= Solution
For a covariant rank-two tensor,
$$
\nabla_\alpha g_{\mu\nu}
=\partial_\alpha g_{\mu\nu}
-\Gamma^\rho_{\alpha\mu}g_{\rho\nu}
-\Gamma^\rho_{\alpha\nu}g_{\mu\rho}.
$$
Impose <metric compatibility> $\nabla g=0$, write this equation and its two cyclic permutations in $\alpha,\mu,\nu$, and use the assumed torsion-free symmetry of the connection. Solving gives the unique Levi--Civita connection
$$
\boxed{
\Gamma^\gamma_{\alpha\beta}
=\frac12g^{\gamma\delta}
\left(\partial_\alpha g_{\beta\delta}
+\partial_\beta g_{\alpha\delta}
-\partial_\delta g_{\alpha\beta}\right).}
$$
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