Školjka

%V0 Given an initial integer $n_0 > 1$, two players, ${\mathcal A}$ and ${\mathcal B}$, choose integers $n_1$, $n_2$, $n_3$, $\ldots$ alternately according to the following rules : I.) Knowing $n_{2k}$, ${\mathcal A}$ chooses any integer $n_{2k + 1}$ such that $$n_{2k} \leq n_{2k + 1} \leq n_{2k}^2.$$ II.) Knowing $n_{2k + 1}$, ${\mathcal B}$ chooses any integer $n_{2k + 2}$ such that $$\frac {n_{2k + 1}}{n_{2k + 2}}$$ is a prime raised to a positive integer power. Player ${\mathcal A}$ wins the game by choosing the number 1990; player ${\mathcal B}$ wins by choosing the number 1. For which $n_0$ does : a.) ${\mathcal A}$ have a winning strategy? b.) ${\mathcal B}$ have a winning strategy? c.) Neither player have a winning strategy?