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)
n = 6 (spoiler)
