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 $n+1$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.