Draw some polygons by joining the dots on a $3$ by $3$ grid.
What is the greatest number of sides that your polygon could have?
What about on a $3$ by $4$ grid, or a $3$ by $5$ grid?
What about on a $3$ by $n$ grid?
Can you explain the pattern by which the 'number of sides' increases?
Explore some polygons on grids that are $4$ dots high.
What is the maximum number of sides a polygon could have on a $4$ by $n$ grid?
Can you explain how you know?
What is the maximum number of sides a polygon could have on a $6$ by $6$ grid?
And on a $6$ by $n$ grid?
With thanks to Don Steward, whose ideas formed the basis of this problem.