Circle packing in a square

Circle packing in a square is a packing problem in recreational mathematics, where the aim is to pack n unit circles into the smallest possible square; or, equivalently, to arrange n points in a unit square for the greatest minimal separation, d_n, between points.^[1] To convert between these two formulations of the problem, the square side for unit circles will be $L=2+\frac{2}{d_n}$.

Solutions (not necessarily optimal) have been computed for every N≤10,000.^[2] Solutions up to N=20 are shown below.:^[2]

Number of circles	Square size (side length)	dn[1]	Number density	Figure
1	2	∞	0.25
2	$2+\sqrt{2}$ ≈ 3.414...	$\sqrt{2}$ ≈ 1.414...	0.172...
3	$2+\frac{\sqrt{2}}{2}+\frac{\sqrt{6}}{2}$ ≈ 3.931...	$\sqrt{6} - \sqrt{2}$ ≈ 1.035...	0.194...
4	4	1	0.25
5	$2+2\sqrt{2}$ ≈ 4.828...	$\frac{1}{2} \sqrt{2}$ ≈ 0.707...	0.215...
6	$2 + \frac{12}{\sqrt{13}}$ ≈ 5.328...	$\frac{1}{6} \sqrt{13}$ ≈ 0.601...	0.211...
7	$4+ \sqrt{3}$ ≈ 5.732...	$4- 2\sqrt{3}$ ≈ 0.536...	0.213...
8	$2 + \sqrt{2} + \sqrt{6}$ ≈ 5.863...	$\frac{1}{2}(\sqrt{6} - \sqrt{2})$ ≈ 0.518...	0.233...
9	6	0.5	0.25
10	6.747...	0.421...	0.220...
11	7.022...	0.398...	0.223...
12	$2 + 15\sqrt{\frac{2}{17}}$ ≈ 7.144...	0.389...	0.235...
13	7.463...	0.366...	0.233...
14	$6 + \sqrt{3}$ ≈ 7.732...	0.348...	0.226...
15	$4 + \sqrt{2} + \sqrt{6}$ ≈ 7.863...	0.341...	0.243...
16	8	0.333...	0.25
17	8.532...	0.306...	0.234...
18	$2 + \frac{24}{\sqrt{13}}$ ≈ 8.656...	0.300...	0.240...
19	8.907...	0.290...	0.240...
20	$\frac{130}{17} + \frac{16}{17} \sqrt{2}$ ≈ 8.978...	0.287...	0.248...

References

1. Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991). Unsolved Problems in Geometry. New York: Springer-Verlag. pp. 108–110. ISBN 0-387-97506-3.
2. Eckard Specht (20 May 2010). "The best known packings of equal circles in a square". Retrieved 25 May 2010.