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: Solution, perhaps? by Charlie)

In addition to the stipulation that no side may have length over 10000, given the conditions that each new triangle must be composed of integer sides (I am assuming this is the case) and that no side length can be repeated, there must be a finite number of possibilities, and it's probably not a huge number.

I'm guessing the key to this would be to find a list of perfect squares (less than or equal to 10,000^2) that can be expressed as the sum of two other perfect squares.  Again, I'm assuming here that each new triangle must have integer sides.  This would put a limit on the possible number of, um...hypotenuses?  ...hypoteni?  (Not sure what the plural of hypotenuse is :P)

I don't know if anyone has a list like that handy, but maybe one of our computer-savvy friends can throw something together to look for that...hopefully I'm not way off base with this idea.

Posted by tomarken on 2006-03-20 13:39:40