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1) xy=6
x-y=3
x^3-y^3=?

2) The hypotenuse of a right triangle with integral sides is 41 in. The shortest leg is ? in.

3) Fibonacci Sequence squared
ex.
1^2+1^2+2^2+3^2+5^2+8^2 =?

1) Use a nifty way of writing [unparseable or potentially dangerous latex formula]

2) Normally, they generate pythagorean triples by squaring an odd number, dividing by [unparseable or potentially dangerous latex formula] and taking the two closest integers to that. So you get [unparseable or potentially dangerous latex formula] from starting with the odd number [unparseable or potentially dangerous latex formula], then looking at [unparseable or potentially dangerous latex formula], taking [unparseable or potentially dangerous latex formula] to get the other two sides. Another example might be [unparseable or potentially dangerous latex formula]. Can you think of some triples that cannot be generated in this way? Are there any?

For this problem, we're given the hypotenuse, which if you use the above algorithm, is always the "[unparseable or potentially dangerous latex formula]" part... going in reverse we can find the smaller leg. [unparseable or potentially dangerous latex formula].

3) What are the first few sums? Look for a pattern, with an educated guess maybe. I must go.

3) It can be shown by induction that [unparseable or potentially dangerous latex formula]

In your specific example, [unparseable or potentially dangerous latex formula].

And, the grand daddy of all pythagorean triple generators:

[unparseable or potentially dangerous latex formula]

Now try to find a triple that cannot be generated!