Suppose we have a rectangle with sides $a$ and $b$, $a, $a,b \in \mathbb R$. What is the minimum number of circles centered in the rectangle with radius $1$ such that each line passing through the rectangle intersects at least one circle in at least one point?
I posed this problem a week ago and tried to solve it, however, I couldn't get anywhere. One obvious thing is that if we restrict the lines to be parallel to the sides of the rectangle, the answer will be $\left \lceil \frac{\left \lceil b \right \rceil}{2} \right \rceil$. Any ideas for the original problem?