My friend Mickey asked me to play the following game with him.
The number $1$ is written on the whiteboard. Me and Mickey take turns to do the following, starting with me.
If the number written on the board is $n$, then replace it by $n+d\le2019$ where $d|n$. The player who can not replace the number written on the whiteboard losses.
But I always lose. Actually, does Mickey has a winning strategy?
Any help is appreciated!
Note that if $n$ is odd then $d$ is odd and $n+d$ is always even. On the other hand, if $n$ is even then $1$ is a divisor of $n$ and $n+1$ is odd. Therefore it should follow that $n$ is a winning position if and only if $n$ is even. Starting from $1$, the second player has always a winning strategy: replace the current (even) number $n$ with $n+1$.