The two researchers weren’t goofing off. With clever computer programming, Kunkle figured out that any Rubik’s Cube can be solved in 26 moves or fewer. The previous record was 27. And Schaeffer proved that if both opponents in a checkers game play flawlessly, the game will always end in a tie.

Playing games and puzzles is a great way to sharpen your problem-solving skills.

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Studying puzzles and games may sound like fun, but the work might also eventually help scientists solve real-world problems.

Cracking the cube

Each side of a Rubik’s Cube is divided into nine squares, like a tic-tac-toe board. When the puzzle is solved, all nine squares (called facelets) on each side are the same color as one another. So, there’s a red side, a green side, and so on. Hinges allow rows of facelets to rotate.

When the puzzle is solved, each side of a Rubik’s Cube contains squares, or facelets, of just one color.

TheCoffee/Wikipedia

A series of random rotations mixes up the colors. To solve the puzzle, you have to make the right series of twists to group the colors.

The facelets of a Rubik’s Cube can be arranged in about 43 quintillion (that’s 43 with 18 zeros after it) possible ways. By hand, it can take a long time to find a solution.

A computer can try every possible move and compare solutions to solve the problem much more quickly. But with so many potential arrangements (also called configurations), even the world’s fastest computer would need a ridiculously long time to solve the problem.

To save time, Kunkle and computer scientist Gene Cooperman of Northeastern University in Boston, Mass., looked for strategies to break the problem into smaller pieces.

First, they calculated how many steps would be required to solve the puzzle using only half-turns, which send a facelet to the opposite side of the cube. They excluded quarter-turns, in which a facelet ends up on the side of the cube right next to where it began.

Their study showed that only 600,000 possible configurations can be solved this way. Using a desktop computer, Kunkle discovered that all these arrangements could be solved in 13 moves or less.

Puzzle pieces

Next, the researchers wanted to calculate how many steps would be necessary to turn any other configuration into one of the special 600,000 presolved arrangements. That required a time-consuming search through 1.4 trillion configurations. To speed the process, Kunkle and Cooperman wrote a program for an extremely fast computer, called a supercomputer.

A mixed-up Rubik’s Cube can take many hours to solve, unless you have the brain of a supercomputer!

Matthew Fisher

It took the supercomputer 63 hours to do the calculations. Results showed that any configuration could be turned into one of the presolved, half-turn configurations in 16 moves or fewer. Remember that it took a maximum of 13 steps to solve one of these special configurations. In sum, the researchers concluded, any configuration could be solved in a maximum of 29 steps.

That result fell shy of the record 27 steps established a year ago by another researcher. Kunkle and Cooperman noticed, however, that only about 80 million configurations (far less than 1 percent of all possibilities) actually needed more than 26 steps to reach a solution. So, the pair focused on those few, hardest arrangements.

This time, instead of searching for ways to turn each tricky configuration into a special configuration, they searched through every possible way of solving each one.

The effort paid off: They set a new record of 26 steps. Researchers think the absolute minimum is just 20 moves, but they have yet to find a way to prove it.

The strategies that Kunkle and Cooperman used to solve the cube can be applied to other complicated problems, especially ones that require searching through lots of possibilities. Scheduling airplane flights to carry millions of people to a variety of destinations as quickly as possible is one example.

Checkerboard solutions

Solving the Rubik’s Cube was a major feat, but Jonathan Schaeffer of the University of Alberta in Edmonton, Canada, faced an even bigger challenge: winning at checkers.

On a traditional 8-square by 8-square checkerboard, each player starts with 12 pieces in his or her own back three rows. All moves are diagonal. During each turn, you slide one of your pieces a distance of one square toward your opponent’s side.

You can capture an enemy piece by jumping over it with one of yours into an open square. When one of your pieces reaches your opponent’s side, it earns the ability to move backward too. If you can remove all enemy pieces, you win.

At the beginning of a game of checkers, each player lines up his or her pieces on one side of the board. Players take turns moving a single piece one diagonal space at a time. (This is a special board with 100 squares and 4 rows of pieces per team instead

Michel32Nl/Wikipdia

No one had ever attempted to write a program to simulate all moves on a checkerboard. That might be because the pieces on a checkerboard can be arranged in more than 500 quintillion ways (that’s a 5 with 20 zeroes after it). Compared to a Rubik’s Cube, a checkerboard has 10 times as many possible configurations.

Like the Rubik’s researchers, Schaeffer and colleagues started with a smaller problem. They imagined two pieces left on the board at the end of a game. For every position that those two pieces could occupy, a computer program simulated every possible outcome.

A game of checkers gets more complicated with each move.

PartsnPieces/Wikipedia

The program went through the same process for 3 pieces, then 4, and so on, up to 10 pieces. At that point, there were 39 trillion possibilities for where the pieces might be.

Checkmate

Whenever Schaeffer added a piece to the board, the time needed for calculations was 10 times as long as the time needed for the previous step. The computer was not powerful enough to continue the process.

So Schaeffer started over from the beginning of a game. His program considered all possible moves and countermoves until only 10 pieces remained the board. Since he had already figured out every way the game could end once there were 10 or fewer pieces left, he was able to combine the two programs to simulate an entire game.

In spite of Schaeffer’s efforts to cut down time, the computers took 18 years to finish the problem. “I’m quite amazed that I had enough patience to stick with this,” Schaeffer says.

It took computers 18 years to come up with a solution for the game of checkers. Chess, shown here, is an even more complicated game.

iStockphoto.com

Like the methods Kunkle developed for the Rubik’s Cube, Schaeffer’s strategies can be applied to practical problems in scheduling and even in human biology. The work might also some day help a computer play a perfect game of chess, which is far more complicated than checkers.

Take it from Kunkle and Schaeffer: Playing games can lead to serious science.

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Word Find: Play This

All I could see was dirt and jagged gray rocks. There wasn’t a touch of green, much less homes or other buildings.

But I knew that I was looking at the remains of an ancient village in Peru—a village so sophisticated that it astounded the archaeologists who recently excavated part of the site. The scientists found that the village, called Buena Vista, contained artifacts that allowed its residents to track the seasons and predict the weather 4,000 years ago.

Today, Buena Vista looks like a pile of dirt and rocks. Archaeologists covered the area they’d dug out to discourage looters.

David Hauer

I visited Buena Vista during a recent trip to Peru. Since none of the ancient village is visible unless you know where to look, Hugo Ludeña, a Peruvian archaeologist working at the site, gave me a tour.

A fierce face

In one area, a metal trapdoor was fixed in the dirt. A worker opened the padlock to let me in. Sunlight poured through the opening, illuminating a mud staircase that led down to an underground chamber.

I walked down the stairs, pausing to allow my eyes to adjust to the dappled light.

I found myself staring at a huge face. It glared at me with scowling eyes. The head, flanked by two foxes, was at the center of a 20-foot-long mud sculpture.

This 20-foot-long mud sculpture, called “The Menacing Disk,” directly faces the setting sun at winter solstice.

Robert Benfer

The sculpture, which the archaeologists named the Menacing Disk, is 4,000 years old. It’s an amazing piece of ancient art, but it’s also more than that. Archaeologists who have studied the sculpture now know that it was part of a solar observatory. The people of Buena Vista used it to keep track of the seasons.

The clue that tipped them off was subtle. The sculpture is almost, but not quite, parallel with the wall behind it. One end of the sculpture is 3 inches farther from the wall than the other.

“That doesn’t happen by accident,” says archaeologist Robert Benfer of the University of Missouri–Columbia, the lead scientist at the site. In other buildings from that period, things such as walls and artworks were very carefully lined up with each other. The researchers figured that the sculpture had been placed in its odd position for a reason.

What are you looking at?

As they continued their research, the scientists discovered that the sculpture faced a 300-foot-long platform on the ridge of a mountain on the other side of the valley in which Buena Vista sits. The platform was covered in ash that had worked its way into the surrounding soil. To scientists, this suggested that fires, perhaps ceremonial ones, had once been set there.

The Buena Vista research team consulted an astronomer to find out whether the platform and the Menacing Disk had astronomical significance. They found out that it did. If someone stood in front of the disk and looked toward the platform on the shortest day of the year—the winter solstice—the setting sun would appear to sink into the platform.

The winter solstice was an important date for these ancient people. It marked the end of the harvest. Somewhat like Thanksgiving in the United States, it was a time of celebration.

The Temple of the Fox

The Menacing Disk wasn’t the only discovery of astronomical significance that archaeologists found at Buena Vista. Elsewhere at the site, atop a 33-foot-tall pyramid, they found a temple. In the entryway of the temple was a mural that showed a fox inside a llama. The researchers named this structure the Temple of the Fox.

Inside the temple, the archaeologists found bits of cotton and burned twigs, which they knew were the remains of ancient sacrifices. By testing the carbon in the cotton and twigs, they found that the temple was in use around 2200 BC.

“The Temple of the Fox is a place where people made payments to the earth, to make sure they would get a good crop,” Benfer says.

Inside the temple is a special chamber for offerings. The chamber’s doorway directly faces a large, carved rock on the horizon. The carving resembles a human face in profile. To someone who looked out of the chamber’s door on the summer solstice, which is the longest day of the year, the sun would appear to set behind the carved face. In Peru, that solstice is December 21.

That date was important to ancient farmers of this land. The area of Peru near Buena Vista receives almost no rainfall. Without rain, the farmers were entirely dependent on the river for water for their crops, which grew in the rich sediment laid down by yearly floods.

The farmers had to know when the floods were coming so that they would know when to plant their crops. December 21 was the date just before the floods occurred. By watching the sunset from the special chamber, they could tell when this date would arrive.

El Niño

However, the weather patterns aren’t always as predictable as the ancient farmers would have liked them to be. Every few years, a shift in winds and ocean currents in the Pacific causes a temporary change in the weather system. It’s called El Niño, and it can cause either terrible droughts or terrible floods.

El Niño has a powerful effect on animal life. Usually, sea animals stay near Peru’s coast all year long. But in an El Niño year, almost all the fishes, seals, and sea lions will have swum away from the coast by December 21. At Buena Vista, however, nothing would be out of the ordinary.

Benfer speculates that the temple priests, armed with the knowledge of the date from their astronomical observations, could have sent a runner down to the ocean on December 21 to check on the sea life. If the animals were gone, the priests would have known it was going to be a bad year, and the whole community could have planned for it.

The researchers believe that the entire community left the valley in bad years and went high into the mountains to plant their crops. Up there, fog provided moisture, and the river couldn’t reach their fields and houses, even if it flooded.

Burying the old

Around 1800 BC, the religion changed at Buena Vista. But the people didn’t want to destroy their temples. So they buried them under dirt and rocks, and then built new temples right on top.

Many times in the centuries after Buena Vista was no longer inhabited, looters raided the site, searching for treasure. They must have been disappointed. Despite their astronomical expertise, the ancient people of Buena Vista were mainly simple farmers. They hadn’t learned to make ceramics, much less to forge gold and silver, so they left few valuables behind for the looters to steal.

When Benfer and his team arrived in 2003, they started digging out the areas where the looters had been. They started here so that they could learn about the site without disturbing any areas that remained untouched.

Barely an inch below where the looters had stopped digging, the researchers discovered the roof of the Temple of the Fox.

A worker digs out a sculpture at the ancient city of Buena Vista.

Robert Benfer

Benfer’s team hated to damage the roof by digging into it. However, they knew if they didn’t, people from the villages nearby might break through it themselves in the hope of finding buried treasure. Therefore, the archaeologists carefully broke through the roof, examined the temple, and then painstakingly reburied it, protecting it for centuries to come.

That meant that when I visited it, all I could see of the Temple of the Fox was a slight indentation in the dirt and rocks where the scientists had done their work. I felt awed, knowing what that moonscape concealed.

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Word Find: Observatory

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Fifty feet below, Jerome Hillaire and Karina Quinteros of the Tambopata Research Center look up from a laptop to admire the bird as she cocks her head at them. The computer’s screen is showing live images of the macaw and her chicks, sent from a tiny camera that the researchers have tucked inside her nest. They hope that understanding how macaws live will help efforts to save the birds.

Macaws normally nest in cavities in very large, old trees. At Tambopata, researchers hang artificial nests from big trees for the birds to use.

David Hauer

Parrots in peril

Like many other kinds of parrots in the wild, macaws are in trouble. People sometimes kill them for their beautiful feathers, or capture the chicks and sell them as pets.

Many parrots, such as this macaw, are endangered.

© Donald Brightsmith, Texas A&M University

More important, the macaws are losing their homes. These birds usually nest in natural cavities in old trees. These large trees are valuable for lumber, so they’re often cut down, and the macaws are left with nowhere to lay their eggs.

Fake nests

So, researchers at Tambopata built artificial nests out of wood or pieces of plastic pipe. They hung the nests from trees and then watched to see whether the macaws would use them.

The scientists were pleased to see that the birds seem to approve of the fake nests. The macaws used them year after year. Since that discovery, people have begun building similar artificial nests in other areas where macaws are struggling to survive.

In addition to helping the birds, the fake nests make life easier for the researchers. They now have lots of subjects to observe near Tambopata. The squawking sounds of macaws fill the air around the research center.

“It’s a parrot laboratory, because there are lots of birds,” says Donald Brightsmith, Tambopata’s research director. “It’s a very good place to learn how parrots work.”

Learning more about macaws

Hillaire and Quinteros, for example, are trying to solve a puzzle about macaw chicks. Macaws often lay three or four eggs at a time, but only one or two of the hatchlings develop to maturity. The other chicks starve during infancy, even if the parents have plenty of food.

Hillaire and Quinteros want to know why.

They note how much time the mother macaw spends with her chicks, how often she feeds them, how often the chicks fight with each other, and how much food each chick gets.

“The third chick is going to die soon,” Hillaire says, as he studies another live image on the computer screen. The first-born chick fills most of the screen, but off in the corner, a scrawny chick trembles. “It’s kind of sad. You watch him day after day, and he fights and fights and fights, and he only gets a little food. He’s wearing himself out with all that fighting.”

This macaw chick is about a month old. The researchers climbed up the tree and removed him from his nest in order to weigh him. His parents swooped down to attack the researchers while they did that!

David Hauer

Cheeky chicos

Early in the research center’s history, scientists decided to try to save the third and fourth eggs and raise the chicks by hand. (The researchers call the young birds “chicos,” which means “kids” in Spanish.) Of the eggs they saved, 26 chicos grew into adult macaws. The birds now live free in the jungle. They’ve found mates, and most have raised chicks of their own.

But because the birds got used to humans, many like to come back and visit. They’ll fly into the center’s open-air dining room, march across the tables, and snatch food from people’s plates. They can be lured with food to stand on someone’s shoulder. They’ll also steal shiny things such as watches or jewelry.

Researchers hand-raised some of the chicks that wouldn’t have survived in their parents’ nests. These birds now live in the jungle, but like to come back for treats.

David Hauer

The researchers stopped hand-raising chicks from the third and fourth eggs a dozen years ago. They didn’t think it was a good idea to create partly tame parrots. What’s more, the macaw population in the area was thriving, even though some chicks were not surviving.

Macaws, which ordinarily mate for life, are often seen flying in pairs.

© Donald Brightsmith, Texas A&M University

Tasty clay

Bird researchers like Tambopata because the area has so many macaws. The birds have a different reason for liking the place. Tambopata has an enormous clay lick, which is a place with a special kind of clay-rich dirt that the birds like to eat. This clay lick, which forms a 1,500-foot-long cliff, is the largest one known in the world.

The researchers get up at 4 a.m. every day. They arrive at the clay lick before dawn to count how many birds show up. The smaller parrots, such as mealy parrots, are usually the first to fly around the lick, gabbling in high voices. Then other birds—blue-headed parrots, chestnut-fronted macaws, dusky-headed parakeets—join in. Eventually, a pair of great scarlet macaws will soar overhead, slowly flapping their enormous wings.

Finally, the birds pick a spot to land. They start scooping up dirt with their beaks. The birds use the clay lick most heavily during the breeding season. They feed clay to their chicks too.

Macaws and other parrots in Tambopata like to eat dirt. The dirt contains special, salty clay that may help the birds get rid of toxins in their diets.

© Donald Brightsmith, Texas A&M University

The researchers think that the birds eat the clay because it contains salt, which the birds need in their diets. They also think the clay protects the birds against toxins in the seeds and fruit they eat. And the birds seem to enjoy eating the clay. It probably tastes good to them, says Brightsmith.

Helping all parrots

The knowledge gathered at Tambopata is helping parrots around the world.

The artificial nests that the researchers have developed are being used to save macaws in immediate danger of extinction. These endangered birds include the great green macaw, which lives in Ecuador and Costa Rica. Brightsmith is working with the Brazilian government to develop a plan to protect Brazil’s wild macaws. And what Tambopata researchers have learned about the diets of wild parrots is even helping improve commercial bird foods such as those you might feed a pet parrot.

Brightsmith hopes that the center’s work will help ensure that macaws continue to flash scarlet streaks through the Amazon jungle for centuries to come.

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Word Find: Macaw

Mathematicians are particularly fond of sharing birthday cake. Not just any birthday cake, but one with lots of icing and various decorations, with nuts here and coconut there. Then they ask, if two people like different parts of the cake better, how can they divide the cake into two pieces so that they’re both satisfied with the piece that they each get?

You and your friend want to divide a cake into two pieces in such a way that each of you is happy with the piece that you get. How would you do it?

There’s an old solution known as “I cut, you choose.” You start by cutting the cake into two pieces that you like equally well. Then your friend picks the one that she prefers.

The two pieces don’t have to be the same size. If you particularly like nuts, for example, you might make the piece with fewer nuts bigger, so that you’d be happy no matter which piece your friend chose. You’d end up with either a smaller piece with lots of nuts or a larger piece with fewer nuts.

But Steven Brams of New York University doesn’t think that’s fair. When you’re done, you get a piece that you might think is worth half the value of the cake. But your friend might think that she got much more than half the value of the cake.

For example, suppose that your friend really likes coconut, and the bigger, less nutty piece has lots of coconut. Then she’ll think that she’s gotten a really great deal. She got not only more cake but also the best part!

Brams says that a division should be considered fair only if two people think they both got pieces of the same value. He’s worked out a new procedure for cake-cutting that makes this happen.

In dividing this cake, A marks the cut where you think the two pieces have equal value. B shows the division where your friend thinks the two pieces are equal. By a new method, you’d get the leftmost piece and your friend would get the rightmost piece, an

E. Roell

Here’s how it works. You and your friend would each tell your mom where you would divide the cake into two pieces. If the two of you happen to pick the same spot, she’d simply divide the cake at that spot. Both of you would be equally happy with your shares.

But suppose the two spots are different. If your spot were to the left of your friend’s spot, you’d get the piece to the left of your spot. Your friend would get the piece to the right of her spot. And there’d be a piece left over in the middle. Your mom would then split the middle section between you and your friend.

That way, you each get a piece that you value equally—plus a bonus!

It’s a neat idea, but is such a procedure practical? Would you use it?

“I don’t know if anyone other than me has actually brought a cake in and tried to divide it,” says James Tanton, a mathematics teacher at St. Mark’s School in Southborough Mass. Such schemes often don’t work in practice. “Human beings are too fuzzy,” he says. “They change their minds.”—J.J. Rehmeyer

Going Deeper:

Rehmeyer, Julie J. 2006. A fair slice: New method makes for equitable eating. Science News 170(Dec. 16):390. Available at http://www.sciencenews.org/articles/20061216/fob7.asp .

Peterson, I. 1999. Fair shares. Muse 3(May/June):28. Available at www.sciencenewsforkids.org/pages/puzzlezone/
muse/muse0599.asp.

______. 1996. Formulas for fairness. Science News 149(May 4):284-285. Available at http://www.sciencenews.org/pages/sn_arch/5_4_96/bob1.htm .

Researchers have now found that microbes living under the ocean floor appear to produce propane and another gas called ethane. These microbes chew up ancient organic material, such as leaves and twigs buried in the sand, and they generate the gases as waste products.

That’s a surprise. Scientists had thought that propane and ethane could be produced only in the same way that petroleum is—by great heat applied to ancient, buried material.

Kai-Uwe Hinrichs examines a sample taken from a cylinder of sediment drilled out of the ocean floor.

Ocean Drilling Program Leg 201 Science Party

A team led by Kai-Uwe Hinrichs of the University of Bremen in Germany went on a research ship equipped with an enormous drill that dug out cylinders of sand or rock thousands of feet long. When the researchers examined these cylinders, they found traces of ethane and propane locked in the sediment.

Normally, to generate these gases, Earth’s heat cooks organic material in sand for many thousands of years. This can happen only at spots above cracks in Earth’s crust, where heat can leak out from inside Earth, and where thick layers of sediment would act like a blanket.

But the samples that Hinrichs and his coworkers had looked at contained thin layers of sediment. Some cylinders had also been obtained from places far from any cracks in Earth’s crust. So where could the gases be coming from?

Scientists already knew that microbes could break down organic material to produce a related, simpler gas called methane. So, undersea microbes were the only thing that made sense.

“When you can’t come up with any geologic source, then biology is an obvious candidate,” Hinrichs says.

The finding may someday lead to practical applications. Propane is valuable as a fuel, and ethane is used to make plastics. Pulling propane and ethane out of sediment is too difficult to be practical. But if scientists can better understand how microbes create the gases, they might be able to use the microbes’ methods to make ethane and propane directly from organic material.—J. Rehmeyer

Going Deeper:

Rehmeyer, Julie. 2006. Gassy bugs: Microbes may produce propane under the sea. Science News 170(Sept. 30):213. Available at http://www.sciencenews.org/articles/20060930/fob4.asp .

You can learn more about propane at en.wikipedia.org/wiki/Propane and ethane at en.wikipedia.org/wiki/Ethane (Wikipedia).

Cutraro, Jennifer. 2006. Microbes at the gas pump. Science News for Kids (April 12). Available at http://www.sciencenewsforkids.org/articles/20060412/Feature1.asp .

Sohn, Emily. 2006. Plant gas. Science News for Kids (Jan. 18). Available at http://www.sciencenewsforkids.org/articles/20060118/Note2.asp .

______. 2004. Drilling deep for fuel. Science News for Kids (Sept. 29). Available at http://www.sciencenewsforkids.org/articles/20040929/Note3.asp .

ScienceFairZone
Harvesting Biogas from Manure
www.sciencenewsforkids.org/articles/
20050504/ScienceFairZone.asp

Archaeopteryx lived 150 million years ago and had teeth and claws like a dinosaur, but wings and feathers like a bird. Feathers also covered its back legs, and a new report argues that these feathered legs acted like small extra wings.

A fresh look at a fossilized Archaeopteryx has shown that this ancient bird had feathers on its legs that may have helped it fly. The inset shows how the bird may have looked with feathers.

Nick Longrich

Nick Longrich of the University of Calgary started wondering if Archaeopteryx might have used leg feathers for flying after researchers in China found a fossil of a ancient bird that had long flight feathers on both its wings and its legs.

Longrich studied an Archaeopteryx fossil that had been found in 1877. When it was first discovered, it had shown hind feathers. But when the fossil was prepared for display, the feathers had been stripped away to show the bones more clearly.

All was not lost, however. When researchers split a rock in two to reveal a fossil, one side has the bones and the other side has an imprint of the fossil. Usually, scientists look at the fossil, but Longrich decided to look at the imprint. It still showed the feathers.

Could the feathers have helped Archaeopteryx fly? Because the feathers were more like the ones that modern birds use for flying than the ones they use just to keep warm, Longrich concluded that Archaeopteryx could have used them for flying, too. He found that the feathered back legs would have allowed the creature to make tighter turns and fly more slowly that it could have without its feathered legs.

But some other scientists don’t think Archaeopteryx could have spread its legs out like wings. They suggest that the hind-leg feathers were like ones on eagles today, which just keep the birds warm and streamlined.—J. Rehmeyer

Going Deeper:

Rehmeyer, Julie. 2006. Flying with their legs: Hind feathers made primitive bird nimble. Science News 170(Sept. 23):197-198. Available at http://www.sciencenews.org/articles/20060923/fob6.asp .

You can learn more about Archaeopteryx at www.ucmp.berkeley.edu/diapsids/birds/archaeopteryx.html (University of California, Berkeley).

Ramsayer, Kate. 2004. Early birds ready to rumble. Science News for Kids (Oct. 27). Available at http://www.sciencenewsforkids.org/articles/20041027/Note3.asp .