Given a foundation that is in form of a square , that is divided into smaller squares. There is a gap of length between any two adjacent squares. The foundation is covered with several layers of bricks of size . Every layer consists of bricks and each brick fully covers exactly one gap of length . Such cover is called *strong*, if every gap is covered by a brick at least in one of the layers. What is the minimum amount of layers that make a strong cover?
(Bogdan Rublyov)
Solution
Consider a square of size , that does not touch the borders of . All sides of it have to be covered by bricks. Moreover, these bricks have to be different for different
Fig. 16
sides, since one half of each brick covers the square . Thus, there have to be at least layers in order for cover to be strong. It suffices to show that it is possible to have a strong cover with layers (Fig. 16).
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