A board game is played on a hexagonal grid of 19 tiles. A 'traveler' token starts on the center tile. Each turn a die is rolled to determine what neighboring tile the traveler moves to (all six directions equally likely). The turn that the traveler leaves the board, the game ends. What is the expected number of turns of the game?
I have tried making a diagram for the problem ..but not sure if this is correct!!
Here is the transition graph for the game:
Here is the transition matrix (where the sequence is 0, 1, 2a, 2b, 3):
$$M = \left( \begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 1/6 & 1/3 & 1/6 & 1/6 & 0 \\ 0 & 1/6 & 0 & 1/3 & 1/2 \\ 0 & 1/3 & 1/3 & 0 & 1/3 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right)$$
Read the row as "from node..." and the column as the "to node..."