5. Each football is sewn from pieces of leather in the shape of regular pentagons and regular hexagons. The ball has a total of 32 leather pieces. Each piece of leather in the shape of a pentagon is connected along its sides only to pieces of leather in the shape of hexagons. Each piece of leather in the shape of a hexagon is connected along its sides to 3 pentagons and 3 hexagons. How many pieces of leather are in the shape of a pentagon, and how many in the shape of a hexagon?
Solution. Let $x$ be the number of hexagonal pieces. According to this, $32-x$ is the number of pentagonal pieces. If we count the sides of the pentagons in total, it is $5(32-x)$. Since on the other hand, the hexagons are also bordered by 3 sides of pentagons, the number of sides of pentagons is $3 x$. Therefore,

$$
3 x=5(32-x) \Rightarrow 3 x+5 x=160 \Rightarrow x=20
$$

This means there were 20 hexagons and 12 pentagons.

## VII section