Square Packing

Find the minimum size Square capable of bounding $n$ equal Squares arranged in any configuration. The only packings which have been proven optimal are 2, 3, 5, and Square Numbers (4, 9, ...). If $n=a^2-a$ for some $a$, it is Conjectured that the size of the minimum bounding square is $a$ for small $n$. The smallest $n$ for which the Conjecture is known to be violated is 1560. The size is known to scale as $k^b$, where

$${\textstyle{1\over 2}}(3-\sqrt{3}\,) < b < {\textstyle{1\over 2}}.$$

$n$	Exact	Decimal
1	1	1
2	2	2
3	2	2
4	2	2
5	$2+{\textstyle{1\over 2}}\sqrt{2}$	2.707...
6	3	3
7	3	3
8	3	3
9	3	3
10	$3+{\textstyle{1\over 2}}\sqrt{2}$	3.707...
11		3.877...
12	4	4
13	4	4
14	4	4
15	4	4
16	4	4
17	$4+{\textstyle{1\over 2}}\sqrt{2}$	4.707...
18	${\textstyle{1\over 2}}(7+\sqrt{7}\,)$	4.822...
19	$3+{\textstyle{4\over 3}}\sqrt{2}$	4.885...
20	5	5
21	5	5
22	5	5
23	5	5
24	5	5
25	5	5
26		5.650...

The best packing of a Square inside a Pentagon, illustrated above, is 1.0673....

References

Erdös, P. and Graham, R. L. ``On Packing Squares with Equal Squares.'' J. Combin. Th. Ser. A 19, 119-123, 1975.

Friedman, E. ``Packing Unit Squares in Squares.'' Elec. J. Combin. DS7, 1-24, Mar. 5, 1998. http://www.combinatorics.org/Surveys/.

Gardner, M. ``Packing Squares.'' Ch. 20 in Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, 1992.

Göbel, F. ``Geometrical Packing and Covering Problems.'' In Packing and Covering in Combinatorics (Ed. A. Schrijver). Amsterdam: Tweede Boerhaavestraat, 1979.

Roth, L. F. and Vaughan, R. C. ``Inefficiency in Packing Squares with Unit Squares.'' J. Combin. Th. Ser. A 24, 170-186, 1978.