  In recent years, second-order topological insulators have been proposed as a
new class of topological insulators. Second-order topological insulators are
materials with gapped bulk and surfaces, but with topologically protected
gapless states at the intersection of two surfaces. These gapless states are
called hinge states. In this paper, we give a general proof that any insulators
with inversion symmetry and gapped surface in class A always have hinge states
when the $\mathbb{Z}_{4}$ topological index $\mu_{1}$ is $\mu_{1}=2$. We
consider a three-dimensional insulator whose boundary conditions along two
directions change by changing the hopping amplitudes across the boundaries. We
study behaviors of gapless states through continuously changing boundary
conditions along the two directions, and reveal that the behaviors of gapless
states result from the $\mathbb{Z}_{4}$ strong topological index. From this
discussion, we show that gapless states inevitably appear at the hinge of a
three-dimensional insulator with gapped surfaces when the strong topological
index is $\mathbb{Z}_{4}=2$ and the weak topological indices are
$\nu_{1}=\nu_{2}=\nu_{3}=0$.
