Regional Competition

We consider the set $B$ of the numbers on the blackboard at the end of the game. It is clear that $B\subseteq \{1,\dots ,2016\}$. Let $m=\min B$ and $n\in B$. We claim that $m\mid n$. Otherwise, write $n=qm+r$ with $0<r<m$. By induction on $k$, we have $n-km\in B$ for $0\le k\le q$ (because no more moves are possible, these numbers must be on the blackboard). Thus $r=n-qm\in B$, which contradicts the minimality of $m$.

We conclude that $m\mid 1=\gcd (2016,47)\in B$. By induction on $c$, we have $n-c\in B$ for $0\le c\le 2015$. This also implies that $B=\{1,\dots ,2016\}$.

As $2$ numbers had been on the blackboard at the beginning of the game, the game ends after $2014$ moves when all other numbers have been written. Therefore, Bob wins after move $2014$.