Quote:
Let $\displaystyle n$ be an even number. Two players play the game by writing one number alternately from a set $\displaystyle \left \{ 6, 7, 8, 9 \right \}$ on the board, until $\displaystyle n$ numbers are written. A second player wins the game if the sum of all written numbers is divided by $\displaystyle 9$, otherwise the first player will win. Which player has a winning strategy if:
a) $\displaystyle n=12$; b) $\displaystyle n=14$?

If one player doesn't want another to win, he will try to keep sum of currently written numbers by modulo $\displaystyle 9$ to be $\displaystyle 4$ or $\displaystyle 5$ (because with numbers from given set another player doesn't have number with which he can make sum to be congruent with $\displaystyle 0$ by modulo $\displaystyle 9$). But I can't make any strategy with this.