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.
- First Curvature
- First Derivative Test
- First Digit Law
- First Digit Phenomenon
- First Multiplier Theorem
- Fischer Groups
- Fischer's Baby Monster Group
- Fish Bladder
- Fisher-Behrens Problem
- Fisher Index
- Fisher Kurtosis
- Fisher's Block Design Inequality
- Fisher's Estimator Inequality
- Fisher's Exact Test
- Fisher Sign Test
- Fisher Skewness
- Fisher's Theorem
- Fisher's z-Distribution
- Fisher's z'-Transformation
- Fisher-Tippett Distribution
- Fitting Subgroup
- Five Cubes
- Five Disks Problem
- Five Tetrahedra Compound
- Fixed