I should mention that my solution for an 8-by-8 square is a special case of a more general spiral construction that, for n = 1, 2, 3, ..., yields a grid-path in [0,n]x[0,n] with signed area 1, 4, 9, 19, 32, 53, 78, 114, 155, 210, ... This is not itself in OEIS, but it is an alternation between two sequences, both of which are given by third-degree polynomials in n, and one of which does occur in the OEIS, under the name "house numbers" (or is it "House numbers"?). But I do not know whether this spiral construction is optimal. Jim Propp On Wed, Mar 23, 2016 at 10:24 AM, James Propp <jamespropp@gmail.com> 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?
I found one with signed area 114; is this best possible? An easy upper bound is 120 (the sum of the upper bounds on the individual winding numbers associated with the 64 grid squares).
Jim Propp