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 integer right triangles.

No side length may be repeated.

If n is the number of triangles in the chain, what is the minimum largest side for n=2, 3, 4, 5, 6, 7, 8, 9, 10.

Well, I'll start.

n   min
--  ---
1    5
2   13
3   15
4   17
5   17

The smallest 5 triangles are:
    3     4     5
    6     8    10
    5    12    13
    9    12    15
    8    15    17

And they can be chained:

n = 1
------
3-4-5

n= 2
------
3-4-5
5-13-12

n = 3
-------
3-4-5
5-13-12
12-9-15

n = 4
-------
3-4-5
5-13-12
12-9-15
15-17-8

n = 5
-------
3-4-5
5-13-12
12-9-15
15-17-8
8-6-10

This problem seems a lot more manageable then  the original pythagorean chain problem, Jer!

Posted by Steve Herman on 2006-06-16 17:06:43