On 23/03/2016 14:24, James Propp wrote:
Every closed path on a square grid has a "signed area" equal to the sum of the winding numbers around the grid squares.
What is the maximum possible signed area of a closed grid-path that lives in [0,8]x[0,8] and doesn't re-use any edges?
The following observation (which may already have been obvious to you, in which case I apologize) may help to clarify arguments like Tom R's: The per-square winding numbers completely determine the lines and directions of the grid path: wherever you have two adjacent squares with equal PSWN there is no path-segment between them; wherever you have two with PSWN differing by 1 there is a path-segment with the higher-PSWN square on its left; there cannot be two with PSWN differing by more than 1. -- g