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Joseph O'Rourke
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Not an answer, just an illustration for $6$ points, $0 \le x+y \le 2$.


      [![Poly3][1]][1]
  • $A,B,C$: Point $3$ cannot connect to $1$ or $6$, so it must connect to $2,5$ or $4,5$ or $2,4$.
  • $B$: Point $6$ is now isolated by $34$ from $1$ and $2$.
  • $C$: Point $1$ is now isolated by $34$ from $5$ and $6$.
  • $D$: $1$ is trapped.
  • $E$: $2$ is trapped.

**(Later.)** Here is a simple polygonal path for the lattice points $0 \le x+y \le 6$:
      [![Path6][2]][2]
Joseph O'Rourke
  • 150.9k
  • 36
  • 358
  • 958