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.

## A simple upper bound for Pi(n)

For any positive integer

(Sketchy) proof, using

To cover the square of side length

Place

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

n: Pi(n) does not exceed sqr(n)+n+1 .einstead of epsilon:n+e:nbynunit squares as a square of side lengthnin 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 byeup and right.econsisting ofn+1 horizontal lines of length 1+eandnvertical lines of length 1-e. Ifeis small enough, it is possible to cover that area with a regular array ofn+1 non-overlapping unit squares such that each of them covers one horizontal line and parts of the one or two connected vertical lines.