13.1. [7-8.7 (20 points), 9.8 (15 points), 10.8 (20 points)] There is a rotating round table with 16 sectors, on which numbers $0,1,2, \ldots, 7,8,7,6, \ldots, 2,1$ are written in a circle. 16 players are sitting around the table, numbered in order. After each rotation of the table, each player receives as many points as the number written on the sector they end up in after the table stops. It turned out that after 13 rotations of the table, player number 5 scored a total of 72 points, and player number 9 scored a total of 84 points. How many points did player number 1 score?
Answer: 20.

Solution. Players No. 5 and No. 9 together scored $72+84=156=12 \cdot 13$ points. In one spin, they can together score no more than 12 points. Therefore, in each of the 13 spins, they together scored 12 points. Note that the 12 points they score can be one of the sums $8+4, 7+5$, $6+6, 5+7$ or $4+8$, when the sector 8 is in front of one of them or between them, i.e., in front of one of the players with numbers $6, 7, 8$. In each of these five cases, player No. 1 scores respectively $4, 3, 2, 1$ or 0 points. This can be expressed by the formula $x_{1}=\frac{x_{5}-x_{9}}{2}+2$, where $x_{n}$ is the number of points scored by the player with number $n$ in this spin. Then the number of points scored by player No. 1 after 13 spins is $\frac{72-84}{2}+2 \cdot 13=20$.