Can we find an (infinite) sequence of disks in the Euclidean plane such that: (1) their centers have no (finite) limit point in the plane; (2) the total area of the disks is finite; and (3) every line in the plane intersects at least one of the disks?

**Solution**

Answer: yes.

Easy.

The fact that the centers have no limit point is a red herring - after all putting them close together would hardly help to ensure that the disks blocked all lines. The easiest strategy is evidently to find a continuous line of overlapping disks. That evidently requires that if their radii are r_{i}, then ∑ r_{i} diverges, but the sum of their areas is finite, so ∑ r_{i}^{2} must converge. That is easy to achieve. For example, take r_{i} = 1/i.

That leaves a little tidying up. We need the line to be infinite in both directions, and we need another line of disks to catch lines parallel to the first line of disks. But multiplying by four leaves convergence and divergence unaffected.

So, define C_{n} = ∑_{1}^{n} 1/i, R_{n} = 1/n. Define disk D_{4n} to have center (C_{n}, 0) and radius R_{n}. That gives us a line of overlapping disks of finite total area, covering the positive x-axis. Similarly, define D_{4n+1} to have center (-C_{n}, 0) and radius R_{n}; they cover the negative x-axis. Define D_{4n+2} to have center (0, C_{n}), radius R_{n}, and D_{4n+3} to have center (0, -C_{n}), radius R_{n}; they cover the y-axis. The complete set D_{1}, D_{2}, D_{3}, ... have the required properties.

© John Scholes

jscholes@kalva.demon.co.uk

21 Sep 1999