Shortest path sequence

[Paul C. Leopardi]

David,
My reply below.
Best regards
On Fri, 25 Mar 2005 03:17 pm, David Wilson wrote:
> It is not difficult to show that the length of the path will be even.
>
> Let P be a polygonal path from (0, 0) to (n, n) with vertices
> having integer coordinates and edges of integer length.
>
> Let (a, b) be an edge of P.  If gcd(a, b) = g > 1, replace (a, b)
> with g vectors (a/g, b/g).  Thus, WLOG, gcd(a, b) = 1.

If I understand you correctly, within your analysis, my examples would be 
transformed as follows:

(0,0) to (0,2) to (8,8) (2 steps: 2+10=12) would become
(0,0) to (0,1) to (0,2) to (4,5) to (8,8) (4 steps: 1+1+5+5=12).

(0,0) to (3,4) to (5,4) to (8,8) (3 steps: 5+2+5=12) would become
(0,0) to (3,4) to (4,4) to (5,4) to (8,8) (4 steps: 5+1+1+5=12).
 
(0,0) to (3,4) to (4,4) to (8,7) to (8,8) (4 steps: 5+1+5+1=12) would remain 
as is.

All three paths would become 4 step paths. Notice how horizontal and vertical 
steps would be split into steps of length 1. 

Also, it may be that with this extra constraint [gcd(a, b) = 1], all minimum 
length paths would have the same number of steps.