Kovaldji A.K.

A football is sewn from 32 patches: white hexagons and black pentagons. Each black patch borders only white ones, and each white one borders three black and three white. How many white patches are there?
Calculate the number of white patches bordering with black ones in two ways.

## Solution

Let the desired number of white patches be denoted by $x$. Then the number of black patches will be 32 - $x$. To form an equation, we will count the number of "borders" between white and black patches in two ways.

Each white patch borders with three black ones. Therefore, the number of borders is 3 $\cdot$ $x$.

On the other hand, each black patch borders with five white ones. Therefore, the number of borders is 5 $\cdot$ (32 - $x$).

We obtain the equation $3 x=5 \cdot(32-x)$. From this, $8 x=160$ and $x=20$.

## Answer

20.