Problem 10.5. One winter day, 43 children were throwing snowballs at each other. Each of them threw exactly one snowball at someone else. It is known that:

- the first threw a snowball at the one who threw a snowball at the second,
- the second threw a snowball at the one who threw a snowball at the third,
- the forty-third threw a snowball at the one who threw a snowball at the first.

What is the number of the one who threw a snowball at the third?
# Answer: 24.

Solution. First, note that not only did each throw exactly one snowball, but each was hit by exactly one snowball. Indeed, from the phrase "the first threw a snowball at the one who threw a snowball at the second," it follows that someone threw a snowball at the second; similarly, it is established that each was hit by at least one snowball. But since the total number of snowballs is the same as the number of children, each must have been hit by exactly one snowball.

Let's represent each child as a point; draw an arrow from point $A$ to point $B$ if child $A$ threw a snowball at $B$. Since exactly one arrow enters each point and exactly one arrow leaves each point, the arrows form one or several cycles.

Consider the cycle that includes the first child. Let the first child throw a snowball at child number $x$. Then child number $x$ threw a snowball at the second, the second at child number $x+1$, and so on. Thus, this cycle is structured as follows:

$$
\ldots \rightarrow 1 \rightarrow x \rightarrow 2 \rightarrow (x+1) \rightarrow 3 \rightarrow (x+2) \rightarrow 4 \rightarrow \ldots
$$

If we follow it with a step of two arrows, we get the numbers $1, 2, 3, 4, \ldots$ - all numbers from 1 to 43. Therefore, the cycle includes all 43 children, and there are no other cycles.

Notice that if we start from 1 and take 22 steps of two arrows, we will complete the entire cycle and one more arrow, that is, we will land on $x$. Therefore, $x=23$. Then the number of the one who threw a snowball at the third is $x+1=24$.