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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

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

$n$ 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...


\begin{figure}\begin{center}\BoxedEPSF{square_pentagon.epsf}\end{center}\end{figure}

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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© 1996-9 Eric W. Weisstein
1999-05-26