The following pictures show n unit circles packed inside the smallest known square (of side length s). Most of these have been proved optimal.

1.

2.

3.

s = 2

Trivial.
s = 2 + √2 = 3.414+

Trivial.
s = 2 + 1/√2 + √6/2 = 3.931+

Trivial.

4.

5.

6.

s = 4

Trivial.
s = 2 + 2√2 = 4.828+

Trivial.
s = 2 + 12/√13 = 5.328+

Proved by Graham in 1963.

7.

8.

9.

s = 4 + √3 = 5.732+

Proved by Schaer in 1964.
s = 2 + √2 + √6 = 5.863+

Proved by Schaer/Meir in 1964.
s = 6

Proved by Schaer in 1964.

10.

11.

12.

s = 6.747+

Proved by De Groot in 1990.
s = 7.022+

Proved by Peikert in 1991.
s = 2 + 15√(2/17) = 7.144+

Proved by Peikert in 1991.

13.

14.

15.

s = 7.463+

Proved by Peikert in 1991.
s = 6 + √3 = 7.732+

Proved by Wengerodt in 1987.
s = 4 + √2 + √6 = 7.863+

Proved by Peikert in 1991.

16.

17.

18.

s = 8

Proved by Wengerodt in 1983.
s = 8.532+

Proved by Peikert in 1991.
s = 2 + 24/√13 = 8.656+

Proved by Peikert in 1991.

19.

20.

21.

s = 8.907+

Proved by Peikert in 1991.
s = 130/17 + 16/17√2 = 8.978+

Proved by Peikert in 1991.
s = 9.358+

22.

23.

24.

s = 9.463+
s = 2 + 2√2 + 2√6 = 9.727+
s = 6 + √2 + √6 = 9.863+