I understood the OP like all turning points have to be counted.

I'd define the length of a path by number of necessary moves. Hence length is 0 if start and end are identical. (This facilitates adding partial solutions)

I think allowing back-moves is necessary to "solve" a 1024 x 1024 grid with an approach based on probability.

Compare this Re: Highest total sum path problem

Cheers Rolf
(addicted to the Perl Programming Language :)
Wikisyntax for the Monastery

In reply to Re^3: Code challenge: Route planning on a 2D grid by LanX
in thread Code challenge: Route planning on a 2D grid by bliako

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