Question 18
Ana and Beto play on a board with $11$ rows and $9$ columns. First Ana divides the board into $33$ zones. Each zone consists of $3$ adjacent cells aligned vertically or horizontally, as shown in the figure.
Then Beto writes in each cell one of the numbers $0$, $1$, $2$, $3$, $4$, $5$, so that the sum of the numbers in each zone equals $5$. Beto wins if the sum of the numbers written in each of the $9$ columns of the board is a prime number. Otherwise, Ana wins. Prove that Beto has a winning strategy.
Then Beto writes in each cell one of the numbers $0$, $1$, $2$, $3$, $4$, $5$, so that the sum of the numbers in each zone equals $5$. Beto wins if the sum of the numbers written in each of the $9$ columns of the board is a prime number. Otherwise, Ana wins. Prove that Beto has a winning strategy.