For any positive integer n: Pi(n) does not exceed sqr(n)+n+1 . (Sketchy) proof, using e instead of epsilon: To cover the square of side length n+e : Place n by n unit squares as a square of side length n in the lower left corner. Move those unit squares on the upper-right side of the diagonal running from the upper left to the lower right corner by e up and right. Now we have one set of unit squares in the lower left and one in the upper right corner. The remaining uncovered area is a Zigzag-path of width e consisting of n+1 horizontal lines of length 1+e and n vertical lines of length 1-e. If e is small enough, it is possible to cover that area with a regular array of n+1 non-overlapping unit squares such that each of them covers one horizontal line and parts of the one or two connected vertical lines.
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A simple upper bound for Pi(n)
For any positive integer n: Pi(n) does not exceed sqr(n)+n+1 .
(Sketchy) proof, using e instead of epsilon:
To cover the square of side length n+e :
Place n by n unit squares as a square of side length n in the lower left corner. Move those unit squares on the upper-right side of the diagonal running from the upper left to the lower right corner by e up and right.
Now we have one set of unit squares in the lower left and one in the upper right corner. The remaining uncovered area is a Zigzag-path of width e consisting of n+1 horizontal lines of length 1+e and n vertical lines of length 1-e. If e is small enough, it is possible to cover that area with a regular array of n+1 non-overlapping unit squares such that each of them covers one horizontal line and parts of the one or two connected vertical lines.