Square Packing
Find the minimum size Square capable of bounding 
equal Squares arranged in anyconfiguration. The only packings which have been proven optimal are 2, 3, 5, and Square Numbers(4, 9, ...). If 
for some 
, it is Conjectured that the size of the minimum boundingsquare is for small . The smallest for which the Conjecture is known to be violated is 1560. Thesize is known to scale as 
, where
| Exact | Decimal |
| 1 | 1 | 1 |
| 2 | 2 | 2 |
| 3 | 2 | 2 |
| 4 | 2 | 2 |
| 5 |  | 2.707... |
| 6 | 3 | 3 |
| 7 | 3 | 3 |
| 8 | 3 | 3 |
| 9 | 3 | 3 |
| 10 |  | 3.707... |
| 11 | 3.877... | |
| 12 | 4 | 4 |
| 13 | 4 | 4 |
| 14 | 4 | 4 |
| 15 | 4 | 4 |
| 16 | 4 | 4 |
| 17 |  | 4.707... |
| 18 |  | 4.822... |
| 19 |  | 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.
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