58th Ukrainian National Mathematical Olympiad

Question

Given a foundation that is in form of a square

4 \times 4

, that is divided into smaller

1 \times 1

squares. There is a gap of length

1

between any two adjacent squares. The foundation is covered with several layers of bricks of size

2 \times 1

. Every layer consists of

8

bricks and each brick fully covers exactly one gap of length

1

. 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)

Accepted Answer

Consider a square A of size 1 1, that does not touch the borders of 4 4. All 4 sides of it have to be covered by bricks. Moreover, these bricks have to be different for different FIGURE_0 **Fig. 16** sides, since one half of each brick covers the square A. Thus, there have to be at least 4 layers in order for cover to be strong. It suffices to show that it is possible to have a strong cover with 4 layers (Fig. 16).