Rules: Each partner gets a board that looks like this, and you win the race by crossing the edge (any edge) of the box before your opponent does.
Inside the circle, you’re going to draw a shape. Maybe you’ll draw a triangle, or a square, or something else. The next thing that you draw is going to be the same shape, but it has to touch all the corners of your original shape.
So, if your first move looks like this:
Then your second move can look like this:
And your third move will be a third square, touching all four points of the second square. If you keep on drawing squares, eventually you’ll hit the edge. If you get there before your opponent does, you win.
I’d have kids play this game a couple of times and have them figure out strategies. I’d encourage them to try various shapes, and see if any of them get them to the edge faster. If a kid thinks that certain shapes are winners, I’d encourage them to explain why.
That’s one possible opener for a series of days investigating inscribed polygons.