Question 155
Alice and Brian play a game. Before they start, Alice chooses a positive integer $n$ and Brian then chooses the initial value of another positive integer $m$. They then place $n$ counters on a board and take turns removing them; Alice always moves first.
Each move, for some player $P$, always has two parts:
(a) $P$ either lets $m$ unchanged or reduces its value by $1$, with the exception that if $m = 1$, then $m$ cannot be changed.
(b) $P$ then removes $m$ counters from the board.
Eventually, there are too few counters left for $P$ to do (b) and $P$ loses. The winner depends on $n$, the initial $m$, and the moves made. For instance, suppose $n = 11$ and initially $m = 5$. Alice should first remove $4$ counters. Brian then removes either $4$ or $3$ counters, after which Alice should remove $3$ counters. With at most one counter now on the table, Brian has no valid move so Alice wins. If instead, Alice initially removed $5$ counters, it is easily verified that Alice is doomed to lose.
Alice and Brian agree to constrain $n$ and the initial $m$: Alice must choose $n \ge 2024$, and Brian is only allowed to choose $m = 2, m = 3$, or $m = 4$. Find the smallest $n$ that Alice can choose to guarantee her a win if she makes the right moves, regardless of Brian's choice of $m$ and his moves.
Each move, for some player $P$, always has two parts:
(a) $P$ either lets $m$ unchanged or reduces its value by $1$, with the exception that if $m = 1$, then $m$ cannot be changed.
(b) $P$ then removes $m$ counters from the board.
Eventually, there are too few counters left for $P$ to do (b) and $P$ loses. The winner depends on $n$, the initial $m$, and the moves made. For instance, suppose $n = 11$ and initially $m = 5$. Alice should first remove $4$ counters. Brian then removes either $4$ or $3$ counters, after which Alice should remove $3$ counters. With at most one counter now on the table, Brian has no valid move so Alice wins. If instead, Alice initially removed $5$ counters, it is easily verified that Alice is doomed to lose.
Alice and Brian agree to constrain $n$ and the initial $m$: Alice must choose $n \ge 2024$, and Brian is only allowed to choose $m = 2, m = 3$, or $m = 4$. Find the smallest $n$ that Alice can choose to guarantee her a win if she makes the right moves, regardless of Brian's choice of $m$ and his moves.