Circle packing in a square is a packing problem

Packing problem

Packing problems are a class of optimization problems in mathematics which involve attempting to pack objects together , as densely as possible. Many of these problems can be related to real life packaging, storage and transportation issues...

 in recreational mathematics

Recreational mathematics

Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...

, where the aim is to pack n unit circles into the smallest possible square

Square (geometry)

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

; or, equivalently, to arrange n points in a unit square for the greatest minimal separation, d_n, between points. To convert between these two formulations of the problem, the square side for unit circles will be .

Optimal solutions have been proven for N≤30. Solutions up to N=20 are shown below.:

Number of circles	Square size	dn	Figure
1	2
2	≈ 3.414...	≈ 1.414...
3	≈ 3.931...	≈ 1.035...
4	4	1
5	≈ 4.828...	≈ 0.707...
6	≈ 5.328...	≈ 0.601...
7	≈ 5.732...	≈ 0.536...
8	≈ 5.863...	≈ 0.518...
9	6	0.5
10	6.747...	0.421...
11	7.022...	0.398...
12	≈ 7.144...	0.389...
13	7.463...	0.366...
14	≈ 7.732...	0.348...
15	≈ 7.863...	0.341...
16	8	0.333...
17	8.532...	0.306...
18	≈ 8.656...	0.300...
19	8.907...	0.290...
20	≈ 8.978...	0.287...