Geodesic
Packing 10 or 11 unit squares in a square
The electronic journal of combinatorics, Tome 10 (2003)
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Let $s(n)$ be the side of the smallest square into which it is possible pack $n$ unit squares. We show that $s(10)=3+\sqrt{1\over 2}\approx3.707$ and that $s(11)\geq2+2\sqrt{4\over 5}\approx3.789$. We also show that an optimal packing of $11$ unit squares with orientations limited to $0$ degrees or $45$ degrees has side $2+2\sqrt{8\over 9}\approx3.886$. These results prove Martin Gardner's conjecture that $n=11$ is the first case in which an optimal result requires a non-$45$ degree packing.
DOI : 10.37236/1701
Classification : 05B40, 52C15 
@article{10_37236_1701,
     author = {Walter Stromquist},
     title = {Packing 10 or 11 unit squares in a square},
     journal = {The electronic journal of combinatorics},
     year = {2003},
     volume = {10},
     doi = {10.37236/1701},
     zbl = {1011.05019},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1701/}
}
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%J The electronic journal of combinatorics
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Walter Stromquist. Packing 10 or 11 unit squares in a square. The electronic journal of combinatorics, Tome 10 (2003). doi: 10.37236/1701

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