>I tried the non-intersecting path problem proposed by JCM (from Ed Pegg's site) and found two 16-move solutions, one symmetric >which I suppose is the intended answer, but the 17-move eludes me.

The 17 moves solution is attached (17.gif)
It is unique.
 
Since I'm also a peg solitaire 'specialist', I tried some solitaire boards, and found unique solutions on the english and french boards.
I also attached the solutions, since they are unique.
 
Ed proposed a new board, and it's possible to find a lot of different 18 moves solution !
Try to find one that is symmetric (there are 2 symmetric solutions !).

About the problem of "finding a non-crossing path of length 18 in the above graph of knight moves", it seems that the mentioned problem is different from the one I suggested.
In fact, the picture you put allows a 18 moves path, and there are a lot of solutions (I attached the 2 most symmetric solutions) !
On the contrary, the english solitaire path has an unique solution.

I ran my program on english and french solitaire boards.
Result:
19 jumps for the english board
20 jumps for the french board
 
The solutions are unique.
 
I tried the continental board (it has a losenge form), but only 20 jumps are possible.
 
Jean-Charles Meyrignac
-----------------------------------
I found the 16-length non-intersecting leaper tour on this board as
shown in the attached picture.

Joseph DeVincentis
---------------------------------------------------
Hi!
d6-b5-d4-b3-a5-c6-e7-f5-g3-e2-c1-d3-f4-e6-c5-e4-c3
Juha Saukkola