Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

May 31, 2007

Infinite Wonders

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 you'll get a bigger number.

Because there's no imaginable "last" number, mathematicians say there are infinitely many whole numbers. When you start thinking about infinity and numbers, you also run into all sorts of weird mind-boggling puzzles and paradoxes. Here's one in which hotel rooms stand in for numbers.

Suppose you walk into an ordinary hotel. If the desk clerk tells you there are no vacancies, there's really nothing you can do. Every room is occupied.

It's very different at the Infinity Hotel, which has an infinite number of rooms. Even if all the rooms are occupied when you arrive, the clerk can still find you a bed. She simply sends a housekeeper to awaken the guests and move them to new rooms. To make sure everyone has a place, the guest in room 1 goes to room 2, the guest in room 2 goes to room 3, the guest in room 3 goes to room 4, and so on, until each occupant has moved one room over. There's an extra room at the end of the line because infinity plus 1 still equals infinity. And with room 1 vacant, you can unpack and settle down for the night.

In fact, you could make room for an infinite number of guests. Move the guest in room 1 to room 2, the guest in room 2 to room 4, the guest in room 3 to room 6, the guest in room 4 to room 8, and so on. This frees all the odd-numbered rooms, and there is an infinite number of them. The new arrivals can move into the vacant rooms, because adding infinity to infinity still gives you infinity.

That's the strangeness of infinity, where a part can be as large as the whole and there's always room for one more.

Muse, February 2004, p. 27.

May 30, 2007

Dog Does Calculus

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 always play fetch. Standing at the water's edge, Pennings throws a tennis ball out into the waves, and Elvis eagerly retrieves it.

Elvis may do calculus, but does he speak Elvish?

When Pennings throws the ball at an angle to the shoreline, Elvis has several options. He can run along the beach until he is directly opposite the ball, then swim out to get it. Or he can plunge into the water right away and swim all the way to the ball. What happens most the time, however, is that Elvis runs part of the way along the beach, then swims out to the ball.

That happens to be a good strategy. Swimming is slow compared to running, so swimming to the ball takes longer even if the route is more direct. On the other hand, the longer Elvis runs along the beach, the farther he must go to get to the ball. The best bet is a compromise between the two—running a certain distance along the beach before plunging into the water.

Figuring out the best plunge point is a problem that belongs to a branch of mathematics called calculus. Pennings found that Elvis usually picked a path that was very close to the one a mathematician would say was the fastest possible one.

Of course, Elvis doesn't actually know calculus. He just has a sixth sense for efficient fetching that was bred in the bone and honed by lots of practice.

Does your friend behave the same way? You could try throwing a ball into deep snow to see where he or she plunges off the sidewalk. Maybe, like Elvis, your friend does calculus without knowing it.

Muse, January 2004, p. 27.

Photo courtesy of Tim Pennings.

May 29, 2007

Magic Squares

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, column, and diagonal add up to 15.

Arrangements of numbers that add up to the same total in every row, column, and diagonal are known as magic squares. Melancholia, an engraving by the German artist Albrecht Dürer, includes a famous magic square. The rows, columns, and main diagonals all sum to 34.

The magic square is hanging on the wall to the upper right. Not only do the rows, columns, and diagonals total 34, so do the numbers in the corner squares and the numbers in the central four squares. Can you find other combinations within the square that add to 34? There are several. For example, If you divide the four-by-four square into four two-by-two squares, each of those squares will add up to 34. What's more, the numbers in the middle bottom squares read 1514, the year Dürer made the engraving.

Why pack so much number magic into one square? Astrologers in Dürer's time associated different types of magic squares with the planets, which, in turn, were thought to influence health. The brooding man is suffering from Saturn's "saturnine," or gloomy, influence. He hopes Jupiter's "jovial" four-by-four magic square will draw down, or decrease, Saturn's influence. (Of course this is all absolutely nutters, but that's the history of ideas for you.)

You don't have to use consecutive numbers or make sure that all the numbers are different. Here's a three-by-three example made up of only odd numbers.

Here's another one that consists of just prime numbers—numbers evenly divisible only by themselves and 1.

People have been looking for magic squares of various sorts for centuries. In colonial times, Benjamin Franklin used to make up magic squares when he got bored listening to political speeches. He's famous among number fanatics for an amazing eight-by-eight magic square containing the numbers from 1 to 64.

More recently, Lee Sallows, an electronics engineer in the Netherlands, discovered a magic square that has truly astonishing properties. Start with the following magic square:

Spell out the English words for each of the numbers:

Count the number of letters in each word, and enter the number in the appropriate space of a blank three-by-three grid:

The result is another magic square, which contains the consecutive numbers from 3 to 11! Such "alphamagic" squares can be found in other languages, too.

There's still a lot more to discover about magic squares, and there are many more to find. Maybe you'll encounter something interesting in searching for your own digital good-luck charm.

Muse, November/December 2003, p. 32-33.

May 28, 2007

Up the Magician's Sleeve

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, following these rules.

Suppose her secret number is 6. The sixth card that she deals becomes a "key" card, and its face value tells her how many more cards she must deal out to get to the next key card. For example, if the key card happens to be 3, she counts from 1 to 3 to find the next key card. She silently repeats this procedure—without pausing, because pauses would tip you off—until she has dealt out all 52 cards. An ace counts as 1, and a king, queen, or jack counts as 5.

At some point, your friend will run out of cards and won't be able to complete the count. Her final key card becomes her secret "chosen" card. Your task as the magician is to read your friend's mind and identify that card.

Here's what you do while your friend is dealing out the deck. You pick your own secret number, then watching the cards, count your way to the end at the same time as your friend. You'll end up with your own "chosen" card. Amazingly, no matter what secret number you pick, you're likely to end up at the same card as your friend.

One way to see what's going on is to deal out a shuffled deck so the cards are in long rows. You can then use different coins, colored poker chips, or other markers to identify the key cards associated with each of the 10 possible starting points.

Suppose your secret number is 1, and the first card is a 10. The first chip goes on the 10. The second chip goes on the 11th card in the row. If that card is an ace, the next chip would go on the next card in line, and so on, until you reach the final key card. Do the same for the other secret numbers, laying down a trail of colored chips for each one.

You'll see that, for nearly all arrangements of cards, every starting point leads to the same "chosen" card. Somewhere along the way, two separate trails of chips meet on the same card, then coincide from that point on. Here's an example.

This prediction trick is known as the Kruskal count, named for mathematician Martin Kruskal of Rutgers University, who discovered it. Mathematicians have studied the trick and have worked out the chances that a magician will guess the "chosen" card correctly. They put the odds of being successful at five times out of six. It helps a little bit of the magician chooses a lower secret number (say, 2 instead of 5) or simply starts with the first card.

The Kruskal count shows how seemingly unconnected or unrelated chains of events can lock together after a while. Such counterintuitive processes underlie many amazing coincidences and startling predictions. They're what the magician has up his sleeve.

Muse, October 2003, p. 42-43.

May 27, 2007

Seeing Spots

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 idea a step further. They've created bug pictures out of tiny pictures, each of which is just a darker or lighter patch in the larger image. They've even made pictures out of dominoes—those black tiles made up of two squares with white dots in them.

A domino with no dots is the darkest tile, and one in which both squares have nine dots is the lightest tile. The trick is to place dominoes within a rectangle in such a way that the light and dark patches—when seen from far enough away—add up to a recognizable image.

Mathematician Robert Bosch of Oberlin College in Ohio has set himself an even bigger challenge. His goal is to create recognizable portraits using complete sets of dominoes. There are 55 two-square tiles in a set of double-nine dominoes. If Bosch works with, say, 24 sets, he must use exactly 24 blank dominoes, 24 dominoes that have a blank square and a square with one dot, and so on.

Clearly, Bosch's method wouldn't work for a snowy Arctic scene. He wouldn't have enough dominoes with many white dots to fill in the picture properly. But it works surprisingly well for portraits of people, which usually have a nice mix of dark and light areas.

Bosch wrote a computer program to help him decide where to put each domino. His program starts by dividing the original black-and-white picture into squares. Some squares are completely white, others are completely black, and the rest are various shades of gray. White squares are assigned the value 9, black squares have the value 0, and gray squares are given in-between values. Clearly, double-blank dominoes would most likely go where the value is 0, and double-nine dominoes would most likely go where the value is 9.

Bosch's software determines where each available domino should go, selecting the arrangement that most closely matches the numbers that describe the original picture. Bosch then builds his domino portrait.

This is a domino version of the famous portrait known as the Mona Lisa. Bosch constructed the portrait from 40 complete sets of dominoes.

Bosch has made more than a hundred, including portraits of Abraham Lincoln, Albert Einstein, John Lennon, and Martin Luther King Jr. You can see some of them at

A classroom project involving first- and second-graders produced this domino portrait of Martin Luther King Jr.

Muse, September 2003, p. 34-35.

Images courtesy of Robert Bosch.

May 26, 2007

One-Cut Angelfish

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 would be possible to make by folding and just cutting once. To simplify things a bit, they assumed that the shapes would have straight edges. A mathematical figure with straight edges is called a polygon. It can be as simple as a triangle or as complicated as a lacy star.

Remarkably, the researchers proved that after just the right set of folds, any straight-line drawing, or polygonal shape, can be cut out of one sheet of paper by a single straight cut, no matter how complicated the shape may be.

The hard part, however, is figuring out how to fold the paper properly and then knowing exactly where to cut it to get the design you want. Demaine and his coworkers have come up with a procedure for converting a design into fold-and-cut instructions.

Although the procedure can get pretty messy, with all sorts of tricky folds, it works, and the researchers have invented many new fold-and-cut designs, including beautiful angelfish, swans, butterflies, turtles, and fancy stars. There's even a way to do your own name or initials in block letters!

Angelfish Instructions: For easier folding, lay the pattern over another piece of paper or on a sheet of cardboard. Using a ruler and ballpoint pen, trace the pattern to score the paper. The dotted lines are "valley" folds and should be folded toward you. The dashed lines are "mountain" folds and should be folded away from you. Once you've made all the folds, you have to "collapse" the completely folded form into a compact wad. This is the frustrating step. Then one snap of the scissors should suffice to cut all the bold lines at once.

For other one-cut cutouts, go to

Muse, July/August 2003, p. 27.

May 25, 2007

What's the Deal?

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 in the order of the cards—patterns that experts and gamblers can take advantage of to win more often than they otherwise would.

When computer-shuffled decks were first used in bridge tournaments, there was an outcry. The players thought there were wild fluctuations in the distribution of cards of different suits. Research showed the problem lay not in the computer but in the players' expectations.

In bridge, cards tend to clump together in groups of four of the same suit, and shuffling often didn't break up these groups. In fact, the intuition of bridge players had been shaped by generations of badly shuffled cards. Books on bridge recommended strategies based on bad shuffles. When computer shuffling was introduced, many of these strategies had to be changed.

How often should you shuffle a deck to be sure that the cards are all mixed up? Many people think three shuffles are enough. They're wrong. Statisticians David Aldous and Persi Diaconis have studied shuffling, and they concluded that it takes about seven riffle shuffles to put 52 cards in random order. In a riffle shuffle, you cut the deck into two packets of cards, then holding one packet in each hand, you run the cards past your thumbs to raggedly interleave the cards.

Curiously, the transition from order to randomness occurs quite abruptly. If you shuffle five times or fewer, the original order disappears. You can see the same sort of sudden transition in your kitchen when you stir together white flour and cinnamon. At first you see thick streaks as the ingredients mingle. After a few more strokes, the whole mixture suddenly smooths to a tan color.

Not everyone agrees that you need as many as seven shuffles. Other mathematicians, using different ways of measuring randomness, say as few as five shuffles may work. Nonetheless, it's pretty clear that three lackadaisical shuffles aren't enough to truly mix up a deck of cards.

Muse, May/June 2003, p. 23.

May 24, 2007

Flipping a Coin

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 table is smooth and flat, you should end up with an impressive cluster of upright coins. Then let the pennies fall over. You might have to bang the table to get them all to topple. Count how many land with tails up and how many land with heads up.
You probably expected about half to be heads and half tails. In fact, that rarely happens. Nearly always, you end up with far more heads than tails.

If you look closely at the U.S. penny, you'll see the sides aren't really flat. Abraham Lincoln's head is carved out of one side and the Lincoln Memorial is inscribed on the other.

In fact, it looks like a little more metal has been removed from the heads side than from the tails side. As a result, the coin tends to fall over with the head on top. In other words, because of its uneven distribution of weight, the penny is biased.
A penny's slight weight bias has very little effect on how it flips in the air, however. The bias becomes important only when the coin lands. So if you flip a penny to decide something, it's wise to catch it before it has a chance to roll, spin, or bounce to a stop.

Are other coins biased? The best way to find out is by doing the penny experiment with them. Some coins won't stand upright, but among coins that will, you'll find some are more biased than others.

You probably knew some tricksters use a two-headed coin and some magicians have learned how to flip a coin so it always comes up heads. Now you know the coin itself isn't fair. One more trusting assumption knocked on its head—or should I say on its tail?

Muse, April 2003, p. 19.

May 23, 2007

It's Not You, It's the Puzzle

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 not just saying that because we couldn't solve it. We have mathematical proof.

Japanese puzzle designer Nob Yoshigahara invented Rush Hour in the late 1970s. For the U.S. edition of the game, Nob and his team developed several sets of puzzles that offer challenges rated from beginner to expert. For some initial arrangements of blocks, it takes only a few moves to free the designated car. In the most challenging cases, it can take as many as 50 moves to free it.

Mathematicians and computer scientists are interested in sliding-block puzzles like Rush Hour because they resemble real-world motion-planning problems. In some parking lots, for example, attendants cram cars together as tightly as possible. When a patron shows up to retrieve his or her car, the attendant must figure out which other vehicles to move to get the required one out as quickly as possible. (In Japan, Rush Hour is called Tokyo Parking Lot.) Engineers face a similar problem when they have to program a robot to shift bulky crates in a crowded obstacle-strewn maze.

By having computers solve the puzzle, researchers showed Rush Hour really is tough. It takes computers a surprisingly long time to find the best possible solution to a Rush Hour setup. And the more vehicles and the larger the grid, the longer it takes. Analysis puts Rush Hour on the same level of difficulty as such demanding games as Othello, although below that of Chess or Go.

So don't feel bad about being stumped. And consider this: Your parents had it worse. They had to wrestle with the fiendish "14-15" puzzle. You know the one. It consists of 15 tiles numbered from 1 to 15 in a square tray large enough to hold 16 tiles. Tiles 14 and 15 start out switched, and the player has to restore all the tiles to numerical order. No one could solve it, and mathematicians soon proved it could not be solved! Doubtless many trusting young minds were warped for life by the experience of trying.

Muse, March 2003, p. 21.

May 22, 2007

Hailstone Numbers

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 calculator.

Pick any whole number. If it's odd, multiply the number by 3, then add 1. If it's even, divide it by 2. Now, apply the same rules to the answer that you just obtained. Do this over and over again, applying the rules to each new answer.

For example, suppose you start with 5. The number 5 is odd, so you multiply it by 3 to get 15, and add 1 to get 16. Because 16 is even, you divide it by 2 to get 8. Then you get 4, then 2, then 1, and so on. The final three numbers keep repeating.

Try it with another number. If you start with 11, you would get 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, and so on. You eventually end up at the same set of repeating numbers: 4, 2, 1. Amazing!

The numbers generated by these rules are sometimes called "hailstone numbers" because their values go up and down wildly—as if, like growing hailstones, they were being tossed around in stormy air—before crashing to the ground as the repeating string 4, 2, 1.

Mathematicians have tried every whole number up to at least a billion times a billion, and it works every time. Sometimes it takes only a few steps to reach 4, 2, 1; sometimes it takes a huge number of steps to get there. But you get there every time.

Does that mean it would work for any whole number you can think of—no matter how big? No one knows for sure. Just because it works for every number we've tried doesn't guarantee that it would work for all numbers. In fact, mathematicians have spent weeks and weeks trying to prove that there are no exceptions, but they haven't succeeded yet. Why this number pattern keeps popping up remains a mystery.

Muse, February 2003, p. 17.

May 21, 2007

Monopoly Cheat Sheet

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 player follows the same strategy: Buy everything in sight.

But mathematicians who have built computer models of the game would tell you otherwise. You don't have an equal chance of landing on every square. Instead, you are more likely to hit some than others. The square players spend the most time on is Jail, which shouldn't be too surprising because there are so many different ways of being sent there. The next-most-popular squares include ones players tend to hit when they leave Jail (Illinois Avenue, Go, B&O Railroad, Free Parking, and Tennessee Avenue). Mediterranean, Baltic, and Park Place, in contrast, get little traffic, something you might want to consider before buying them.

Although it is tempting to race around the board buying up every available property, you can gain an advantage by being choosy. Which properties are best depends on the stage of the game. At first, railroads and utilities are good investments because a lot of players land on them. But as players gain monopolies and build houses and hotels, railroads and utilities become less important because they pay back much more slowly than other properties.

So what is the best real-estate investment? The orange properties get a lot of traffic from players leaving Jail, they don't cost much, and they have relatively high rents. Together this means they are the fastest to pay back the money you put into buying them. It takes a lot longer for green or purple properties to break even.

On the other hand, if you're certain your fellow players are going to stick it out to the bloody end, you might want to go for green. The green properties are relatively expensive but they pay the most per visitor. So once you've paid them off, you can sit back and rake it in. (Avoid purple at all costs. Those properties take forever to break even, and they pay the least per visitor.)

Of course there's a lot more to the game than rolling dice and buying property. You have to know when to make deals and what to bid in property auctions. In those situations you might want to rely on skills computers can't mimic: schmoozing and sneakiness.

* Just so you know: according to Hasbro, the maker of Monopoly, the longest game ever played lasted 70 straight days. (Is that with or without sleep breaks? we'd like to know.)

Muse, January 2003, p. 18-19.

May 19, 2007

Gambling Dogs

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 treat every time it performs.

In the beginning, you should reward the behavior you want each and every time it occurs. Later, as the dog gets the idea, you offer treats most of the time, then about half the time, then less and less often. Otherwise, as soon as the dog stops getting rewards, it stops sitting. In the end, you'll be offering treats only occasionally, but the dog will still perform on command. The dog doesn't know if you'll come across with the cookie, so it sits just to be on the safe side.

As Treat Distribution Officer, you're more like a slot machine at a gambling casino than a soda machine. Every time you put money in a soda machine, you expect a reward—a can of soda. If the can doesn't tumble down the chute, you get very annoyed. On the other hand, when you put money in a slot machine, reels spin, then stop at some combination of symbols that may or may not reward you with a jackpot. You know you won't get a prize every time, so you keep on putting in coins in the hope of eventually winning something. Companies that manufacture slot machines use mathematics to work out, on average, how often they need to pay out to keep you putting in.

Random rewards have a similar effect on dogs, people—and even pigeons. The power of random rewards was first discovered in experiments in which pigeons got a bit of grain if they pecked a key. If a pigeon got grain every time it pecked a key, and the grain was then cut off, it would peck the key only 50 to 100 times before giving up. But if it had been rewarded only occasionally for pecking, it would peck 4,000 to 10,000 times without any reward.

It seems all creatures are gamblers at heart. So it's probably a bad idea to take your dog to Las Vegas. (Or your pigeon, either.)

Muse, November/December 2002, p. 45.

May 18, 2007

Tricky Choices

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 procedure you use. And the most popular choice might not even win.

How could that be? Here's an example. Suppose your class of 15 kids must decide which one of three snacks (pizza, popcorn, or cookies) to get for a party. Six kids like pizza best, cookies next, then popcorn. Five kids like popcorn best, cookies next, then pizza. Four kids like cookies best, popcorn next, then pizza.

If each person voted only for his or her favorite snack, pizza would win, popcorn would come second, and cookies would come third.

Would everyone be happy? If you take a close look at everyone's preferences, you'll see that nine kids actually like popcorn more than they like pizza. Similarly, nine kids like cookies more than pizza, and ten kids like cookies more than popcorn. So if you get a pizza, you voted, but the majority did not rule.

How could you get a truer vote? Suppose you dropped the choice with the fewest votes in each round and voted again until the top choice had more than half the votes. In this case, cookies would be dropped first. In the runoff, however, popcorn would beat pizza!

The winner has changed, and, hey, what about cookies? More people like them than like popcorn.

This is very confusing. What if you try a voting system in which you give two points to your first choice, one point to your second choice, and no points to your third choice. The winner is the one with the most points. In this case, cookies would win. The answer changed again, and you ended up with cookies even though most kids wanted pizza or popcorn rather than cookies!

So just by changing the voting procedure, you can get a different result—and still not satisfy the majority of the voters.

Mathematicians who have studied voting procedures involving three or more choices have come to a startling conclusion. It turns out that no matter which system you use, it's always possible to get results that don't look fair.

It can happen in all sorts of voting when you have a lot of choices—whether you're picking pizza or a president.

More choices (Oct. 27, 2004)

Lots of things can go wrong in elections. In national, state, and local contests, people worry about voting machines that don't work properly, poorly designed ballots that make it hard to vote for the candidates you want, and votes that are miscounted.

Officials are trying to fix these kinds of problems to make sure elections are fair and run smoothly. In some places, they're even trying out different voting rules.

For example, instead of picking just one candidate from a list, voters in San Francisco will now rank their choices, from first to third. These votes are counted in a special way to determine the winner.

Here's what happens in elections for San Francisco's Board of Supervisors.

• Voters pick three candidates from a list. They rank their choices, from first to third.
• All the top choices are counted first. If any candidate gets more than 50 percent of the vote, that candidate wins.
• If no candidate has a majority, the candidate with the fewest first-place votes is eliminated. Voters who marked the losing candidate as their first choice will have their votes counted for their second-choice candidate.
• The process continues until one candidate receives a majority of the vote.

Does that sound confusing? Will the system discourage people from voting? Does it look like it might take a long time to get the final results? Will the final result be any fairer?

The first test of the San Francisco system will take place in November. It'll be interesting to see what happens.

Muse, October 2002, p. 24.

May 13, 2007

Global Views

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 you. His answer was to paint the scene not on a flat canvas but on the surface of a large plastic ball. Using geometry, he worked out a way to translate the view from inside a sphere to the outside of one.

In Termes's sphere painting of a fishbowl, you can even see beyond the table and cat to glimpse the rest of the room and a kitchen to one side, where someone has apparently just finished eating fish for supper and left only bones on the plate.

Hung from a rod, the painted ball slowly rotates, presenting six different viewpoints. Amazingly, as you stare at the revolving sphere, it appears to pop inside out. You feel as if you are sucked inside to get a weirdly distorted, inside-out view of the painted scene. It's a remarkable optical illusion.

Termes (above) has painted all sorts of scenes on the surface of spheres, including the interiors of famous buildings, many different geometric patterns, and various imaginary "dream" worlds. His largest "termesphere" started out as an orange, rotating Union 76 gas station sign (below), seven-and-a-half feet in diameter, and ended up at the Wyoming Law Enforcement Academy in Douglas, Wyoming.

Termes's sphere paintings even give you a rough idea of what your surroundings might look like if you were moving at nearly the speed of light—300,000 kilometers (about 186,000 miles) per second. Physicists have shown that no object can travel faster than this speed, and as you get closer to the universal speed limit, the appereance of your surroundings becomes distorted, much as does Sainte Chapelle (below) in one of Termes's sphere paintings.

Termes enjoys thinking about mathematical patterns, and he loves explaining and illustrating what he does so that we can all see the world from new, unusual perspectives.

Muse, September 2002, p. 42.

Dick Termes has a Web site at

Images courtesy of Dick Termes.

May 12, 2007

Unfolding Wonders

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 an artist, engineer, architect, and inventor, Hoberman spends a lot of time thinking about simple geometric shapes—circles, triangles, spheres, and so on. Like a magician, he is fascinated by the notion of making something vanish, then suddenly reappear, and he loves the idea of one shape smoothly transforming into another. He has a passion for designing things that not only look interesting but also act in a surprising way.

Hoberman's mother was a children's book author and his father an architect. As you might expect, he played with building blocks, Lincoln logs, and Erector sets when he was young. But he says he preferred drawing and painting. He spent hours with his brother making comic books. An art teacher at school made him aware of how important it was to observe carefully and draw exactly what you see.

In college, he was given the assignment of making a sculpture that moved and came up with an artwork made of plastic sheets that unrolled to reveal interesting patterns. That got him interested in how gears, levers, pulleys, struts, and cables work together in mechanisms, and he studied mechanical engineering to learn more.

Early in his career, Hoberman worked at a robotics company, where he learned how to use computers for design and to work with machinery. On his own, he explored different ways of folding origami-like paper constructions into compact units. He then moved on to metal and plastic structures that could balloon into surprisingly large forms.

Hoberman now holds several patents for ways of packing large structures into small spaces. He has used those ideas not only to create ingenious toys but also to make practical items, including a briefcase that collapses to the size of a book, a portable tent made from a single sheet of fabric, and medical devices that can sneak into tight spaces.

You can see large-scale versions of his unfolding structures at several science centers, including the Liberty Science Center in Jersey City, New Jersey, and the California Science Center in Los Angeles. Photographs of these and other Hoberman creations can be found on the Web at

One of the most spectacular of Hoberman's creations was a latticework aluminum arch 36 feet tall that stood earlier this year on the stage at the Olympics Medals Plaza for the 2002 Winter Olympics in Salt Lake City. His arch opened like the iris of an eye, revealing the stage.

To Hoberman, math is not just numbers or formulas on a page. Math is about shapes and relationships among shapes—which he makes beautiful, surprising, and fun.

Muse, July/August 2002, p. 34.

Images: Hoberman

May 11, 2007

Batting Streaks

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 likely to be about half the number of games he played. In the end Bonds played 153 games and walloped 73 homers.

Here's an interesting question. Bonds had a 50 percent chance of hitting a homer over the course of the season, but were those his odds for any particular game as well? What if he had a hot streak? What if you went to a game when he was in the middle of his hot streak? Wouldn't the chances of his hitting a homer be higher than even-steven?

Another way to ask this question is to ask whether Bonds was hitting homers like a tossed coin falls. When you toss a fair coin, it has a 50 percent chance of coming up heads. Even if you get heads three times in a row, the next time you tossed the coin it would still be even odds it would come up heads. Was Bonds's ability to hit as random as a coin toss, or could he, by eating his Wheaties or spitting on his hands, force a long run of good luck?

For each game during last summer's baseball season, economist Paul Sommers of Middlebury College in Vermont checked whether Bonds hit one or more home runs and looked for streaks. Bonds had one stretch of 13 games in which he failed to homer. He had two stretches of six games in which he hit home runs every time.

You can get streaks when you toss a coin, too. Suppose you toss a fair coin 250 times. You will probably get two runs of six heads or more, and one run of seven heads or more. Most people are surprised by this. When they are asked to write down a long string of heads and tails that they believe is random, they rarely include four or five heads in a row, even though such runs are likely to occur.

Sommers used a statistical formula to find out whether Bonds's hot streaks were as random as home runs in a coin toss. They were. How could this be? Maybe so many factors affected his hitting—the weather, game time, the ballpark, whether the pitcher was right- or left-handed, and so on—that it wasn't really possible to predict whether he'd hit a homer in a particular gasme. All you could say is, based on past performance, his odds were about 50-50.

Somehow that sort of takes the magic out of a streak. But it doesn't mean Bonds isn't a good player. A lesser player might have as much chance of hitting a homer as a single die has to come up as a three (one chance in six).

Muse, May/June 2002, p. 35.

May 10, 2007

Mental Math

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 know the distance to a building, he peers through a telescope called a surveyor's level, moving it until it is lined up with the building. He can then read out the angle between this line and his line of sight to the distant horizon. Knowing this angle and the height of the telescope, he can calculate the distance to the building.

The cool thing is you do this yourself all the time. As you look around, you constantly make decisions about how far away things are—whether it's your pal down the street or a tree in the distance. Of course you don't really do trig. That's great for drawing a map or locating a building, but it's more work than you want to do for an on-the-fly estimate.

Instead you guess how far away the object is by where it falls in your field of view. If you have to look down toward your feet, you can assume that it's close to you. If you have to peer toward the horizon to see it, it's far away. So, the farther away an object is on the ground, the higher it is in your field of view. This is actually more or less the same calculation the surveyor does. You're using an angle to figure out a distance.

Scientists now have evidence confirming that people really do use this angle to decide how far away things are. Ten Leng Ooi of the Southern College of Optometry in Memphis, Tennessee, and her coworkers asked volunteers to wear prism goggles. The prism changed the direction in which light travels, making objects appear lower in the field of view than they really were. The volunteers missed when they tried to walk to the objects blindfolded or throw beanbags at them.

Volunteer (above) viewing a distant object without crazy prism goggles.

When people were allowed to get used to the goggles beforehand, they judged distances correctly. (This proves the brain has an amazing ability to adapt to weird input.) Then, when they took off the goggles, they temporarily went to the opposite extreme, overestimating distances and overshooting objects when they tried to walk to them.

Volunteer (above) viewing a distant object with crazy prism goggles.

So, even if you haven't studied trigonometry yet, your brain is using it all the time. But beware of optometrists looking for volunteers. They are apt to make you look like a total idiot.

Muse, April 2002, p. 44.

May 9, 2007

What a Coincidence!

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 fall on any of 364 days. That means the probability that Allen and Brandi have different birthdays is 364/365. Now Carol enters the room. There are only 363 unused birthdays, so the probability that her birthday is different from the other two is 363/365. The probability that all three birthdays are different is 364/365 times 363/365.

For four children, you mulitply 364/365 times 363/365 times 362/365. Amazingly, by the time you have 23 kids in the room, the chance that two of them have the same birthday is higher than 50-50. Or to put it another way, if you checked all classrooms with 23 kids, about half of the classes would have a duplicate birthday.

Try it out on your class. If there are more than 23 kids, the chances are even better that you'll find matching birthdays. In fact, because birthdays aren't really spread evenly throughout the year, you're likely to find a duplicate birthday in about half of all classes with as few as 20 children.

If this seems surprising, consider this twist: The chance that two people have the same birthday is pretty high, but the chance that someone else has your birthday is pretty low. Those probabilities are trickier than they seem.

Muse, March 2002, p. 43.

May 8, 2007

Multicolored Maps

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 of states without using a fourth color. Can you figure out what other states force the use of a fourth color?

Would you ever need five colors? In 1976, two mathematics professors at the University of Illinois proved that four colors would be enough for any map that could be drawn on a flat piece of paper.

Curiously, four colors are needed whenever a state is surrounded by three or more states and the number of surrounding states is an odd number. You see that not only in a U.S. map but also in a map of South America. The country Bolivia, for example, is surrounded by an odd number of countries: Peru, Brazil, Paraguay, Argentina, and Chile. What other country in South America is in the same situation?

Simpler maps sometimes use fewer than four colors. For example, if you had a map that looked like a checkerboard or some other sort of grid, you would need only two colors.

What about a pattern made up of tiles shaped like kites and darts or a pattern made of diamonds like the one below?

Mathematicians have recently proved that you need at most three colors for such a map.

Muse, February 2002, p. 44.

Answer to U.S. map: Kentucky, which is surrounded by seven states; and West Virginia, which is surrounded by five states.

Answer to South American map: Paraguay, which is surrounded by three countries.

May 7, 2007

Poe's Secrets

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 go about solving the puzzle?

The clever treasure seeker in "The Gold-Bug," William Legrand, assumed that symbols stood for letters of the alphabet, with no spaces left between the words of the message. He noticed that the character "8" appears 33 times, far more often than any other character. In the English language, the letter that occurs most often is "e." Starting with that clue, he went on to look for combinations of three characters that might represent "the"—a very common word in English. Legrand could then guess that the semicolon represents "t" and 4 represents "h."

Following such hints, Legrand deciphered the secret message (see end). Clues contained in the mysterious message eventually led him to a fortune in gold and jewels.

For a short time, Poe was also editor of a magazine. In a contest that lasted six months, he invited his readers to submit coded messages that they thought would stump him. Poe then published two puzzles of his own and challenged readers to solve them. The puzzles were so hard that the first one wasn't solved until 1992, and the second one wasn't solved until 2000—and then only with the help of computers.

Here is Poe's first puzzle (above).

The second message was tough to decipher because it used several different symbols for each letter. Moreover, the number of symbols for a given letter depended on how often that letter appears in English text. For example, there were fourteen symbols standing for "e" and just two symbols standing for "z."

Here is Poe's second puzzle (above).

Poe would have been delighted to know how long he had managed to mystify his readers!

What the Gold-Bug Message Said

A good glass in the bishop's hostel in the devil's seat twenty-one degrees and thirteen minutes northeast by north main branch seventh limb east side shoot from the left eye of the death's-head a bee line from the tree through the shot fifty feet out.

Roughly translated, the message means that if you sat in a scooped out hollow in a rock formation called the Bessop's Castle and looked northeast through a telescope you would see something in a distant tree. This turned out to be a skull nailed to the tree's seventh limb. Dropping a weight through one eye socket of the skull marked a point on the ground. If you drew a line through that point starting at the tree trunk, you'd find the treasure 50 feet away from the tree along that line.

Muse, January 2002, p. 44.

Poe's magazine puzzles are deciphered at

May 5, 2007

Bunching Buses

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 instead of at regular intervals?

Some people claim that bus bunching doesn't happen very often. They say that passengers tend to remember the few times when more than one bus arrives at a stop and to forget the many more times when a bus arrives alone.

Mathematicians who study traffic, however, say that bunching really can happen.

The problem is the people at the stops. If there were always the same number of people, the buses would keep to their schedule. But usually there are many people at a few stops and no one at others.

Suppose many people happen to gather at a particular stop. It takes longer than usual for the passengers to board the first bus that arrives, so it gets delayed and the bus behind it catches up a bit. When the second bus arrives at the same stop, there has been less time for passengers to assemble, so the bus goes on its way quickly. Meanwhile, the first bus arrives at its next stop a little later than usual, so there's more time for passengers to join the crowd, and so on. After several more stops, the second bus catches up with the first.

Once one bus catches up with the other bus, the two buses end up traveling together. If the route is a long one, a third bus could eventually catch up with the first two.

Check it out the next time you hop a bus!

Muse, December 2001, p. 39.

May 4, 2007

Decoding Bar Codes

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 8

The first digit, 0, gives the product category—in this case, general groceries. The next five digits (16000) identify the manufacturer (General Mills). The following five digits (66610) identify the specific product: a 15-ounce box of Cheerios. A larger or smaller box would have a different number.

When you check out, the bar code is scanned and the store's computer retrieves the price for that product. The computer also does some arithmetic to help make sure the bar code scanned correctly. That's where the final digit of a UPC comes in. It's called a check digit.

In the tiny amount of time before the price appears on a cash register screen, the computer performs the following calculation using the first 11 digits of the UPC.

• Adds the six digits in the odd positions: 0 + 6 + 0 + 6 + 6 + 0 = 18
• Multiplies the result by 3: 18 x 3 = 54
• Adds the five digits in the even positions: 54 + 1 + 0 + 0 + 6 + 1 = 62
• Subtracts the sum from the next-highest multiple of 10: 70 – 62 = 8

If the result matches the UPC check digit, as it does in this case, the computer sends the price to the cash register. If it doesn't, the cashier hears a beep.
Try the calculation yourself on the UPC of your favorite product. It's amazing how much math goes on behind the scenes—even at a grocery store!

Muse, November 2001, p. 33.