Wow! That was fast! I verified Alan's claim by writing down all the winding numbers: 11111110 12222221 12333221 12344321 12344321 12333321 11222221 11111111 and checking that they add up to 117. Can anyone beat this, or prove that it's optimal? (Of course n=8 is just a special case; it'd be nice to know the answer for general n.) Jim On Wed, Mar 23, 2016 at 12:17 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I have one that I think is 117.
00 e8 n7 w7 s5 e5 n3 w3 s2 e2 n3 w3 s5 e5 n7 w7 s8.
63 + 35 + 15 + 4 = 117.
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).
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