You've probably heard that it's impossible to fold a sheet of paper in half more than seven or eight times. Usually you're also told that it doesn't matter how big or thin the sheet is.

Try folding a sheet of notebook paper. You'll probably find that it is pretty tough to get beyond eight folds. However, just because people—even experts—say something's impossible doesn't mean it is. That's what high-school student Britney Gallivan discovered when she succeeded in folding a sheet in half an unheard-of 12 times. She had to solve the problem to get extra credit in one of her math classes.

Britney Gallivan and her folded paper sheet after the eleventh fold. Photo by James Gallivan.

Why is it hard to get past eight folds? Suppose you're just folding in one direction instead of turning the paper 90 degrees between folds. Each time you fold, the thickness of the folded wad doubles and its width is halved. If you start with a standard sheet of paper, after seven folds, the wad is thicker that it is wide, and it takes too much strength to fold it again.

Analyzing the problem this way, however, you might begin to wonder whether you could beat the limit by folding something very, very thin or something very, very wide.

At first, Britney tried thin. She spent hours trying to fold paper sheets, newspapers, and any other flat material that she could get her hands on. Paper didn't appear to work, so she decided to use gold foil—only 11 millionths of an inch thick. Working with soft artists' brushes, rulers, and tweezers, she managed to fold a 4-inch-by-4-inch square of gold foil in half 12 times without tearing the extremely delicate sheet.

But that wasn't good enough. Britney's teacher said the problem was to fold a sheet of paper—not gold foil—12 times.

Determined to solve the problem, Britney tried again. This time she decided to go for width. If she used paper that was the same thickness as regular paper, she calculated, she would need a roll that was nearly 4000 feet long (about three-quarters of a mile) to be able to fold it 12 times. She found special toilet paper that met these requirements and bought a roll for $85.

Equipped with her jumbo roll, Britney went to a shopping mall in her hometown of Pomona, California. She unrolled the paper and marked the halfway point. It took three people (Britney and her parents) 7 hours, mostly on hands and knees, to complete the folding.

"The problem was a lot of work, a lot of frustration, a lot of fun, and I learned a lot from it," Britney later wrote in a booklet describing her accomplishment. "The world was a great place when I made the twelfth fold."

You can order a copy of Britney's booklet at pomonahistorical.org/12times.htm (Historical Society of Pomona Valley).

Muse, July/August 2004, p. 33.

## MatheMUSEments

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

## June 4, 2007

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