20. In the city, there are 10 parallel streets running from north to south, and another 10 streets running from west to east, all forming a $9 \times 9$ square. At each intersection, there is a square (a total of 100 squares). The distance from each square to the next along any street is 1 km. A pharmacy within walking distance is defined as a pharmacy that can be reached by walking no more than 3 km along the streets. The Ministry of Health wants to install pharmacies on some squares so that any person walking along any street, regardless of the direction, would always be getting closer to some pharmacy within walking distance (i.e., the distance between them is decreasing). Would 12 pharmacies be enough for this?
20. Answer. Not enough.

Solution. The position of each moving person is characterized by a segment of the street between squares (there are $2 \cdot 10 \cdot 9=180$ such segments) and the direction of movement - a total of 360 positions. Each pharmacy uses 36 positions (i.e., in 36 positions, this pharmacy will be within walking distance, and the person will be heading towards it). Therefore, if we do not consider the edge cases, 10 pharmacies would just suffice.

But let's consider the edges. Suppose a person is heading towards a corner intersection. Let this intersection be the southeastern one, and the person is walking south on the easternmost street, with less than 1 km remaining to the corner. Then, they are approaching a pharmacy only if this pharmacy is located on the southernmost street and within 3 km of the easternmost street. Similarly, there should be a pharmacy on the easternmost street near the southern street. (These pharmacies can coincide if they are at the corner). In any case, at least 24 positions are lost at each corner. Therefore, 12 pharmacies control a maximum of $36 \cdot 12 \cdot 4 \cdot 24 < 360$ positions, so these pharmacies are not enough.