Bakayev E.v.

10 children of different heights are standing in a circle. From time to time, one of them runs to another place (between some two children). The children want to stand in order of increasing height as quickly as possible, in a clockwise direction (from the shortest to the tallest). What is the minimum number of such moves they will definitely need, regardless of how they are initially standing?

#
Number the children in ascending order of height $-1, 2, \ldots, 10$.

Estimation. Suppose they initially stood in reverse order.

First method. If there were fewer than eight sprints, then some three children remained in their places, and their order is opposite to the required one.

Second method. Let's call the complexity the number of turns we make when walking counterclockwise from the 1st to the 2nd, then to the 3rd, ..., from the 10th to the 1st. Initially, the complexity is 1, and at the end, it is 9. It is not difficult to verify that when one child sprints, the complexity can change by no more than 1. Therefore, at least eight sprints are needed.

Example with eight sprints: the 1st and 2nd remain in their places, the 3rd sprints and stands behind the 2nd, then the 4th - behind the 3rd, and so on.

## Answer

8 sprints.