Question 54
Players $A$, $B$ and $C$ are playing the following game. Initially, the number $1$ is written on a blackboard. On their move, each player replaces the number $n$ currently on the blackboard with either $n+1$, $7n+7$, or $4n^3 + 3n + 4$ at their own choice, under the condition that the new number must not be larger than $10^9$. The player $A$ makes the first move, then $B$ takes turn, then $C$, after him $A$ again etc., until some player cannot make a legal move. The player who makes the last move wins. Can any of the players win the game against every legal play by the opponents and if yes then who?