  Given an integer $1\leq j <n$, define the $(j)$-coloring of a $n$-dimensional
hypercube $H_{n}$ to be the $2$-coloring of the edges of $H_{n}$ in which all
edges in dimension $i$, $1\leq i \leq j$, have color $1$ and all other edges
have color $2$. Cheng et al.
  [Proper distance in edge-colored hypercubes, Applied Mathematics and
Computation 313 (2017) 384-391.] determined the number of distinct shortest
properly colored paths between a pair of vertices for the $(1)$-colored
hypercubes. It is natural to consider the number for $(j)$-coloring, $j\geq 2$.
In this note, we determine the number of different shortest proper paths in
$(j)$-colored hypercubes for arbitrary $j$.
