**Paper Folding**

(This 805th *Buffalo Sunday News* column was first published on
September 3, 2006.)

Doubling has a way of getting out of hand.

Perhaps the best illustration of this is the
story about the inventor of chess. Called before his appreciative Vizier, he
was asked what reward he sought. After thinking for a moment, he said,
"Just some wheat, sire. If you would have your minions place one grain of
wheat on the first square of a chessboard and double that amount on each
succeeding square, I will be completely satisfied."

"Let it be done," replied the Vizier.

But it could not be done. The doubling series, 1
+ 2 + 4 + 8 +... gets large rather fast and there are 64 squares on a
chessboard. By the end of the second row, there would already be 65,535 grains.
Indeed, the number of grains of wheat that would have fulfilled the Vizier's
order is 18,446,744,073,709,551,615. For comparison, this country's
out-of-control national debt today is about $8.5 trillion. The chess reward is
over 20,000 times as many grains of wheat than cents in that national debt. One
mathematician estimates that the chess inventor's reward would have filled
about five billion 15'x30'x6' swimming pools.

We are not told the Vizier's response when he
learned of this.

A related problem is the choice between $1
million and a cent the first day of the month with twice as much each following
day. Take the second choice. Even in February you would come out ahead with
$2,684,354.55. Wait until October to take the bet and you would get close to
eight times as much. (Be sure, however, to demand that payment not be made in
pennies.)

But halving works in the opposite direction. You
get very small, very fast. I suggest an experiment to show just how fast. Take
a full two-page sheet of this newspaper -- not the page on which this column is
written, of course; that should be preserved -- and see how many times you can
fold it in half always in the same direction. (The best way to do this is to
fold over the long way, using the first crease that is already there.)

But before you start, guess how many times you
can fold the paper this way.

Most people are shocked at how few folds they
can perform following this procedure. In fact, there have been many claims that
seven or eight folds is the maximum anyone can do.

Thus it may have been as a kind of joke that a
math teacher challenged his Pomona, California eleventh grade honors class to
fold some material in half this way a dozen times. He promised to assign an A
to anyone who could do this.

Britney Gallivan, "loving challenges more
than math", responded. In the process she extended our knowledge.

She first did some research, finding many
accounts claiming that the size and thickness of the material being folded made
no difference.

After many trials, Britney decided that the
accepted wisdom was wrong. She was able to obtain a 4-inch by 4-inch square of
gold foil to test this. Gold foil is among the thinnest of materials, only 11
millionths of an inch thick. By painstaking effort, using artist's brushes,
tweezers and rulers, she was able to fold this square the required dozen times.

Britney
Gallivan: One Fold to Go

She
demonstrated her solution to her teacher who was impressed, but he weaseled on
his bet, telling Britney that the challenge was to do this with paper.

Britney,
instead of hiring a lawyer, set out once again. She realized that thinness was
only one aspect of the solution. The length of the paper was the other. Using a
complicated formula she was able to determine that, with the kind of thin
tissue used in lavatories, she would need a roll about 3622 feet long. She was
able to find a 4000-foot Jumbo roll.

She
and a number of helpers headed for the local mall, where she lay out the paper
and began folding. She accomplished her task. Her achievement has been well
received by mathematicians and is recorded, for example, as a new entry in an
Encyclopedia of Integer Sequences.

I
hope that Britney then got her A.

To
obtain Britney's published account and more information visit the website of the
Pomona
Historical Society.-- Gerry Rising