10 In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, edge $A B=6, B C=B B_{1}=$ $\sqrt{2}$, point $P$ is a moving point on line segment $B C_{1}$. Then the minimum value of $A P+P B_{1}$ is $\qquad$
(10) $5 \sqrt{2}$ Hint: As shown in the figure, rotate $\triangle B B_{1} C_{1}$ around $B C_{1}$ to be coplanar with plane $A B C_{1}$, resulting in $\triangle B B_{2} C_{1}$, then $\angle A B B_{2}=135^{\circ}$, hence
$$
\begin{aligned}
& A P+P B_{1} \\
= & A P+P B_{2} \geqslant A B_{2} \\
= & \sqrt{6^{2}+(\sqrt{2})^{2}-2 \times 6 \times \sqrt{2} \cos 135^{\circ}} \\
= & 5 \sqrt{2},
\end{aligned}
$$

Equality holds if and only if $P$ is the intersection point of $A B_{2}$ and $B C_{1}$.