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.

Why am I still talking to myself? When does Charlie weigh in?

Well, I'll start.

n min

-- ---

1 5

2 13

3 15

4 17

5 17

6 25

The smallest 8 triangles are:

3 4 5

6 8 10

5 12 13

9 12 15

8 15 17

12 16 20

7 24 25

15 20 25

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

n = 6

-------

3-4-5

5-13-12

12-16-20

20-25-15

15-17-8

8-10-6

(note that 20 is not an achievable minimum if n = 6, because the a side of 12 comes up in 3 of the first 6 triangles)

(note that I formed chains 2,3,4,5 by adding one more to the previous chain. for n = 6, I needed to rearrange)