Take a right triangle with integer sides A, B, & C.
(C need not be the hypotenuse.)

To side C attach another right triangle with integer sides C, D & E.

On this new triangle attach another right triangle to either side D or E. Continue the process of attaching a new right triangle to the previous; creating a chain of right triangles.

Three further rules:
1. No side length may be repeated.
2. No triangles may overlap.
3. No side may have length over 10000.

How many triangles can you make in this chain?

(In reply to Ideas... by tomarken)

The formulas for generating triplets are very useful.  Unfortunately, they also 'ignore' integer multiples of smaller triplets.  For instance, in your example, another option after (5,12,13) would be (12,16,20) and then (20,48,52).  This chain doesn't seem to grow as fast, but obviously loses odd numbers right off the bat (and hence A LOT of potential triplets along the way).

Posted by Rollercoaster on 2006-03-20 15:55:00