, 2010

Player $B$ does not have a winning strategy, since on the first move player $A$ can write the name $A$ on his sheet, after that the name $B$ is not on any of the sheets. Similarly player $C$ does not have a winning strategy.

To prove that even player $A$ does not have a winning strategy, we show that players $B$ and $C$ have a joint strategy which guarantees that among the names written on the sheets there are at least two different names. Namely, if player $A$ on his move writes a name on the sheet of player $B$, then $B$ writes a name on the sheet of player $A$, otherwise on the sheet of player $C$. Player $C$ always writes a name on the sheet of player $B$.

In the beginning both players $B$ and $C$ have names different from the name on the sheet of player $A$. Hence $A$ cannot win in one move. Independent of which name $A$ changes on his move, after $B$ moves, the name on the sheet of $C$ differs from the name on the sheet of $A$, and after $C$ moves, both $B$ and $C$ have names on their sheets different from the one on the sheet of $A$, as in the beginning. So the cycle repeats.

Denote the players starting from any player in the order of their turns by $X$, $Y$, and $Z$. Show that the players $Y$ and $Z$ can together always keep $X$ from winning. Indeed, $X$ can win only on his move because $Y$ and $Z$ can always play so that their move does not result immediately in $X$ winning. $X$ can win on his turn only if before his move he and somebody else have his name on their sheets. The player $Z$ cannot prevent this situation only if the same situation occurred already before his move and his sheet has the name of $X$ on it. But after $Y$ moves, then either $X$ or $Z$ has the same name on their sheets as $Y$ has, and so $Y$ can always prevent both $X$ and $Z$ having the same name on their sheets. Thus none of the three players has a winning strategy.