Question 160, As shown in the figure, the edge length of the cube $A B C D-E F G H$ is 2. On the incircle of the square $A B F E$, take any point $P_{1}$; on the incircle of the square $B C G F$, take any point $P_{2}$; on the incircle of the square $E F G H$, take any point $P_{3}$. Find the minimum and maximum values of $\left|P_{1} P_{2}\right|+\left|P_{2} P_{3}\right|+\left|P_{3} P_{1}\right|$.
Question 160, Solution: As shown in the figure, with the center of the cube as the origin, the DA direction as the positive direction of the $x$-axis, the $\overrightarrow{\mathrm{DC}}$ direction as the positive direction of the $x$-axis, and the $\overline{\mathrm{DH}}$ direction as the positive direction of the $z$-axis, a spatial rectangular coordinate system is established. Let $\mathrm{P}_{1}\left(1, \cos \alpha_{1}, \sin \alpha_{1}\right), \mathrm{P}_{2}\left(\sin \alpha_{2}, 1, \cos \alpha_{2}\right), \mathrm{P}_{3}\left(\cos \alpha_{3}, \sin \alpha_{3}, 1\right)$ (where the subscript is taken modulo 3). Then
$$
\begin{array}{l}
\left|\mathrm{P}_{1} \mathrm{P}_{2}\right|+\left|\mathrm{P}_{2} \mathrm{P}_{3}\right|+\left|\mathrm{P}_{3} \mathrm{P}_{1}\right| \\
=\sum_{i=1}^{3} \sqrt{\left(1-\sin \alpha_{i+1}\right)^{2}+\left(\cos \alpha_{i}-1\right)^{2}+\left(\sin \alpha_{i}-\cos \alpha_{i+1}\right)^{2}} \\
\text { On the one hand, according to the quadratic mean inequality, we have: } \\
\left|\mathrm{P}_{1} \mathrm{P}_{2}\right|+\left|\mathrm{P}_{2} \mathrm{P}_{3}\right|+\left|\mathrm{P}_{3} \mathrm{P}_{1}\right| \\
\geq \sum_{i=1}^{3} \sqrt{\frac{\left(\left(1-\sin \alpha_{i+1}\right)+\left(1-\cos \alpha_{i}\right)^{2}\right.}{2}} \\
=\frac{\sum_{i=1}^{3}\left(2-\sin \alpha_{i+1}-\cos \alpha_{i}\right)}{\sqrt{2}} \\
=\frac{6-\sum_{i=1}^{3}\left(\sin \alpha_{i}+\cos \alpha_{i}\right)}{\sqrt{2}} \\
\geq \frac{6-3 \sqrt{2}}{\sqrt{2}}=3 \sqrt{2}-3
\end{array}
$$

When $\alpha_{1}=\alpha_{2}=\alpha_{3}=\frac{\pi}{4}$, the equality holds.
On the other hand, according to the power mean inequality, we have:
$$
\begin{array}{l}
\left|P_{1} P_{2}\right|+\left|P_{2} P_{3}\right|+\left|P_{3} P_{1}\right| \\
\leq \sqrt{\sum_{i=1}^{3}\left(\left(1-\sin \alpha_{i+1}\right)^{2}+\left(\cos \alpha_{i}-1\right)^{2}+\left(\sin \alpha_{i}-\cos \alpha_{i+1}\right)^{2}\right)} \\
=\sqrt{12-2 \sum_{1=1}^{3} \sin \alpha_{i}-2 \sum_{i=1}^{3} \cos \alpha_{1}-2 \sum_{i=1}^{3} \sin \alpha_{i} \cos \alpha_{i+1}} \\
=\sqrt{18-2 \sum_{i=1}^{3}\left(1+\sin \alpha_{1}\right)\left(1+\cos \alpha_{i+1}\right)} \\
\leq \sqrt{18}=3 \sqrt{2}
\end{array}
$$

When $\alpha_{1}=\alpha_{2}=\alpha_{3}=\pi$, the equality holds.
In summary, the minimum value of $\left|\mathrm{P}_{1} \mathrm{P}_{2}\right|+\left|\mathrm{P}_{2} \mathrm{P}_{3}\right|+\left|\mathrm{P}_{3} \mathrm{P}_{1}\right|$ is $3 \sqrt{2}-3$, and the maximum value is $3 \sqrt{2}$.