Here is a true story of a ninth-grader. He was thinking about the game of Go which is played on a grid and was wondering how many intersection points there were on the board. So he wanted to know what was, but did not have a piece of paper handy. He wondered if he could figure out how much needed to be subtracted from to get . He mentally envisioned the following table:

He saw that the square of *n*+1 was obtained by adding 2*n*+1 to the
square of *n*. Thus so .

Later, when he wrote this down, he recognized that he had simply rediscovered
the formula , which he had seen before. But now he
began to think of something else: When going from to the
amount added was 9, which just happened to be a perfect square. So
, which means that (4,3,5) is a Pythagorean triple.
Might not other Pythagorean triples be found this way? They can for
integer values of *n* for which 2*n*+1 is a perfect square. Or,
working backwards, start with a perfect square , set it equal to
2*n*+1, solve for *n*, and see if *n* is an integer. A little
reflection convinced him that this works if and only if *m* is odd.
In this case, and you have the Pythagorean triple
(*m*,*n*,*n*+1).

This gave him a method of generating some Pythagorean triples:

He noticed that not all Pythagorean triples were generated this way;
for example, the triple (6,8,10) would be absent. But he realized
he could make more triples using similar formulas. For example, he
could start with . If 4*n*+4 happened to be a
perfect square , then he could solve for *n*, getting
and the triple (*m*,*n*,*n*+2). He realized that *n* would be an integer
if and only if *m* were even. So he generated more triples:

Finally, he generalized this procedure by using the formula
. If he started with a perfect square , set
it equal to , and solved for *n*, he got .
If *n* turns out to be an integer, the Pythagorean triple
(*m*,*n*,*n*+*f*) results. By choosing any number *m*, running through all
possibilities of *f* from 1 to *m*, he realized that all Pythagorean
triples starting with *m* could be found.

He wrote up this investigation as a science fair project, received the grand prize in his school and an honorable mention in his county.

Wed Jan 6 11:37:02 EST 1999