Shapovalov A.V.

A row of new recruits stood facing the sergeant. On the command "left," some turned left, while the rest turned right. It turned out that six times more soldiers were looking at the back of their neighbor than in the face. Then, on the command "about face," everyone turned in the opposite direction. Now, seven times more soldiers were looking at the back of their neighbor than in the face. How many soldiers are in the row?
Let's assume that the sergeant has lined up the soldiers between two posts. After the first command, each recruit either looks at the back of the head of the neighbor or at the face, except for the two soldiers at the ends, who can look at the posts.

If a soldier is looking at the back of the head of a neighbor, then after the turn, this neighbor will be looking at the back of the head of the first soldier. Therefore, the number of those looking at the back of the head will not change.

An end soldier who was looking at the post will not do so after the turn; conversely, if he was not looking at the post, he will be after the turn. Thus, the number of those looking at the post will either remain the same (was 1 and will remain 1), or increase by 2 (was 0, will become 2), or decrease by 2 (was 2, will become 0).

Since the total number of soldiers is constant, the number of those looking at the face will either remain the same, or increase or decrease by 2.

According to the problem, the number of soldiers looking at the face was initially one-sixth of the number looking at the back of the head, and then one-seventh. Therefore, their number decreased (and, therefore, decreased by 2). On the other hand, it changed by $1/6 - 1/7 = 1/42$ of the constant number of those looking at the back of the head. That is, there were $2 \cdot 42 = 84$ people looking at the back of the head, and 14 people looking at each other before the turn, which is $84 / 6 = 14$. There were no soldiers looking at the posts. Thus, the total number of recruits is $84 + 14 = 98$.

## Answer

98.