Question 199
Three players $A$, $B$ and $C$ play the following game. At the beginning of the game, each player has a sheet of paper with the name of the player written on it. Player $A$ chooses one of the other players and replaces the name on this player's sheet with the name on his own sheet. Then player $B$ makes a similar move, then player $C$ and after that the turn to move goes to player $A$ again. The game ends when all the sheets have the same name written on them and the winner is the player whose name it is. Does any of the players have a winning strategy (i.e., a strategy that allows a player to win no matter what his opponents play)? (Grade 10.)