Question 38
The nonnegative integers $2000$, $17$ and $n$ are written on a blackboard. Alice and Bob play the following game: Alice begins, then they play in turns. A move consists in replacing one of the three numbers by the absolute difference of the other two. No moves are allowed where all three numbers remain unchanged. A player in turn who cannot make a legal move loses the game.
* Prove that the game will end for every number $n$.
* Who wins the game in the case $n = 2017$?
* Prove that the game will end for every number $n$.
* Who wins the game in the case $n = 2017$?