179. The Tennis Federation has assigned qualification numbers to all its tennis players: the strongest player gets the first number, the next strongest gets the second number, and so on. It is known that in matches where the qualification numbers differ by more than 2, the player with the lower number always wins. A tournament involving the 1024 strongest tennis players is held according to the Olympic system: participants in each round are randomly paired, and the winner of each pair advances to the next round, so the number of participants is halved after each round. Thus, after the tenth round, the winner will be determined. What is the highest number he can have?
179. Answer! The highest possible number of the winner is 20.

Since a tennis player with number $k$ can lose (not counting stronger ones) only to the $(k+1)$-th and $(k+2)$-th tennis players, the number of the strongest winner after each round cannot increase by more than 2. Thus, the number of the winner of the entire tournament cannot be more than 21. However, we will show that the 21st tennis player cannot become the winner. For this, after the first round, the 1st and 2nd players must be eliminated, losing to the 3rd and 4th (otherwise, the number of the winner would be less than 21), in the second round, the 3rd and 4th must be eliminated, and the 5th and 6th must win against them, and so on up to the 9th round, where the 19th and 20th must win against the 17th and 18th. Thus, the 21st tennis player does not make it to the final, where two players meet.

It remains to provide an example of a tournament where the 20th player wins. For this, we will divide all the players into two groups of 512 people each. The first group will include the 19th, 20th, and 510 weaker players. We will organize the tournament in this group so that the 20th wins (which is obviously possible). The second group will include the 1st, 2nd, ..., 18th, and the remaining weaker players, and we will organize the tournament so that the 18th wins. This can be done by organizing the tournament as described above: in the first round, the 3rd and 4th win against the 1st and 2nd, in the second round, the 5th and 6th win against the 3rd and 4th, and so on until the eighth round, when the 17th and 18th win against the 15th and 16th, after which, in the ninth round, the 18th wins against the 17th. In the final, the 20th and 18th meet, and thus the 20th can win.

$\nabla$ It can be proven by induction that in the case of $2^n$ players, the highest number of the winner is $2n$.