Problem 4. 13 children sat at a round table and agreed that boys would lie to girls, but tell the truth to each other, and girls, on the contrary, would lie to boys, but tell the truth to each other. One of the children said to their right neighbor: "The majority of us are boys." The latter said to their right neighbor: "The majority of us are girls," and he to his right neighbor: "The majority of us are boys," and so on, until the last child said to the first: "The majority of us are boys." How many boys were at the table?

[6 points] (A. V. Khachatryan)
# Answer: 7.

Solution. It is clear that there were both boys and girls at the table. Let's see how the children were seated. After a group of boys sitting next to each other comes a group of girls, then boys again, then girls, and so on (a group can consist of just one person). Groups of boys and girls alternate, so their number is even. The incorrect statements were made at the transitions from one group to another, so their number is also even. Since seven statements of "most of us are boys" were made, six of the statements "most of us are girls" must be incorrect, and there were also six groups.

The alternation of correct and incorrect statements means that there were two children in each group. Only the first and last child sitting next to each other said the same thing, so there were three people in their group. These are boys, as they are in the majority. In total, there were $2+2+2=6$ girls and $2+2+3=7$ boys sitting at the table.

The diagram shows exactly how the children were seated at the table. The first speaker is outlined in a frame.

![](https://cdn.mathpix.com/cropped/2024_05_06_031fac0a0a7981eac22cg-05.jpg?height=302&width=307&top_left_y=723&top_left_x=1017)