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## Experimenting With Patterns

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 2n+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 2n+1 is a perfect square. Or, working backwards, start with a perfect square , set it equal to 2n+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 4n+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.

Next: Polya's Example Up: Guessing Formulas Previous: Guessing Formulas

Carl Lee
Wed Jan 6 11:37:02 EST 1999