  Multiple Hook Removing Game (MHRG for short) is an impartial game played in
terms of Young diagrams. In this paper, we give a characterization of the set
of all game positions in MHRG. As an application, we prove that for $t \in
\mathbb{Z}_{\geq 0}$ and $m, n \in \mathbb{N}$ such that $t \leq m \leq n$, and
a Young diagram $Y$ contained in the rectangular Young diagram $Y_{t,n}$ of
size $t \times n$, $Y$ is a game position in MHRG with $Y_{m,n}$ the starting
position if and only if $Y$ is a game position in MHRG with $Y_{t,n-m+t}$ the
starting position, and also that the Grundy value of $Y$ in the former MHRG is
equal to that in the latter MHRG.
