[ Extremal properties (other) $)]$

In a city, there are 10 streets parallel to each other and 10 streets intersecting them at right angles. What is the smallest number of turns a closed bus route can have, passing through all intersections?

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A closed route passing through all intersections can have 20 turns (see figure). It remains to prove that such a route cannot have fewer than 20 turns. After each turn, there is a transition from a horizontal street to a vertical one or vice versa. Therefore, the number of horizontal segments of the closed route is equal to the number of vertical segments and is equal to half the number of turns.

Suppose the closed route has fewer than 20 turns. Then there will be streets of both directions that the route does not pass through. Therefore, the route does not pass through the intersection of these streets.

![](https://cdn.mathpix.com/cropped/2024_05_06_55a07599adfd5314dc75g-02.jpg?height=780&width=786&top_left_y=1685&top_left_x=636)

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