Three players, $A$, $B$ and $C$, repeatedly toss a standard six-sided die. If the die lands 1 or 2 on a particular toss, then $B$ and $C$ each give one dollar to $A$. If the die lands 3 or 4 on a particular toss, then $A$ and $C$ each give one dollar to $B$. If the die lands 5 or 6 on a particular toss, then $A$ and $B$ each give one dollar to $C$. Players $A$, $B$ and $C$ start the game with respective numbers of dollars $a$, $b$ and $c$. Let $X_n, Y_n$ and $Z_n$ denote the respective amounts of money that players $A,B$, and $C$ have after the $n$-th toss, $n \geq 1$, and let $X_0 = a$, $Y_0 = b$, and $ Z_0 = c$.
a) I'm trying to prove that $W_n = X_nY_nZ_n + n(a + b + c - 2)$ is a martingale.
b) And the expected # of tosses required to finish the game.
For part a, we're essentially looking for a fair game where no knowledge of past events can help to predict future winnings. So we can condition on what?