Disk covering problem

The disk covering problem asks for the smallest real number ${\displaystyle r(n)}$ such that ${\displaystyle n}$ disks of radius ${\displaystyle r(n)}$ can be arranged in such a way as to cover the unit disk. Dually, for a given radius ε, one wishes to find the smallest integer n such that n disks of radius ε can cover the unit disk.^[1]

Unsolved problem in mathematics

What is the smallest real number ${\displaystyle r(n)}$ such that ${\displaystyle n}$ disks of radius ${\displaystyle r(n)}$ can be arranged in such a way as to cover the unit disk?

More unsolved problems in mathematics

The best solutions known to date are as follows.^[2]

n	r(n)	Symmetry
1	1	All
2	1	All (2 stacked disks)
3	${\displaystyle {\sqrt {3}}/2}$ = 0.866025...	120°, 3 reflections
4	${\displaystyle {\sqrt {2}}/2}$ = 0.707107...	90°, 4 reflections
5	0.609382... OEIS: A133077	1 reflection
6	0.555905... OEIS: A299695	1 reflection
7	${\displaystyle 1/2}$ = 0.5	60°, 6 reflections
8	0.445041...	~51.4°, 7 reflections
9	0.414213...	45°, 8 reflections
10	0.394930...	36°, 9 reflections
11	0.380083...	1 reflection
12	0.361141...	120°, 3 reflections

Method

The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6. One of the covering disks is placed central and the remaining five in a symmetrical way around it.

While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively.^[2] The corresponding angles θ are written in the "Symmetry" column in the above table.