# 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, dn, 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. ^ a b 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. ^ a b Eckard Specht (20 May 2010). "The best known packings of equal circles in a square". Retrieved 25 May 2010.