I can't comment on the question, so I will suggest an approach here. Resize the targets (partial checkerboards in R^2) by dividing by n, and stop when the union of the resizings covers the plane (or enough of it). Hint: don't expect a maximum for n.
Now I notice n is an arbitrary real, and not an integer. I believe there is an upper bound for n, simply because the squares block all lines of sight from the origin. 3 is looking reasonable as a bound now.