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 re: Ideas... by Rollercoaster)

The equations I gave do find the multiples of the smaller triplets.  For example, for p = 4 and q = 2:

2pq = 16

p^2 - q^2 = 12

p^2 + q^2 = 20

Anyway, I hope that the solution to this puzzle doesn't require an exhaustive computer search of all the possible triangles, and instead makes use of some relationship between formulas like those above.  I'm probably not the best person to try as I have no formal math training and my brain is completely fried at this point on a Monday afternoon...I'll try to clear it out and come back to this one later tonight. :)

Posted by tomarken on 2006-03-20 16:52:12