Construct a simple polygon on a grid of equal-distanced points such that:
- all the polygon's vertices are grid points,
- there are exactly $i~(\geq 0)$ lattice points in the interior, and
- there are exactly $b~(\geq 3)$ lattice points on the boundary.
As a practice, construct it for:
- $i = 5$ and $b = 12$
- $i = 0$ and $b = 10$
- $i = 9$ and $b = 3$
As for examples:
- On the left side, it's a polygon with $5$ interior points and $16$ boundary points.
- On the right side, it's a polygon with $9$ interior points and $10$ boundary points.
This puzzle was submitted (but not accepted) for The 31st International Olympiad in Informatics (IOI 2019).