  We prove the following result which is due to the third author. Let $n \geq
1$. If $\boldsymbol\Pi^1_n$ determinacy and $\Pi^1_{n+1}$ determinacy both hold
true and there is no $\boldsymbol\Sigma^1_{n+2}$-definable $\omega_1$-sequence
of pairwise distinct reals, then $M_n^\#$ exists and is $\omega_1$-iterable.
The proof yields that $\boldsymbol\Pi^1_{n+1}$ determinacy implies that
$M_n^\#(x)$ exists and is $\omega_1$-iterable for all reals $x$. A consequence
is the Determinacy Transfer Theorem for arbitrary $n \geq 1$, namely the
statement that $\boldsymbol\Pi^1_{n+1}$ determinacy implies
$\Game^{(n)}(<\omega^2 - \boldsymbol\Pi^1_1)$ determinacy.
