included in his story:It looks like a crazy math equation! How would you

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-35786016557998668182007-05-05T11:03:00.000-05:002007-05-05T11:16:00.355-05:00

You're standing at a bus stop, waiting for a bus to arrive. You wait and you wait. There's supposed to be a bus every 10 minutes, but you haven't seen one for at least 20 minutes. Finally, a bus arrives. It's full of people. Just as you try to squeeze yourself in, you spot another bus coming down the street, and another one behind it!What's going on? Why do buses always seem to come in bunches

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-25103630887772706472007-05-04T18:50:00.000-05:002007-05-04T19:02:08.787-05:00

Just about every package you buy at the supermarket has a small label made of wide and narrow bars. This pattern stands for a 12-digit number, called a Universal Product Code (UPC), that identifies the product.A 15-ounce box of Cheerios, for example, has the following number:0 16000 66610 8The first digit, 0, gives the product category—in this case, general groceries. The next five digits (16000)

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-45966221394278911022007-04-29T18:17:00.000-05:002007-04-29T18:25:26.485-05:00

About 1700 years ago, the Roman Empire was under attack, and Emperor Constantine had to decide where to station his diminished forces. Constantine organized his legions into four field armies. He needed to protect eight regions with these forces. The trick was to place the armies so that each region was either occupied by an army or was only one step away from an army. But an army could be sent

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-25885128539666398342007-04-27T20:26:00.000-05:002007-04-27T21:07:57.383-05:00

You're at the Yahoo! or Google Web site signing up for a new email account. You fill out a form, typing in your name and other information. When you come to the end, you find you have to enter a code, which you get from a string of letters and numbers shown just below the entry form. The weird thing is that the letters and numbers are all twisted and angled, with squiggly lines running through

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-76180768367047954842007-04-26T20:23:00.000-05:002007-04-26T20:38:09.025-05:00

An ordinary bottle has an inside and an outside. If an adventurous ant with sticky feet were to walk along that bottle's surface to get from the outside to the inside, it would have to cross the lip that forms the bottle's mouth. Mathematicians have come up with a bizarre, mind-bending object they call a Klein bottle that has no such lip, or edge. What appears to be the bottle's inside is

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-47638023579630248892007-04-25T19:51:00.000-05:002007-04-25T19:59:25.450-05:00

The familiar game of Dots and Boxes seems simple. But it's really a lot trickier than it looks.Mathematician Elwyn Berlekamp of the University of California at Berkeley first learned to play the game when he was in grade school. He has been studying it ever since and has even written a book about strategies for playing it.The playing field is a rectangular or square grid of dots. You and your

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-77990827844014492602007-04-24T18:47:00.000-05:002007-04-24T18:59:46.960-05:00

Much of the fun of an amusement park ride is its stomach-churning, mind-jangling unpredictability. The Tilt-A-Whirl, for example, spins its passengers in one direction, then another. The ride's cars sometimes hesitate between moves and at other times swing suddenly from one motion to another. You never know what to expect next.These surprising movements arise from a simple geometry. A rider sits

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-4222981912978890652007-04-23T19:46:00.000-05:002007-04-23T20:03:12.863-05:00

The tiles you see in your bathroom or on a kitchen floor are usually square, though you may also come across tiles that are shaped like hexagons or octagons.But tiles don't have to be polygons. Many of the drawings by M.C. Escher, a Dutch artist who lived from 1898 to 1972, contain interlocking tiles in the shape of birds, fish, reptiles, butterflies, and other living things.As a young boy,

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-86518843617133090512007-04-22T09:08:00.000-05:002007-04-22T09:41:56.305-05:00

The amazing thing about origami is the enormous number of different objects you can make by folding a square sheet of paper. No glue or scissors allowed! You can make airplanes, flowers, butterflies, and noisemakers, or flapping birds, fierce devils, and fully equipped lobsters.Robert Lang's incredible origami lobster. Courtesy of Robert Lang.Tom Hull, a mathematician at Merrimack College in

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-90548541027301526982007-04-15T07:15:00.000-05:002007-04-15T07:26:06.144-05:00

An art gallery isn't the first place you would think to look for math, let alone a mathematical puzzle. But Barry Cipra, a mathematician and writer in Northfield, Minnesota, found a puzzle in a set of large drawings by American artist Sol LeWitt.LeWitt is famous for the use of simple patterns and basic geometric shapes in his drawings, sculptures, and paintings. In 1973, he composed an artwork

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-89580364950831803332007-04-14T09:43:00.000-05:002007-04-14T09:54:11.991-05:00

When Viviana Risca was a high school student in Port Washington, New York, she found a way to hide secret messages among DNA molecules. Heaps of DNA strands sit like microscopic spaghetti inside plant and animal cells. They are a kind of secret code already, only the secret message they normally carry is instructions for making a living creature, such as a clam or a peacock.How did Risca use DNA

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-91009169345671298982007-04-11T19:43:00.000-05:002007-04-11T19:47:25.975-05:00

Imagine stepping into a pitch-black room. How might you figure out the room's size in the dark? You could, for example, carefully follow the walls and count how many steps you took going from one corner to the next all the way around. That would give you a rough idea of how big the room was.Certain ants have their own method for estimating size in the dark. Ants of the species Leptothorax

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-88343071814504100662007-04-09T19:06:00.000-05:002007-04-09T19:24:17.855-05:00

When you're riding along on your bike or in a car, you sometimes see the word "ONLY" painted on the roadway just before an intersection. The white letters look normal from where you're sitting. But if you were standing beside the "ONLY" instead of riding toward it, the letters would look stretched out. It's only when you look at them at the proper angle that they don't look distorted.Artists have

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-74876777742402430742007-04-08T09:29:00.000-05:002007-04-09T19:06:17.184-05:00

What do you think the world would look like if you and everything in it were squished flatter than a pancake? Like shadows, you and your friends would freely flit about the surface. But if you couldn't rise above or below the surface, objects with any thickness would look very strange to you.That's the idea behind a book called Flatland, written more than 100 years ago by Edwin A. Abbott. Head of

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-16026953244505601252007-04-07T14:00:00.000-05:002007-04-07T14:22:55.675-05:00

Ever caught someone staring off into space and wondered what the heck they are looking at? To find out, you might follow their gaze, study their expression, or hey, just ask them, right? But now there's an even better, more precise way: Observe what is being reflected on the surface of their eyes.Here's how to try this out: Get close and look right into a friend's eyes. You'll see a curved

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com2tag:blogger.com,1999:blog-75893127086540976.post-33753959899017438752007-04-04T19:53:00.000-05:002007-04-04T20:10:55.801-05:00

Madeleine L'Engle, who wrote these words in A Wrinkle in Time, used tesseract to mean a shortcut through space and time. In her story, space-time wrinkles, or folds onto itself, creating new paths that allow characters to tesser, or travel from one end of the galaxy to the other in an instant.Mathematicians also use the word tesseract, but they mean something different. A tesseract is another

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-23905784655685416162007-04-03T19:04:00.000-05:002007-04-03T19:23:11.440-05:00

The shape of a billiard table has a lot to do with the types of shots you can make in a game of billiards. Which of these odd tables do you think would be your best bet for hitting another ball?With the help of a little geometry, an expert billiards player can figure out exactly where a ball will go. Unless it's a trick shot, the ball will travel in a straight line until it hits a cushion. When

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-542195576202581042007-04-02T18:47:00.000-05:002007-04-02T18:55:04.610-05:00

The dice game known as Piggy punishes the greedy player. In Piggy, you and your opponent take turns rolling a pair of ordinary dice. Your score is the sum of the face values of the dice, so if you roll a three and a four, you get seven points. The first player to reach 100 points wins.Here's the fiendish part. You can roll as many times as you want in a turn, but as soon as you roll doubles, you

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-61755966904677575462007-04-01T11:45:00.000-05:002007-04-01T12:13:46.145-05:00

To Arthur Silverman, a sculptor in New Orleans, tetrahedrons, or triangular pyramids, are very special. He's been creating sculptures based on the tetrahedron for more than 20 years. You might see examples on display in plazas and office buildings in New Orleans, San Francisco, and other cities in the United States.Until the age of 50, Silverman had been a successful surgeon. He gave that up,

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-50994445545671348432007-03-31T12:23:00.000-05:002007-03-31T12:47:02.388-05:00

Look into a mirror, what do you see? One image of yourself. But if you stand between two mirrors that are parallel to each other, one in front of you and one behind, you see countless images of yourself. Light rays bounce back and forth between the mirrors to create that endlessly repeated pattern.What happens when two mirrors are not parallel to each other? You can find out by

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-85703722451846396502007-03-30T19:14:00.000-05:002007-03-30T19:39:07.146-05:00

Playing catch is easy. It's not hard to follow a single ball thrown back and forth between two people. But add another ball or two (or more), and take away one of the catchers. You end up with something that looks quite magical. The juggled balls seem to take on a life of their own.Many of the roughly 3,000 members of the International Jugglers Association are mathematicians or computer

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-63604091948320181682007-03-29T19:16:00.000-05:002007-03-29T19:33:14.179-05:00

There's trouble in the schoolyard. A kid has accused you and your pals of doing something you shouldn't have. You have to decide whether to stick with the gang or go your own way. You're not completely sure you can trust your friends, so you face a tough choice. Mathematicians and economists call this type of problem the prisoner's dilemma: It forces you decide whether you cooperate with others

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-75482371844678075752007-03-28T19:36:00.000-05:002007-03-28T19:47:50.586-05:00

You've probably looked at your distorted reflection in one of the shiny ornaments that decorate shops and homes during the holiday season. But you may be surprised to learn what the reflections in a pyramid of silvery balls are like.You can try this at home with four round Christmas tree ornaments and a small flashlight. Large, silver balls work best.Place three of the balls on a table so they

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-67466629408818895952007-03-27T19:19:00.000-05:002007-03-27T19:52:12.360-05:00

If you've ever looked for a four-leafed clover, you know that nature rarely delivers such a curiosity. Nearly every clover plant you check has the usual three leaves. If you study the flowers in your garden or in the countryside, you'll discover the most common number of petals is five. Buttercups, geraniums, pansies, primroses, rhododendrons, tomato blossoms, and many more all have five

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-43970103878524606322007-03-26T19:27:00.000-05:002007-03-26T19:40:34.725-05:00

One of your sneakers has a broken shoelace. You want to replace it, but the only laces you have around are much shorter. Is there a way to lace your shoes so that you can use the shorter laces?There are three common ways to lace shoes: American (or standard) zigzag, European straight, and quick-action shoe store. The lacing style you happen to use is generally the one you learned as a

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-17866688867247439812007-03-25T11:48:00.000-05:002007-03-25T11:54:58.065-05:00

You tear open a package of M&M's chocolate candies. Fifty-seven little candies spill out. You notice right away that certain colors are more common than others. In fact, you might count 7 brown, 17 red, 18 yellow, 6 green, 5 orange, and 4 blue candies in this bag. Does every package contain exactly the same amount of each color?A second package turns out to hold 55 candies: 12 brown, 12 red, 13

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-75893127086540976.post-47145624016706647942007-03-24T18:07:00.000-05:002007-03-24T18:18:52.106-05:00

Have you ever wondered why the cover of a manhole is nearly always round? Why couldn't it be oval or square?The usual answer is that a circular lid, unlike a square or an oval cover, won't fall through the opening. There's no way to position a round lid so that it would slip through a slightly smaller hole of the same shape. That's because the circle has a constant width—it's the same width all

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-32065160000725771342007-03-23T18:25:00.000-05:002007-03-24T18:16:39.665-05:00

The birthday party is over, and one chunk of thickly frosted, richly decorated cake is uneaten. Your mother insists that you and your sister slice the cake into two equal pieces, so that she doesn't have to listen to you fight over which is bigger. What should you do?The simplest strategy is to let one person cut the cake into two pieces, then let the other person choose first. That's sure to

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-49123589753510655452007-03-22T19:49:00.000-05:002007-03-22T20:01:33.980-05:00

The rocky Coast Range of California is the setting for an unusual game played by the brightly colored side-blotched lizard (Uta stansburiana). Each male of the species has one of three throat colors, and each color of lizard has a different mating behavior.The lizards' antics are like the game of rock-paper-scissors. In the playground version, each of the two players holds a hand behind his or

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-14852399951218096932007-03-21T19:17:00.000-05:002007-03-21T19:44:39.738-05:00

Have you ever watched a magician tie a humongous knot and then, as if by magic, make it fall apart? Sometimes what looks like an impressive knot isn't a knot at all. Magicians and escape artists are experts at tying phony knots.Here's an example you can use to impress your friends. Get a piece of rope—a few feet long—and follow these steps.A tug should easily undo the resulting tangle.Though

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-49528510925210165182007-03-20T19:25:00.000-05:002007-03-21T19:11:05.713-05:00

Riding around on a flat tire is no fun. It feels really bumpy. But a square wheel may be the ultimate flat tire. There's no way it can roll over a flat, smooth road without jolting the rider again and again.Stan Wagon, a mathematician at Macalester College in St. Paul, Minnesota, has a bicycle with square wheels. It's a weird contraption, but he can ride it perfectly smoothly. His secret is the

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-51795612856538024992007-03-18T12:22:00.000-05:002007-03-18T12:38:10.916-05:00

Recognize the symbol on the garbage bin? It looks like three bent arrows chasing each other around a triangular loop. You often find it printed on cardboard cartons, envelopes, greeting cards, packages, and trash containers. It stands for recycling.It's also an intriguing mathematical object. Imagine joining the tip of one of the arrows to the tail of the next all the way around to create a

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-14260942002833744132007-03-16T19:03:00.000-05:002007-03-16T19:10:15.892-05:00

When he was a kid, Arthur Benjamin liked to show off. Now, he's a math professor at Harvey Mudd College in Claremont, California—and a professional magician.Unlike other magicians, Benjamin doesn't pull rabbits out of hats, make coins appear and disappear, or perform rope tricks. Instead, he amazes audiences by multiplying numbers in his head faster than someone using a calculator.Benjamin says

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-75893127086540976.post-70146887182223692742007-03-10T11:03:00.000-05:002007-03-21T19:16:22.416-05:00

You desperately need a photo of a spider for a paper due tomorrow, so you search the Internet for a good image. Your search turns up all sorts of pictures, not only of spiders but also of sports cars, comic-book characters, card games, and other stuff.A lot of the pictures are wrong because many of the billions of images on the Web aren't labeled very well. Some aren't labeled at all. And

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com3