A token is placed on the corner square of a 3×3 chess board. The token is moved either up, down, left, or right, with equal probability. If the token would move off the edge of the board it "wraps around'' the board and moves to the opposite side instead. For example, if the token was in the bottom right corner and moves down, it would move to the top right corner. What is the expected number of moves it will take for the token to end up on the center square of the board for the first time?

10

To find the expected number of moves for the token to reach the center square of the board, we can use a Markov Chain approach.

First, let's define the states of the Markov Chain:
- S1: Starting state, where the token is on the corner square.
- S2: Transition state, where the token is on one of the four edge squares adjacent to the center square.
- S3: Final state, where the token is on the center square.

Next, we need to determine the transition probabilities between these states. Since the token moves in one of the four directions (up, down, left, right) with equal probability, the transition probabilities can be calculated as follows:

- P(S1 → S2) = 1/4, as there is an equal chance of moving in any direction from the corner square.
- P(S2 → S3) = 1/4, as there is an equal chance of moving in any direction from the transition state.
- P(S2 → S1) = 3/4, as there are three out of four possible directions that would move the token back to the corner square.

From this information, we can construct a transition matrix with the probabilities for each state transition. In this case, the transition matrix would be:

| P(S1 → S1) P(S1 → S2) P(S1 → S3) |
| P(S2 → S1) P(S2 → S2) P(S2 → S3) |
| 0 0 1 |

Next, we need to find the expected number of moves to reach the final state (S3) starting from the initial state (S1). This can be done by calculating the expected number of moves for each state.

Let E(S1) represent the expected number of moves starting from state S1 to reach the final state (S3). Similarly, let E(S2) represent the expected number of moves starting from state S2 to reach the final state.

Using the transition matrix, we can set up the following system of equations:

E(S1) = 1 + (1/4) * E(S2)
E(S2) = 1 + (1/4) * E(S3) + (3/4) * E(S1)

Since E(S3) = 0 (as it represents the final state), we can substitute this into the second equation:

E(S2) = 1 + (1/4) * 0 + (3/4) * E(S1)
E(S2) = 1 + (3/4) * E(S1)

Now, let's solve for E(S1) using these equations:

E(S1) = 1 + (1/4) * E(S2)
E(S1) = 1 + (1/4) * (1 + (3/4) * E(S1))
E(S1) = 1 + (1/4) + (3/16) * E(S1)
(13/16) * E(S1) = 1 + (1/4)
(13/16) * E(S1) = (5/4)

Simplifying further:

E(S1) = (5/4) * (16/13)
E(S1) = (80/52)
E(S1) = 1.538461538

Therefore, the expected number of moves for the token to end up on the center square for the first time is approximately 1.54 moves.