In Anchuria, a checkers championship is taking place in several rounds. The days and cities for the rounds are determined by a draw. According to the championship rules, no two rounds can take place in the same city, and no two rounds can take place on the same day. Among the fans, a lottery is organized: the main prize goes to the person who correctly guesses before the start of the championship in which cities and on which days all the rounds will take place. If no one guesses correctly, the main prize will go to the championship organizing committee. There are eight cities in Anchuria, and the championship is allocated a total of eight days. How many rounds should the championship have so that the organizing committee has the highest probability of getting the main prize?
In an $8 \times 8$ table, you need to select $k$ cells such that no more than one cell is selected in any row or column. The number of tours $k$ should be chosen so that the number $N k$ of possible selection options is maximized.

$$
N_{k}=C_{8}^{k} A_{8}^{k}=\frac{8!\cdot 8!}{(8-k)!(8-k)!k!}\left(C_{8}^{k}-\text { the number of ways to choose } k\right. \text { cities from eight without considering }
$$

the order, $A_{8}^{k}$ - the number of ways to choose $k$ days from eight considering the order, as each day must correspond to one of the cities).

Notice that $\frac{N_{k}}{N_{k-1}}=\frac{(9-k)^{2}}{k}=\frac{m^{2}}{9-m}$, where $m=9-k$. Since $m^{2}N_{7}>N_{8}$.

## Answer

6 tours.