16. (12 points) As shown in Figure 1, in the cube $A B C D$ $-A_{1} B_{1} C_{1} D_{1}$, $O$, $E$, $F$, $G$ are the midpoints of $B D$, $B B_{1}$, $A_{1} D_{1}$, $D_{1} C_{1}$ respectively, and $A B=1$. Find the volume of the tetrahedron $O E F G$.
16. Connect $B_{1} D_{1}$ and $F G$ intersecting at point $H$. Clearly, points $F$ and $G$ are symmetric with respect to the plane $B B_{1} D_{1} D$. Then
$$
\begin{array}{l}
V_{O-E F G}=2 V_{F-O E H}=\frac{2}{3} S_{\triangle O E H} \cdot F H \text {. } \\
\text { Also, } S_{\triangle O E H}=S_{\text {trapezoid } O B B_{1} H}-S_{\triangle E B_{1} H}-S_{\triangle O B E} \\
=\frac{5 \sqrt{2}}{8}-\frac{\sqrt{2}}{8}-\frac{3 \sqrt{2}}{16}=\frac{5 \sqrt{2}}{16} \text {. } \\
\end{array}
$$

Therefore, $V_{O-E F C}=\frac{2}{3} \times \frac{5 \sqrt{2}}{16} \times \frac{\sqrt{2}}{4}=\frac{5}{48}$.