You are throwing a fair, $6$-sided dice till $N = 7$ most recent rolls sum up to $s = 28$. Then, the game ends.
What is the expected number of throws required to end the game?
I have solved the problem numerically resolving system of equations of the expected values, conditioned by previous $N-1$ states and obtained expected number of throws $= 24.7213325$.
However, I am interested in other approaches to solve the problem, perhaps via generating functions, smartly designed absorbed Markov chains?