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 experiment.

Carefully join two small mirrors with a strip of tape. Stand the pair of mirrors on a table so that the tape is vertical and the reflecting surfaces face you. Make the angle between the mirrors 90 degrees, and place an object between them. Count the number of reflections you see.

You may have noticed such an arrangement of mirrors in the corner of a fitting room at a clothing store. You see the object (you in new clothes) and three images of the object.

What happens to the number of reflections as you make the angle between the mirrors larger? Smaller?

At certain angles, the patterns are especially beautiful. When the angle is 60 degrees, for example, you see the object plus five images, with no overlap. That's the kind of pattern you typically see in a kaleidoscope.

Suppose you looked into a special kaleidoscope and found a pattern that consisted of one object and seven images of that object. Can you figure out what the angle between the mirrors inside the kaleidoscope must be?

You can experiment with your hinged mirrors, or you can use a mathematical formula to answer the question. Simply take the number of images you want, add one for the object itself, and divide 360 degrees by this number. Three hundred sixty divided by 8 is 45. So the angle between the mirrors must be 45 degrees.

Artists, toymakers, and other people have built all sorts of kaleidoscopes, including some you can actually climb inside and some that are works of art.

The possibilities are limitless. Instead of two mirrors, you can use three, fastened together to form a triangular enclosure, to create an endless tapestry of repeated shapes. Instead of a simple object, you can use pieces of colored glass, beads, coins, or even lines and shapes drawn on paper.

Muse, March 2000, p. 18.

Mess around with a computer kaleidoscope at http://aleph0.clarku.edu/~djoyce/pix/kaleido.html.

The Exploratorium in San Francisco has a kaleidoscope that you can really get into: http://www.exploratorium.edu/imagery/stills/Duck_Into_Kaleidoscope.jpg.

The kaleidoscope illustrations for this article were created using KaleidoMania! software. See http://www.keypress.com/x6173.xml.

## MatheMUSEments

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

## March 31, 2007

## March 30, 2007

### Juggling by Number

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 scientists. Mathematician Joe Buhler started juggling years ago when he was in college. He did it just because it was fun. On a good day, he can keep as many as seven balls going in a pattern called a cascade.

The cascade pattern requires an odd number of balls. Each ball follows a looping path that resembles a figure 8 on its side. Once the balls are in motion, the juggler never holds more than one ball at a time. The world record for a sustained cascade is nine balls for 60 catches in a row.

With a little coaching and some practice, just about anyone can learn how to juggle, Buhler says. Once learned, juggling—like bicycle riding—is almost impossible to forget.

Mathematicians have invented a way to write down juggling patterns as sets of numbers. They look at the order in which balls are tossed into the air and then caught. Each toss or catch happens on a particular beat, as if the juggler were keeping time to music with a definite rhythm.

In the three-ball cascade, for example, ball 1 is thrown at beat 0, again on beat 3, then on beat 6, and so on. Ball 2 follows the same pattern. It's thrown at beat 1, then thrown again on beats 4, 7, and so on. Ball 3 is thrown at times 2, 5, 8, and so on. Because each ball is tossed on every third beat, mathematicians give the cascade pattern the label 3.

By combining numbers according to a few rules, they can invent new patterns for jugglers to try. One of the new (but tricky) juggling patterns is 441. It requires three balls, and each ball travels high twice, then low once, and then the sequence repeats.

The math captures only the order in which balls are tossed. It ignores crowd-pleasing features such as the style of throws and catches and their location—behind your back or between your legs, for example. The types of objects also make a difference. Buhler likes juggling with clubs, which look like oversized bowling pins. You might also see fruit, rings, plates, axes, flaming torches, or chainsaws sailing through the air!

Muse, February 2000, p. 26.

Many of the roughly 3,000 members of the International Jugglers Association are mathematicians or computer scientists. Mathematician Joe Buhler started juggling years ago when he was in college. He did it just because it was fun. On a good day, he can keep as many as seven balls going in a pattern called a cascade.

The cascade pattern requires an odd number of balls. Each ball follows a looping path that resembles a figure 8 on its side. Once the balls are in motion, the juggler never holds more than one ball at a time. The world record for a sustained cascade is nine balls for 60 catches in a row.

With a little coaching and some practice, just about anyone can learn how to juggle, Buhler says. Once learned, juggling—like bicycle riding—is almost impossible to forget.

Mathematicians have invented a way to write down juggling patterns as sets of numbers. They look at the order in which balls are tossed into the air and then caught. Each toss or catch happens on a particular beat, as if the juggler were keeping time to music with a definite rhythm.

In the three-ball cascade, for example, ball 1 is thrown at beat 0, again on beat 3, then on beat 6, and so on. Ball 2 follows the same pattern. It's thrown at beat 1, then thrown again on beats 4, 7, and so on. Ball 3 is thrown at times 2, 5, 8, and so on. Because each ball is tossed on every third beat, mathematicians give the cascade pattern the label 3.

By combining numbers according to a few rules, they can invent new patterns for jugglers to try. One of the new (but tricky) juggling patterns is 441. It requires three balls, and each ball travels high twice, then low once, and then the sequence repeats.

The math captures only the order in which balls are tossed. It ignores crowd-pleasing features such as the style of throws and catches and their location—behind your back or between your legs, for example. The types of objects also make a difference. Buhler likes juggling with clubs, which look like oversized bowling pins. You might also see fruit, rings, plates, axes, flaming torches, or chainsaws sailing through the air!

Muse, February 2000, p. 26.

## March 29, 2007

### Nice Guys Finish First (Sometimes)

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 or take advantage of them.

Imagine you and a friend are prisoners. You're in separate cells. Your captors offer each of you the same deal. If you rat on your partner and he stays mum, you go free and he gets 10 years in prison. If you both stay silent, you both get 6 months. If you both rat, you both get 5 years.

You start thinking about the deal. If my partner rats and I stay mum, I'll do 10 years. If he rats and I rat, I'll do 5 years. If he stays mum and I stay mum, I'll do 6 months. If he stays mum and I rat, I'll go free. Hmmmm. No matter what he does, I'm actually better off betraying him.

If you follow this logic, each turns in the other. You both serve 5 years, which is far worse than the 6 months you would have served if you had trusted each other and said nothing.

The same thing can happen when two competing stores cut prices to lure customers or when two countries are in an arms race. Even though the two parties make the best possible choice from their own viewpoint, they end up worse off than they would if they had cooperated.

What about cooperation on the schoolyard? You'll often find yourself in the prisoner's dilemma, and since you learn which kids you can trust, you can figure out better strategies than always ratting. One is called tit for tat. You start off nice, then next time do what your opponent did the last time.

Even though it may sometimes pay to be nasty, over the long term, being nice actually works better. But there is no hard-and-fast rule that guarantees success every time. Strategies that work well in some situations can fail miserably in others. It's tough out there on the schoolyard.

Muse, January 2000, p. 20.

Imagine you and a friend are prisoners. You're in separate cells. Your captors offer each of you the same deal. If you rat on your partner and he stays mum, you go free and he gets 10 years in prison. If you both stay silent, you both get 6 months. If you both rat, you both get 5 years.

You start thinking about the deal. If my partner rats and I stay mum, I'll do 10 years. If he rats and I rat, I'll do 5 years. If he stays mum and I stay mum, I'll do 6 months. If he stays mum and I rat, I'll go free. Hmmmm. No matter what he does, I'm actually better off betraying him.

If you follow this logic, each turns in the other. You both serve 5 years, which is far worse than the 6 months you would have served if you had trusted each other and said nothing.

The same thing can happen when two competing stores cut prices to lure customers or when two countries are in an arms race. Even though the two parties make the best possible choice from their own viewpoint, they end up worse off than they would if they had cooperated.

What about cooperation on the schoolyard? You'll often find yourself in the prisoner's dilemma, and since you learn which kids you can trust, you can figure out better strategies than always ratting. One is called tit for tat. You start off nice, then next time do what your opponent did the last time.

Even though it may sometimes pay to be nasty, over the long term, being nice actually works better. But there is no hard-and-fast rule that guarantees success every time. Strategies that work well in some situations can fail miserably in others. It's tough out there on the schoolyard.

Muse, January 2000, p. 20.

## March 28, 2007

### Glitter Trap

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 touch one another and form a triangle. Position the fourth ball on top of the other three to create a pyramid. In a darkened room, shine the flashlight through the opening in between the ornaments and into the center of the pile. You'll see an intricate pattern of light and dark patches among the balls.

University of Maryland student David Sweet has taken this procedure one step further. He's photographed the patterns that appear when the four-ball pyramid is lit from below and has blue and red poster boards placed outside two of the faces. The reflections of the colors bounce back and forth among the mirror-like balls and create striking patterns of blue, red, white, and black.

It turns out that the boundary between the colored patches is very complicated. If you magnify a part of the boundary, you see a patchy pattern that looks like the original, unmagnified image. Zooming in further, you see even more of the same patchy pattern. A pattern that looks similar at different magnifications is known as a fractal.

But this "glitter trap" is more than a toy for creating colorful patterns. Among other things, the behavior of light in a stack of mirrored balls can help mathematicians and physicists picture how electrons wander through the materials used to make transistors, computer chips, and other electronic devices.

Muse, December 1999, p. 37.

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 touch one another and form a triangle. Position the fourth ball on top of the other three to create a pyramid. In a darkened room, shine the flashlight through the opening in between the ornaments and into the center of the pile. You'll see an intricate pattern of light and dark patches among the balls.

University of Maryland student David Sweet has taken this procedure one step further. He's photographed the patterns that appear when the four-ball pyramid is lit from below and has blue and red poster boards placed outside two of the faces. The reflections of the colors bounce back and forth among the mirror-like balls and create striking patterns of blue, red, white, and black.

It turns out that the boundary between the colored patches is very complicated. If you magnify a part of the boundary, you see a patchy pattern that looks like the original, unmagnified image. Zooming in further, you see even more of the same patchy pattern. A pattern that looks similar at different magnifications is known as a fractal.

But this "glitter trap" is more than a toy for creating colorful patterns. Among other things, the behavior of light in a stack of mirrored balls can help mathematicians and physicists picture how electrons wander through the materials used to make transistors, computer chips, and other electronic devices.

Muse, December 1999, p. 37.

## March 27, 2007

### Nature's Numbers

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 petals.

Five also shows up in arrangements of seeds. Cut an apple in half across its core (rather than the usual way down the core from the stem), and you'll see the seeds arranged in a beautiful five-pointed star. What numbers do cucumbers, tomatoes, pears, and lemons feature?

Pineapples have eight rows of scales, seen as roughly diamond-shaped markings, sloping in one direction and 13 sloping in the other. Pine cones show the same sort of feature.

The head of a sunflower highlights other numbers. In a perfect head, the tiny flowers, or florets, that will become seeds are arranged in two spirals, one winding clockwise and the other counterclockwise. Depending on the species of sunflower, you might find 34 and 55, 55 and 89, or even 89 and 144 florets along a spiral.

Similarly, floret spirals at the center of certain types of daisies feature the numbers 21 and 34. You can look for similar patterns on brocolli or cauliflower.

Take a look at the numbers that nature seems to like (at least in plants): 3, 5, 8, 13, 21, 34, 55, 89, and 144. Can you find a pattern?

Here's a clue: Start with 1 + 1 = 2, and then add the two numbers on each side of the equal sign. Keep on doing this with each new equation that you get.

1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21, 13 + 21 = 34, and so on.

The sums you get are all members of a famous sequence of numbers named for the mathematician Leonardo of Pisa, also known as Fibonacci, who studied them about 800 years ago. Scientists have long wondered why these number come up in plants. The answer may have something to do with the way plants grow, especially the way petals or buds space themselves to gather the most sunlight and nutrients.

Wherever you look, nature certainly has a way with numbers.

Muse, November 1999, p. 25.

You can learn more about nature and Fibonacci numbers at http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html.

Photo credits:

I. Peterson

Five also shows up in arrangements of seeds. Cut an apple in half across its core (rather than the usual way down the core from the stem), and you'll see the seeds arranged in a beautiful five-pointed star. What numbers do cucumbers, tomatoes, pears, and lemons feature?

Pineapples have eight rows of scales, seen as roughly diamond-shaped markings, sloping in one direction and 13 sloping in the other. Pine cones show the same sort of feature.

The head of a sunflower highlights other numbers. In a perfect head, the tiny flowers, or florets, that will become seeds are arranged in two spirals, one winding clockwise and the other counterclockwise. Depending on the species of sunflower, you might find 34 and 55, 55 and 89, or even 89 and 144 florets along a spiral.

Similarly, floret spirals at the center of certain types of daisies feature the numbers 21 and 34. You can look for similar patterns on brocolli or cauliflower.

Take a look at the numbers that nature seems to like (at least in plants): 3, 5, 8, 13, 21, 34, 55, 89, and 144. Can you find a pattern?

Here's a clue: Start with 1 + 1 = 2, and then add the two numbers on each side of the equal sign. Keep on doing this with each new equation that you get.

1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21, 13 + 21 = 34, and so on.

The sums you get are all members of a famous sequence of numbers named for the mathematician Leonardo of Pisa, also known as Fibonacci, who studied them about 800 years ago. Scientists have long wondered why these number come up in plants. The answer may have something to do with the way plants grow, especially the way petals or buds space themselves to gather the most sunlight and nutrients.

Wherever you look, nature certainly has a way with numbers.

Muse, November 1999, p. 25.

You can learn more about nature and Fibonacci numbers at http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html.

Photo credits:

I. Peterson

## March 26, 2007

### How to Lace Like an Ace

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 kid.

Different lacing patterns require different lengths of lace. You might wonder which one of the three common lacing patterns requires the shortest laces. Notice that, in all three cases, the lace passes through each hole (or eyelet) just once, alternately crossing from one side of the shoe to the other.

Here's an experiment you can try. Using the same lace and shoe each time, follow the lacing patterns shown in the diagrams. Measure the total length of shoelace hanging loose above the top eyelets in each pattern. Which lacing saves you the most lace?

Mathematicians have proved that, if your shoe has four or more pairs of eyelets, the American style requires the shortest laces, followed by the European, then the shoe-store styles. If your shoe has only three pairs of eyelets, the American lacing remains shortest, but the European and shoe-store lacings are of equal length. Amazingly, the American style also wins when the eyelets are irregularly spaced instead of being neatly arranged in two neat rows.

Shorter lacings are possible if the lace doesn't have to pass alternately through the holes on the left and right side of the shoe. Here are some alternative lacings you could try. Black means the laces are on top; red means the laces are underneath.

The first two work only if your shoes have an even number of eyelet pairs.

Watch out, though. You might find that by saving shoelace length you end up with shoes that slip off your feet more easily or laces that break more often.

Muse, October 1999, p. 33.

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 kid.

Different lacing patterns require different lengths of lace. You might wonder which one of the three common lacing patterns requires the shortest laces. Notice that, in all three cases, the lace passes through each hole (or eyelet) just once, alternately crossing from one side of the shoe to the other.

Here's an experiment you can try. Using the same lace and shoe each time, follow the lacing patterns shown in the diagrams. Measure the total length of shoelace hanging loose above the top eyelets in each pattern. Which lacing saves you the most lace?

Mathematicians have proved that, if your shoe has four or more pairs of eyelets, the American style requires the shortest laces, followed by the European, then the shoe-store styles. If your shoe has only three pairs of eyelets, the American lacing remains shortest, but the European and shoe-store lacings are of equal length. Amazingly, the American style also wins when the eyelets are irregularly spaced instead of being neatly arranged in two neat rows.

Shorter lacings are possible if the lace doesn't have to pass alternately through the holes on the left and right side of the shoe. Here are some alternative lacings you could try. Black means the laces are on top; red means the laces are underneath.

The first two work only if your shoes have an even number of eyelet pairs.

Watch out, though. You might find that by saving shoelace length you end up with shoes that slip off your feet more easily or laces that break more often.

Muse, October 1999, p. 33.

## March 25, 2007

### Food Counts

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 yellow, 9 green, 7 orange, and 2 blue. You can see that packages don't necessarily contain the same number of candies. And the number of each color can be different, too. At the same time, you can see hints of a pattern. For example, there are fewer blue candies than red ones. To investigate further, you can open a few more packages, or, better yet, team up with your friends to check out even more. You can then add together the counts for each color and calculate what percentage of the total number of candies each color represents.

Mars, the maker of M&M's, says that it produces the colored candies in the following proportions: 30 percent brown, 20 percent red, 20 percent yellow, 10 percent green, 10 percent orange, and 10 percent blue. The different colors are then all mixed together before packaging. In a perfect bag of 50, you'd have 15 brown, 10 red, 10 yellow, 5 green, 5 orange, and 5 blue.

How do your counts compare with the official figures?

If you were blindfolded and picked about 50 candies out of a huge vat of mixed M&M's, you'd probably get different results each time. On the other hand, if you were to check all the colored M&M's in the vat, you would get figures that exactly matched what went into the vat in the first place. In general, the more samples you took from the vat, the closer you would get to those original numbers.

Once in while, you may find that your counts are off even when you have checked a lot of packages. This could mean one of two things: either the vat wasn't mixed very well, or the company changed the color proportions.

So what do you do with all the candies once you've counted them? You can probably figure out the answer to that question pretty easily. Sampling M&M's is a tasty case of having your data and eating it, too.

Muse, September 1999, p. 34.

A second package turns out to hold 55 candies: 12 brown, 12 red, 13 yellow, 9 green, 7 orange, and 2 blue. You can see that packages don't necessarily contain the same number of candies. And the number of each color can be different, too. At the same time, you can see hints of a pattern. For example, there are fewer blue candies than red ones. To investigate further, you can open a few more packages, or, better yet, team up with your friends to check out even more. You can then add together the counts for each color and calculate what percentage of the total number of candies each color represents.

Mars, the maker of M&M's, says that it produces the colored candies in the following proportions: 30 percent brown, 20 percent red, 20 percent yellow, 10 percent green, 10 percent orange, and 10 percent blue. The different colors are then all mixed together before packaging. In a perfect bag of 50, you'd have 15 brown, 10 red, 10 yellow, 5 green, 5 orange, and 5 blue.

How do your counts compare with the official figures?

If you were blindfolded and picked about 50 candies out of a huge vat of mixed M&M's, you'd probably get different results each time. On the other hand, if you were to check all the colored M&M's in the vat, you would get figures that exactly matched what went into the vat in the first place. In general, the more samples you took from the vat, the closer you would get to those original numbers.

Once in while, you may find that your counts are off even when you have checked a lot of packages. This could mean one of two things: either the vat wasn't mixed very well, or the company changed the color proportions.

So what do you do with all the candies once you've counted them? You can probably figure out the answer to that question pretty easily. Sampling M&M's is a tasty case of having your data and eating it, too.

Muse, September 1999, p. 34.

## March 24, 2007

### Covering Up

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 the way around.

In contrast, an oval is longer than it is wide. You can always find a way to slip an oval lid through a hole of the same shape. That's also true of a square or a six-sided, or hexagonal, cover.

Amazingly, the circle isn't the only shape that would work safely as a manhole cover. Another possibility is the Reuleaux triangle, named after engineer Franz Reuleaux, who was a teacher in Berlin, Germany, more than a hundred years ago. An example of a Reuleaux triangle can be found in your medicine cabinet. If you turn a bottle of NyQuil or Pepto-Bismol upside down, the shape you see looks like a Reuleaux triangle.

One way to draw a Reuleaux triangle is to start with an equilateral triangle, which has three sides of equal length. Place the pointed end of a pair of compasses at one corner of the triangle and stretch the arms until the pencil reaches another corner. Then draw an arc between two corners of the triangle. Draw two more arcs centered on the triangle's other corners.

This "curved triangle," as Reuleaux called it, has a constant width—just like a circle. It would certainly work as a manhole cover.

In fact, you can make a manhole cover out of any shape with an odd number of sides. Beginning with a five-sided shape called a pentagon, for example, you can construct a rounded pentagonal shape that has a constant width.

Imagine walking down the street and finding differently shaped manhole covers on every block!

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 the way around.

In contrast, an oval is longer than it is wide. You can always find a way to slip an oval lid through a hole of the same shape. That's also true of a square or a six-sided, or hexagonal, cover.

Amazingly, the circle isn't the only shape that would work safely as a manhole cover. Another possibility is the Reuleaux triangle, named after engineer Franz Reuleaux, who was a teacher in Berlin, Germany, more than a hundred years ago. An example of a Reuleaux triangle can be found in your medicine cabinet. If you turn a bottle of NyQuil or Pepto-Bismol upside down, the shape you see looks like a Reuleaux triangle.

One way to draw a Reuleaux triangle is to start with an equilateral triangle, which has three sides of equal length. Place the pointed end of a pair of compasses at one corner of the triangle and stretch the arms until the pencil reaches another corner. Then draw an arc between two corners of the triangle. Draw two more arcs centered on the triangle's other corners.

This "curved triangle," as Reuleaux called it, has a constant width—just like a circle. It would certainly work as a manhole cover.

In fact, you can make a manhole cover out of any shape with an odd number of sides. Beginning with a five-sided shape called a pentagon, for example, you can construct a rounded pentagonal shape that has a constant width.

Imagine walking down the street and finding differently shaped manhole covers on every block!

*Muse*, July/August 1999, p. 36.## March 23, 2007

### Fair Shares

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 give a result that appears fair to both people.

This divide-and-choose method of settling arguments over the division of cake, goods, or land goes back thousands of years, to biblical times. Mathematicians became interested in the problem of fair division about 60 years ago when they began to wonder what to do when three people want to divide a cake fairly. Getting a fair result for three people turns out to be surprisingly complicated. Here's one way you might do it. Mathematicians call it the "moving knife" strategy.

Suppose that a knife floats above a rectangular cake. Starting at the cake's left end, it moves slowly toward the right. Three people are all told to shout "Stop!" as soon as the knife reaches a point that, in their opinion, is one third of the cake. The first one to shout gets the first piece, and the remaining two people divide the rest using the "I cut, you choose" scheme.

The moving-knife strategy will also work for more than three people. As the knife moves to the right, more and more people drop out with what they think is a fair share until only two people are left to divide the last morsel.

Mathematicians have figured out other methods of sharing things, many of which don't involve a moving knife. Such methods are now used to help handle disputes between people and have even been used among nations to determine offshore mining rights. It's all in the math of fair cake cutting!

Muse, May/June 1999, p. 28.

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 give a result that appears fair to both people.

This divide-and-choose method of settling arguments over the division of cake, goods, or land goes back thousands of years, to biblical times. Mathematicians became interested in the problem of fair division about 60 years ago when they began to wonder what to do when three people want to divide a cake fairly. Getting a fair result for three people turns out to be surprisingly complicated. Here's one way you might do it. Mathematicians call it the "moving knife" strategy.

Suppose that a knife floats above a rectangular cake. Starting at the cake's left end, it moves slowly toward the right. Three people are all told to shout "Stop!" as soon as the knife reaches a point that, in their opinion, is one third of the cake. The first one to shout gets the first piece, and the remaining two people divide the rest using the "I cut, you choose" scheme.

The moving-knife strategy will also work for more than three people. As the knife moves to the right, more and more people drop out with what they think is a fair share until only two people are left to divide the last morsel.

Mathematicians have figured out other methods of sharing things, many of which don't involve a moving knife. Such methods are now used to help handle disputes between people and have even been used among nations to determine offshore mining rights. It's all in the math of fair cake cutting!

Muse, May/June 1999, p. 28.

## March 22, 2007

### Lizard Game

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 her back. A fist means rock, spread fingers mean paper, or two fingers in a "V" mean scissors. On the count of three, the players reveal their hands. The following rules determine the winner: Scissors cut papter, paper wraps rock, and rock breaks scissors. If both players make the same gesture, the game is a tie.

The lizards play a similar game. Each type of lizard has its own strategy for mating with females. As in rock-paper-scissors, sometimes one strategy wins, sometimes another.

Strategy #1: Have a Lot of Territory

The Orange-Throated Lizard: These males establish large territories, with several females. The more females the more often they can mate.

Strategy #2: Guard Your Mate

The Blue-Throated Lizard: These males defend small territories holding just a few females. Because the territories are so small, they can guard their mates carefully.

Strategy #3: Be Sneaky

The Yellow-Striped-Throated Lizard: These males are sneaky and can mimic the markings and behavior of females.

So, orange-throated males are able to grab territory and females from blue-throated lizards when orange-throated lizards are rare.

But, blue-throated males can take over a population of yellow-striped-throated males when blue-throated lizards are rare.

Completing the cycle, yellow-striped-throated lizards can sneak into the orange-throated territories when yellow-striped-throated lizards are rare.

Biologists Barry Sinervo of the University of California at Santa Cruz and Curt M. Lively of Indiana University have studied how the populations of the three different types of lizards change from year to year. They found a six-year cycle, with each color sometimes dominating. When a population hits a low, that type of lizard has the most babies the next year, helping to keep the cycle going.

Stuck in an endless cycle, side-blotched lizards keep on playing their never-ending mathematical game of survival.

Muse, April 1999, p. 26-27.

Learn more about the amazing lizard game at bio.research.ucsc.edu/~barrylab/lizardland/game.html.

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 her back. A fist means rock, spread fingers mean paper, or two fingers in a "V" mean scissors. On the count of three, the players reveal their hands. The following rules determine the winner: Scissors cut papter, paper wraps rock, and rock breaks scissors. If both players make the same gesture, the game is a tie.

The lizards play a similar game. Each type of lizard has its own strategy for mating with females. As in rock-paper-scissors, sometimes one strategy wins, sometimes another.

Strategy #1: Have a Lot of Territory

The Orange-Throated Lizard: These males establish large territories, with several females. The more females the more often they can mate.

Strategy #2: Guard Your Mate

The Blue-Throated Lizard: These males defend small territories holding just a few females. Because the territories are so small, they can guard their mates carefully.

Strategy #3: Be Sneaky

The Yellow-Striped-Throated Lizard: These males are sneaky and can mimic the markings and behavior of females.

So, orange-throated males are able to grab territory and females from blue-throated lizards when orange-throated lizards are rare.

But, blue-throated males can take over a population of yellow-striped-throated males when blue-throated lizards are rare.

Completing the cycle, yellow-striped-throated lizards can sneak into the orange-throated territories when yellow-striped-throated lizards are rare.

Biologists Barry Sinervo of the University of California at Santa Cruz and Curt M. Lively of Indiana University have studied how the populations of the three different types of lizards change from year to year. They found a six-year cycle, with each color sometimes dominating. When a population hits a low, that type of lizard has the most babies the next year, helping to keep the cycle going.

Stuck in an endless cycle, side-blotched lizards keep on playing their never-ending mathematical game of survival.

Muse, April 1999, p. 26-27.

Learn more about the amazing lizard game at bio.research.ucsc.edu/~barrylab/lizardland/game.html.

## March 21, 2007

###
Knot Magic *Not* Magic

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 magicians like knots that can fool people, mathematicians are mostly interested in real knots, especially knots that can't be undone. A mathematician's knot is always in a piece of rope that has both ends stuck together. Without cutting the rope, there's no way you can get the knot out.

You can make a mathematician's knot by plugging an electrical extension cord into itself and making a single loop. That's called an unknot.

But if you cross and wind the cord around itself a few times before plugging it in, you'll likely end up with a true knot. How many different knots can you make? To tell one from another, you can start putting them into groups by counting how many times the cord crosses itself when you lay each knotted loop down on a table. The simplest possible knot, called a trefoil, has three crossings.

Mathematicians started to describe all the different kinds of knots more than 100 years ago. They now know there are exactly 1,701,936 knots with 16 or fewer crossings. (Imagine trying to draw pictures of all of them!) Along the way, however, they were fooled again and again. Sometimes two knots looked different, but were really the same. Other times, something that looked like a knot was really an unknot. To keep from getting fooled, mathematicians looked for formulas that would serve as shortcuts for telling a knot from an unknot and one knot from another. They're still looking for a single formular that covers all possible knots.

Most mathematicians study knots just because they enjoy it. But figuring out knots can help others understand more about how the world works. Biologists, for example, study knots so tiny you need an electron microscope to see them. Heaps of DNA strands sit like microscopic spaghetti inside plant and animal cells. DNA carries the code that tells each cell what to do.When scientists started to sort out the DNA strands found in cells, they discovered some unexpected loops and knots.

Mathematicians came to the rescue, showing how it was possible to snip a DNA strand in two, then rejoin several split strands to make each of the loops and knots the scientists had seen. That tells biologists a lot about the way cells manipulate and duplicate DNA. They end up with a better understanding of how drugs, viruses, and other things can alter DNA's elaborate tangles.

There's a lot more to knots than tying your shoelaces!

Image credits

Second from bottom and bottom: Knotted strands of DNA. Nicholas R. Cozzarelli, University of California, Berkeley, and Steven A. Wasserman, University of Texas at Austin

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 magicians like knots that can fool people, mathematicians are mostly interested in real knots, especially knots that can't be undone. A mathematician's knot is always in a piece of rope that has both ends stuck together. Without cutting the rope, there's no way you can get the knot out.

You can make a mathematician's knot by plugging an electrical extension cord into itself and making a single loop. That's called an unknot.

But if you cross and wind the cord around itself a few times before plugging it in, you'll likely end up with a true knot. How many different knots can you make? To tell one from another, you can start putting them into groups by counting how many times the cord crosses itself when you lay each knotted loop down on a table. The simplest possible knot, called a trefoil, has three crossings.

Mathematicians started to describe all the different kinds of knots more than 100 years ago. They now know there are exactly 1,701,936 knots with 16 or fewer crossings. (Imagine trying to draw pictures of all of them!) Along the way, however, they were fooled again and again. Sometimes two knots looked different, but were really the same. Other times, something that looked like a knot was really an unknot. To keep from getting fooled, mathematicians looked for formulas that would serve as shortcuts for telling a knot from an unknot and one knot from another. They're still looking for a single formular that covers all possible knots.

Most mathematicians study knots just because they enjoy it. But figuring out knots can help others understand more about how the world works. Biologists, for example, study knots so tiny you need an electron microscope to see them. Heaps of DNA strands sit like microscopic spaghetti inside plant and animal cells. DNA carries the code that tells each cell what to do.When scientists started to sort out the DNA strands found in cells, they discovered some unexpected loops and knots.

Mathematicians came to the rescue, showing how it was possible to snip a DNA strand in two, then rejoin several split strands to make each of the loops and knots the scientists had seen. That tells biologists a lot about the way cells manipulate and duplicate DNA. They end up with a better understanding of how drugs, viruses, and other things can alter DNA's elaborate tangles.

There's a lot more to knots than tying your shoelaces!

*Muse*, March 1999, p. 26-27.Image credits

Second from bottom and bottom: Knotted strands of DNA. Nicholas R. Cozzarelli, University of California, Berkeley, and Steven A. Wasserman, University of Texas at Austin

## March 20, 2007

### Square Wheel

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 shape of the road over which the wheels roll.

A square wheel can roll smoothly if it travels over evenly spaced bumps of just the right shape. That special shape is called an inverted catenary. A catenary is the curve formed by a chain or rope hanging loosely between two supports.

Turn the curve upside down, and you get an inverted catenary—just like one of the bumps in Wagon's road. Make the road out of a whole bunch of those bumps all in a row, and you can take your square-wheeled bike for a quick spin. Click on the image below to see the square wheel roll on its bumpy road.

It turns out that for every possible wheel shape, there's a road that produces a smooth ride. And for every road, there's an appropriate wheel.

Wagon has used mathematics and computers to investigate many wheel-and-road combinations without actually building them. Wheels shaped like pentagons (five sides), hexagons (six sides), and octagons (eight sides) also roll smoothly over bumps made up of pieces of inverted catenaries.

In fact, as the wheel you use gets more and more sides, the catenary segments required for the road get shorter and flatter. Ultimately, the wheel has so many sides that it looks like a circle and its road is practically flat. It rides just like a normal wheel.

That's not all. You can find roads for other wheel shapes, including an ellipse (which looks like a flattened circle), a cardioid (like a heart with a rounded tip), and a rosette (like a flower with four petals). You can also start with a road profile and find the wheel shape that rolls smoothly across it.

There's one little problem. A weird-wheeled bike has to travel in a straight line along its special road. You can't go left or right!

Muse, February 1999, p. 29.

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 shape of the road over which the wheels roll.

A square wheel can roll smoothly if it travels over evenly spaced bumps of just the right shape. That special shape is called an inverted catenary. A catenary is the curve formed by a chain or rope hanging loosely between two supports.

Turn the curve upside down, and you get an inverted catenary—just like one of the bumps in Wagon's road. Make the road out of a whole bunch of those bumps all in a row, and you can take your square-wheeled bike for a quick spin. Click on the image below to see the square wheel roll on its bumpy road.

It turns out that for every possible wheel shape, there's a road that produces a smooth ride. And for every road, there's an appropriate wheel.

Wagon has used mathematics and computers to investigate many wheel-and-road combinations without actually building them. Wheels shaped like pentagons (five sides), hexagons (six sides), and octagons (eight sides) also roll smoothly over bumps made up of pieces of inverted catenaries.

In fact, as the wheel you use gets more and more sides, the catenary segments required for the road get shorter and flatter. Ultimately, the wheel has so many sides that it looks like a circle and its road is practically flat. It rides just like a normal wheel.

That's not all. You can find roads for other wheel shapes, including an ellipse (which looks like a flattened circle), a cardioid (like a heart with a rounded tip), and a rosette (like a flower with four petals). You can also start with a road profile and find the wheel shape that rolls smoothly across it.

There's one little problem. A weird-wheeled bike has to travel in a straight line along its special road. You can't go left or right!

Muse, February 1999, p. 29.

## March 18, 2007

### Chasing Arrows

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 continuous band. What's unusual about the resulting shape?

You can make the same shape from a long strip of paper. Just twist one end halfway around before gluing or taping the strip's two ends together.

Although the resulting band looks like it has two sides, there's really only one connected surface. Try drawing a line down its center until you return to your starting point. You won't be able to find a "side" that has no line. Cut the strip along the line you have just drawn. You end up with a new band twice as long as the original. It has two twists—and two sides!

The one-sided object is called a Möbius strip, named for the German astronomer and mathematician August Ferdinand Möbius, who discovered it in 1858. When you lay it down and flatten it into a triangle, you end up with a shape that has the same geometry as the recycling symbol.

Möbius strips even have practical value. Take a look at the belt that drives a car's radiator fan. Ordinarily, friction wears the belt out more quickly on the inside than the outside. But if a belt were made with a half twist, like the Möbius strip, it would have only one side and wears more evenly and slowly.

If you examine printed recycling symbols, you'll sometimes see a version that doesn't look like a standard Möbius strip. How could that happen?

One possibility is that someone drew just one bent, twisted arrow, made two copies of it, and put the three arrows in a triangle pattern.

In this case, the chasing arrows form a band that includes three half twists instead of just one. If you were to lay a string along its edge until the ends met and pulled the string tight, you would end up with a knot in the string. If you did this with a standard Möbius strip, you wouldn't get a knot.

You never know what sort of cool math you'll find in a trash heap or anywhere else around you.

Muse, January 1999, p. 27-28.

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 continuous band. What's unusual about the resulting shape?

You can make the same shape from a long strip of paper. Just twist one end halfway around before gluing or taping the strip's two ends together.

Although the resulting band looks like it has two sides, there's really only one connected surface. Try drawing a line down its center until you return to your starting point. You won't be able to find a "side" that has no line. Cut the strip along the line you have just drawn. You end up with a new band twice as long as the original. It has two twists—and two sides!

The one-sided object is called a Möbius strip, named for the German astronomer and mathematician August Ferdinand Möbius, who discovered it in 1858. When you lay it down and flatten it into a triangle, you end up with a shape that has the same geometry as the recycling symbol.

Möbius strips even have practical value. Take a look at the belt that drives a car's radiator fan. Ordinarily, friction wears the belt out more quickly on the inside than the outside. But if a belt were made with a half twist, like the Möbius strip, it would have only one side and wears more evenly and slowly.

If you examine printed recycling symbols, you'll sometimes see a version that doesn't look like a standard Möbius strip. How could that happen?

One possibility is that someone drew just one bent, twisted arrow, made two copies of it, and put the three arrows in a triangle pattern.

In this case, the chasing arrows form a band that includes three half twists instead of just one. If you were to lay a string along its edge until the ends met and pulled the string tight, you would end up with a knot in the string. If you did this with a standard Möbius strip, you wouldn't get a knot.

You never know what sort of cool math you'll find in a trash heap or anywhere else around you.

Muse, January 1999, p. 27-28.

## March 16, 2007

### Arithmagic

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 that anyone—young or old—can learn to do the same feats of mental arithmetic. You, too, can look like a genius without really trying, he insists. All it takes is a memory for numbers and the trick of calculating from left to right, opposite to the way you were probably taught.

Suppose you want to multiply 27 by 8. Starting from the left, you would multiply 20 by 8 to get 160. Then, 7 times 8 equals 56, which is added to 160 to give 216.

How would you multiply 378 by 7? Multiply 300 by 7 to get 2,100. In the next step, 70 times 7 equals 490, which is added to 2,100 to give 2,590. Finally, 8 times 7 equals 56, which is added to 2,590 to give the answer 2,646.

One advantage of using the left-to-right method is that you can start saying your answer while you're still calculating it, Benjamin says. That's very handy for a mathemagician intent on impressing his audience.

Squaring a number means multiplying it by itself. Here's a quick way to square a two-digit number, such as 37. Pick a nice round number that's close to 37, such as 40. The number 40 is 3 more than 37. Calculate the number that is 3 less than 37, which is 34. Use left-to-right multiplication to calculate 34 times 40, which is 1,360. Then add the difference (3) multiplied by itself (9) to get the final answer: 1,369.

The trick is to choose the difference so that the multiplication is easy. For example, to square 59, choose a difference of 1. Go up to 60 and down to 58. Multiply 60 times 58 to get 3,480. Multiply 1 by itself to get 1, and add that to 3,480 to get the answer 3,481.

With practice, you can square two-digit numbers in your head faster than you can do it using a calculator. Benjamin uses similar tricks to square three-digit, four-digit, and even five-digit numbers. Imagine multiplying 79,635 by 79,635 in your head.

There are lots of tricks you can use for quick mental arithmetic. Can you find any other magical shortcuts?

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 that anyone—young or old—can learn to do the same feats of mental arithmetic. You, too, can look like a genius without really trying, he insists. All it takes is a memory for numbers and the trick of calculating from left to right, opposite to the way you were probably taught.

Suppose you want to multiply 27 by 8. Starting from the left, you would multiply 20 by 8 to get 160. Then, 7 times 8 equals 56, which is added to 160 to give 216.

How would you multiply 378 by 7? Multiply 300 by 7 to get 2,100. In the next step, 70 times 7 equals 490, which is added to 2,100 to give 2,590. Finally, 8 times 7 equals 56, which is added to 2,590 to give the answer 2,646.

One advantage of using the left-to-right method is that you can start saying your answer while you're still calculating it, Benjamin says. That's very handy for a mathemagician intent on impressing his audience.

Squaring a number means multiplying it by itself. Here's a quick way to square a two-digit number, such as 37. Pick a nice round number that's close to 37, such as 40. The number 40 is 3 more than 37. Calculate the number that is 3 less than 37, which is 34. Use left-to-right multiplication to calculate 34 times 40, which is 1,360. Then add the difference (3) multiplied by itself (9) to get the final answer: 1,369.

The trick is to choose the difference so that the multiplication is easy. For example, to square 59, choose a difference of 1. Go up to 60 and down to 58. Multiply 60 times 58 to get 3,480. Multiply 1 by itself to get 1, and add that to 3,480 to get the answer 3,481.

With practice, you can square two-digit numbers in your head faster than you can do it using a calculator. Benjamin uses similar tricks to square three-digit, four-digit, and even five-digit numbers. Imagine multiplying 79,635 by 79,635 in your head.

There are lots of tricks you can use for quick mental arithmetic. Can you find any other magical shortcuts?

## March 10, 2007

### Picture Game

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 computers aren't very good at figuring out images, so they can't automatically create their own labels or find exactly what you're looking for.

You, on the other hand, have no trouble telling the difference between a tarantula and a sports car, even if the image is bad. So, what's the solution? Hire some people to start labeling images? That would cost too much, and they'd probably get bored pretty quickly. And even so, there'd be billions of unlabeled images.

Computer scientist Luis von Ahn of Carnegie Mellon University has come up with a great solution—and a playful one at that. He's turned the labeling of images into an Internet game, which he calls the ESP Game. To play von Ahn's game, you sign in at his Web site and get paired with another visitor. The two of you see the same image. Each of you then types in a word or phrase that describes the image. If the words match, both of you earn points. If the words don't agree, you keep on trying or switch to a new image. Your score is based on how many matches you get in 150 seconds.

Amazingly, von Ahn has found that people will spend hours playing this game, just to rack up points. No prizes. He collects the "winning" words or phrases, which then become keywords for the images. Von Ahn estimates that, given the popularity of his ESP Game in some tests, he could obtain labels for most of the Web's images in just a few weeks. Last fall, Google adopted von Ahn's idea and created its own label-making game.

Millions of people around the world spend hours playing computer solitaire every day, von Ahn says. So, why not harness all this game-playing energy and do something computers have a tough time doing on their own? Now, von Ahn is busy inventing new games to locate objects in images, write paragraphs to describe scenes, and collect commensense knowledge.

Who knows? The next game you play on the Internet may be useful as well as fun. And someday, you just might use a game like this to label all those digital photos you've stored and forgotten about on your computer.

Muse, March 2007, p. 37.

ESP Game: http://www.espgame.org/

Google Image Labeler: http://images.google.com/imagelabeler/

Image credits

Top: Lynx spider. Photo by John R. Nickles/U.S. Fish & Wildlife Service

Bottom: The ESP Game. Courtesy of Luis von Ahn

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 computers aren't very good at figuring out images, so they can't automatically create their own labels or find exactly what you're looking for.

You, on the other hand, have no trouble telling the difference between a tarantula and a sports car, even if the image is bad. So, what's the solution? Hire some people to start labeling images? That would cost too much, and they'd probably get bored pretty quickly. And even so, there'd be billions of unlabeled images.

Computer scientist Luis von Ahn of Carnegie Mellon University has come up with a great solution—and a playful one at that. He's turned the labeling of images into an Internet game, which he calls the ESP Game. To play von Ahn's game, you sign in at his Web site and get paired with another visitor. The two of you see the same image. Each of you then types in a word or phrase that describes the image. If the words match, both of you earn points. If the words don't agree, you keep on trying or switch to a new image. Your score is based on how many matches you get in 150 seconds.

Amazingly, von Ahn has found that people will spend hours playing this game, just to rack up points. No prizes. He collects the "winning" words or phrases, which then become keywords for the images. Von Ahn estimates that, given the popularity of his ESP Game in some tests, he could obtain labels for most of the Web's images in just a few weeks. Last fall, Google adopted von Ahn's idea and created its own label-making game.

Millions of people around the world spend hours playing computer solitaire every day, von Ahn says. So, why not harness all this game-playing energy and do something computers have a tough time doing on their own? Now, von Ahn is busy inventing new games to locate objects in images, write paragraphs to describe scenes, and collect commensense knowledge.

Who knows? The next game you play on the Internet may be useful as well as fun. And someday, you just might use a game like this to label all those digital photos you've stored and forgotten about on your computer.

Muse, March 2007, p. 37.

ESP Game: http://www.espgame.org/

Google Image Labeler: http://images.google.com/imagelabeler/

Image credits

Top: Lynx spider. Photo by John R. Nickles/U.S. Fish & Wildlife Service

Bottom: The ESP Game. Courtesy of Luis von Ahn

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