We characterize, in terms of determinacy, the existence of $0^{\sharp\sharp}$ as well as the existence of each of the following: $0^{\sharp\sharp\sharp}, 0^{\sharp\sharp\sharp\sharp},0^{\sharp\sharp\sharp\sharp\sharp}, \ldots$. For $k \in \omega$, we define two classes of sets, $(k \ast \Sigma^0_1)^\ast$ and $(k \ast \Sigma^0_1)^\ast_+$, which lie strictly between $\bigcup_{\beta < \omega^2}(\beta-\Pi^1_1)$ and $\Delta(\omega^2-\Pi^1_1)$. We also define $0^{1\sharp}$ as $0^\sharp$ and in general, $0^{(k + 1)\sharp}$ as $(0^{k\sharp)^\sharp}$. We then show that the existence of $0^{(k + 1)\sharp}$ is equivalent to the determinacy of $((k + 1) \ast \Sigma^0_1)^\ast$ as well as the determinacy of $(k \ast \Sigma^0_1)^\ast_+$.