In a chess tournament, all players faced each other twice, scoring 1 point for a win, half a point for a draw, and zero points for a loss. The winner was the one who scored the most points in the competition. Luíza, a curious mathematician, found a piece of paper stating that the total score of all participants was 210 points.

a) How many players participated in the tournament?

b) The following image was the logo of the tournament and was on the paper that Luíza found.

![](https://cdn.mathpix.com/cropped/2024_05_01_09dbad5281c800d666d6g-21.jpg?height=536&width=529&top_left_y=1543&top_left_x=867)

Luíza paused for a few seconds observing it and noticed that there was a knight in the center, with the black squares illustrating the ones the knight could move to in one move. She then wondered:

- If this board were infinite, and the knight maintained the movement pattern, how many squares could the knight reach in two moves?

Initially, she thought "64," but quickly realized she was wrong. Now, it's up to you, answer Luíza's question correctly.
c) Luíza received a message from a friend who participated in the tournament, informing her that he scored 12 points. Luíza replied: “- Don't be sad, you'll win the next championship!". How did Luíza know that her friend had not won the tournament?
Solution

a) Let $x$ be the number of chess players. Since in each match exactly one point is contested, the total number of points is equal to the number of games. Each athlete will play $(x-1) \cdot 2$ matches, as they will play against all the others twice. However, the matches $a \times b$ and $b \times a$ will be counted both in the list of games of " $a$ " and in that of " $b$ ". Thus, the total number of games can be obtained by the equation

$$
\begin{array}{r}
\frac{x \cdot(x-1) \cdot 2}{2}=210 \\
x \cdot(x-1)=210
\end{array}
$$

Since $x$ is a positive integer, we can analyze all 8 ways to write 210 as a product of 2 positive integers: $1 \cdot 210, 2 \cdot 105, 3 \cdot 70, \ldots, 14 \cdot 15$ and conclude that only one of them is formed by consecutive numbers. Therefore,

$$
\begin{aligned}
x \cdot(x-1) & =15 \cdot 14 \\
x & =15
\end{aligned}
$$

Thus, there were 15 players in the tournament.

b) First, notice that with two moves, regardless of the chosen square, the knight can return to the starting square, which is the common destination possible from the eight highlighted squares after the first move. Each square, by repeating the move made, will have a unique square that can only be reached by it. Finally, imagining a circle centered on the central square and containing all the knights on its boundary, we can see that every pair of knights can have a common square besides the central square, except for the knights that are diametrically opposite. Since each knight reaches 8 squares on the second move, but 7 excluding the central square and 6 of them are accessible to another knight, then on the second move, a total of $\frac{8 \cdot 6}{2}+8$ new positions are obtained, in addition to the central square. That is,

$$
1+8+24=33 \text { options for squares. }
$$

The following diagram shows all the squares that can be reached on the second move.

![](https://cdn.mathpix.com/cropped/2024_05_01_09dbad5281c800d666d6g-23.jpg?height=735&width=709&top_left_y=381&top_left_x=777)

c) Luíza found the average points per player as $\frac{210}{15}=14$ points. Since the average is 14, unless everyone had the same score (which did not happen, as Carlos scored 12), there must have been at least one chess player with more than 14 points. Thus, Luíza concluded that her friend had not won the competition.