1. (10 points) 16 children of different heights stood in a circle, all facing the center. Each of them said: “My right neighbor is taller than my left neighbor.” What is the minimum number of children who could be lying?

![](https://cdn.mathpix.com/cropped/2024_05_06_5eae8220555784fd7f73g-1.jpg?height=52&width=680&top_left_y=642&top_left_x=254)
a) 0 ;

b) 1 ;

c) 2 ;

d) more than 2 .
Answer: b).

Solution: Let's number the positions of the children in the circle from 1 to 16 counterclockwise. We will divide the children into 2 groups: those standing in positions with odd numbers - the first group, and those standing in positions with even numbers - the second group. Note that the statements of children from the first group refer only to children from the second group, and vice versa. Since all children are of different heights, there is the tallest child in each group. Thus, at least 2 statements are true: the statement of the child from the first group standing to the left of the tallest in the second group, and the statement of the child from the second group standing to the left of the tallest in the first group.

Let's provide an example of an arrangement where exactly 2 statements are true. Arrange the children in descending order of their heights (i.e., the first will be the tallest, and the 16th - the shortest). It is easy to see that in such an arrangement, only the statements of the children standing in the 1st and 16th positions will be true.