Question 179
Given positive integers $a, b, c, p, q, r$, where $p, q, r \ge 2$. Let $Q = \{(x, y, z) \in \mathbb{Z}^3 \mid 0 \le x \le a, 0 \le y \le b, 0 \le z \le c\}$ be the game board.
Initially, put $M$ game pieces on $Q$ (no restriction on the number of pieces in each position). There are three types of legal moves:
(1) Remove $p$ pieces from $(x, y, z)$ and place one piece at $(x-1, y, z) \in Q$;
(2) Remove $q$ pieces from $(x, y, z)$ and place one piece at $(x, y-1, z) \in Q$;
(3) Remove $r$ pieces from $(x, y, z)$ and place one piece at $(x, y, z-1) \in Q$.
Find the minimum value of $M$, such that no matter how the $M$ game pieces are placed initially, one can always take a sequence of moves to get one piece placed at $(0, 0, 0)$.
Initially, put $M$ game pieces on $Q$ (no restriction on the number of pieces in each position). There are three types of legal moves:
(1) Remove $p$ pieces from $(x, y, z)$ and place one piece at $(x-1, y, z) \in Q$;
(2) Remove $q$ pieces from $(x, y, z)$ and place one piece at $(x, y-1, z) \in Q$;
(3) Remove $r$ pieces from $(x, y, z)$ and place one piece at $(x, y, z-1) \in Q$.
Find the minimum value of $M$, such that no matter how the $M$ game pieces are placed initially, one can always take a sequence of moves to get one piece placed at $(0, 0, 0)$.