9.1. (13 points) The surface of a round table is divided into 9 identical sectors, in which the numbers from 1 to 9 are written sequentially clockwise. Around the table sit 9 players with numbers \(1, 2, \ldots, 9\), going clockwise. The table can rotate around its axis in both directions, while the players remain in place. The players are seated at equal distances from each other, so when the table stops rotating, exactly one player is opposite each sector and receives the number of coins written on that sector. It is known that after 11 rotations of the table, player ㄱo4 received 90 coins, and player № 8 received 35 coins. How many coins did player № 1 receive?
Answer: 57.

Solution. If, as a result of spinning the table, one coin went to someone with numbers from 5 to 8, then player No. 8 received 5 fewer coins than player No. 9, and if one coin went to someone else, then No. 8 received 4 more coins than player No. 4. Let the number of spins where one coin went to someone with numbers from 5 to 8 be exactly $k$. Then we get the equation $-5k + 4(11 - k) = 35 - 90$, from which $k = 11$. This means that in all 11 spins, each player from 9 to 4 (if moving clockwise) received exactly 11 more coins than the previous one. Therefore, the first player has $35 + 11 + 11 = 57$ coins.