How many squares can be packed into a circle?

Question

How many squares of size a×a can be packed into a circle of radius R?

I don\'t need a solution. I just need some kind of a starting idea.

Answer 1

Rasterise the circle using something like the midpoint circle algorithm. The number of filled pixels is the number of squares you can fit in the circle. Of course, since you're not actually filling the pixels, just counting them, this should take time proportional to the circumference of the circle, not its area.

You'll have to pick the radius for rasterisation carefully, so that you only count pixels that are strictly inside the circle.

Edit: This may not be exactly correct, as it's possible that applying a sub-pixel offset to the grid could change the result. I'll leave the answer here as it may provide a useful starting point for an exact solution.