"Very Frustrating Game" has six levels. When a level is attempted, the player goes to the next level if they succeed, but back to the previous level if they fail (or if they are on level 1 they restart).

a) Aerith has a $\frac{1}{2}$ success rate on all levels. How many level attempts on average would it take her to complete the game?

b) Bob has a $\frac{1}{3}$ success rate on all levels. How many level attempts on average would it take him to complete the game?

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a) Let $A(x)=42-\left(x+x^{2}\right)$. One can check that $A(x)=1+\frac{1}{2} A(x+1)+\frac{1}{2} A(x-1)$, that $A(0)=A(-1)$, and that $A(6)=0$. The answer when starting on level $n+1$ is therefore $A(n)$. Plugging in $n=0$ gives 42 .

b) Let $B(x)=360-3\left(2\left(2^{x}-1\right)-x\right)$. One can check that $B(x)=1+\frac{1}{3} B(x+1)+$ $\frac{2}{3} B(x-1)$, that $B(0)=B(-1)$, and that $B(6)=0$. The answer when starting on level $n+1$ is therefore $B(n)$. Plugging in $n=0$ gives 360 .