These ballpark estimates can often be done without counting. That’s because humans are born with the ability to approximate, or closely guess, the number of items in a group. Researchers refer to this trait as a person’s “number sense.”

Scientists have discovered that this inborn sense of numbers may influence learning and achievement in the classroom. Studies with teenagers show that students who excel at estimating quantities also did well on standard math achievement tests, going as far back as kindergarten.

These results suggest a “strong and significant relationship” between a person’s inborn number sense and his or her ability to learn mathematics in school, says psychologist Justin Halberda of Johns Hopkins University in Baltimore.

Researchers already knew that humans have a natural grasp of numbers. The ability to make rough approximations can be found in infants as young as 4 months old, and even in some animals. This inborn numerical sense reaches back millions of years, researchers say, and has been used by humans and animals to help guide everyday behaviors such as hunting for food.

But sometimes an approximation just won’t do. Most mathematical calculations carried out in the classroom and in day-to-day transactions require an exact number. To succeed in formal mathematics requires verbal reasoning, not to mention hours of homework and training.

To see how a person’s inborn, or intuitive, number sense might be linked to mathematical performance in the classroom, Halberda and his colleagues ran some tests.

In a new study, 14-year-olds had a fraction of a second to identify the more numerous of two sets of colored dots, such as those in the images shown here.

Halberda

The scientists asked 64 14-year-olds to look at images of yellow and blue dots that flashed on a computer screen for a fraction of a second. Each image contained between 10 and 32 dots that varied in size.

Some images contained twice as many blue dots as yellow dots. In other images, however, the number of blue and yellow dots was nearly equal. For each image, the students were asked to estimate which color had more dots.

The scientists found a wide variation in how well students could pick the color with the most dots. Some students could correctly approximate images with nearly equal numbers of dots. But others found it difficult to make such estimates, even when the ratio, the number of one color of dots compared to the number of another color of dots, wasn’t as close.

The scientists then looked at the students’ math scores dating back to kindergarten. Children that performed best in the image test also scored the highest in standard math achievement tests.

The same finding held true at the other end of the spectrum. Students who didn’t score well on the image test tended to receive lower math scores, even after factors, such as IQ levels, were taken into account.

The study was the first to show a link between a person’s inborn number sense and his or her achievement in formal math training.

Does this connection mean that one cannot be good in math if they have a weak number sense? Or that having a strong number sense is a guarantee for good grades in math? The answers are not clear.

While scientists continue looking at the possible links between a person’s number sense and math achievement, one thing is certain: Doing lots of math homework will boost your chances of success.

Going Deeper:

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.

iStockphoto.com

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.

Going Deeper:

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Psychologists from the University of Nottingham in England recently tested kids from a variety of backgrounds to make sure that wealth or level of education didn’t get in the way of their results.

In the new study, kids faced approximate-arithmetic problems involving addition, such as this one.

Camilla K. Gilmore

The first study involved 20 5- and 6-year-olds from wealthy, well-educated families. The kids sat in front of computers that showed a series of three-part math problems. One problem, for example, showed a girl’s face in one box and a boy’s face in another box. Above the girl’s face was a bag labeled “21.” Words on the screen read, “Sarah has 21 candies.”

The next screen showed a bag labeled “30″ above the girl. Words read, “She gets 30 more.” Finally, a bag marked “34″ appeared above the boy. Words read, “John has 34 candies. Who has more?”

Nearly three-quarters of kindergartners got the answer right. If the kids had just guessed who had more candies, only half of them would have been correct.

In a second experiment, the scientists tested 37 kindergartners from poor and middle-class families. The kids had to answer questions in a hallway outside their public school classroom, meaning there were more distractions than in the first study. Still, almost two-thirds of these kids got the answers right.

Subtraction problems presented to kids in the study included this one.

Camilla K. Gilmore

In a final experiment, 27 kindergartners from wealthy backgrounds faced a subtraction problem and a comparison problem. Again the text was accompanied by boxes showing girls and boys. Subtraction questions looked like this: “Sarah has 64 candies. She gives 13 of them away. John has 34 candies. Who has more?”

Comparison questions asked things like: “Sarah has 51 candies. Paul has 64 cookies. John has 34 candies. Who has more candies, Sarah or John?”

Again, the young math whizzes came through. They correctly answered two-thirds of the subtraction problems and four-fifths of the comparison problems.

The results of these tests suggest that kids have a natural ability to estimate numbers. Scientists have already observed similar abilities in other animals.

Knowing their students have such math skills might help teachers better teach arithmetic.

“The teachers . . . were skeptical about our experiments,” says lead researcher Camilla K. Gilmore. But in the end, she adds, teachers were “surprised both by their students’ success and by their enjoyment of the tasks.”—Emily Sohn

Going Deeper:

Bower, Bruce. 2007. Take a number: Kids show math insights without instruction. Science News 171(June 2):341-342. Available at http://www.sciencenews.org/articles/20070602/fob6.asp .

Sohn, Emily. 2005. Wired for math. Science News for Kids (Dec. 7). Available at http://www.sciencenewsforkids.org/articles/20051207/Feature1.asp .

______. 2005. Monkeys count. Science News for Kids (July 13). Available at http://www.sciencenewsforkids.org/articles/20050713/Note3.asp .

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 .