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?

Well, 4 is certainly not the maximum chain length. Here's a 6 chain: 4-3-5 5-13-12 12-16-20 20-15-25 25-7-24 24-30-18 Note that I'm not close to the 10000 limit.  Seems to be room for a longer chain ... For instance, I can multiply the 6-chain by 3, and then attach a new triangle to the start of the chain, forming: 20-16-12 12-9-15 15-39-36 36-48-60 60-45-75 75-21-72 72-90-54 Voila!  A 7-chain Still seems to be a lot of room...

Edited on June 9, 2006, 6:14 pm

Posted by Steve Herman on 2006-03-20 18:17:44