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Subject: Geometry
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Posted by: Martin Erickson
on: July 18th, 2011
Conjecture   For any integer $ n \geq 1 $, it is impossible to cover a square of side greater than $ n $ with $ n^2+1 $ unit squares.

Alexander Soifer in [S] raises the question of the smallest number $ \Pi (n) $ of unit squares that can cover a square of side $ n+\epsilon $. He shows the asymptotic upper bound $ n^2+o(1)n+O(1) $, and the small values $ \Pi (1)=3 $, $ 5 \leq \Pi (2) \leq 7 $, and $ 10 \leq \Pi (3) \leq 14 $. He conjectures the asymptotic lower bound $ n^2+O(1) $.

Bibliography

[S] Soifer, Alexander, "Covering a square of side n+epsilon with unit squares," J. of Combinatorial Theory, Series A 113 (2006):380-383.


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