TxO Math Bowl 2024 - Team Contest - Problem 16

Cities A, B, C, D, W, and L are located in the country of Portaland. A one-way magic portal exists in each of the cities A, B, C, and D. A portal code shows a portal's destinations and their corresponding probability. The format of portal codes is Portal M: (N, P, Q), which represents that by taking the portal in city M, the probability of travelling to cities N, P, Q are \frac{1}{2}, \frac{1}{3}, and \frac{1}{6}, respectively. The portal codes of the four portals are as follows:

• Portal A: (B, C, D)
• Portal B: (C, W, L)
• Portal C: (D, W, L)
• Portal D: (A, W, L)

If Mike starts in city A and continually takes portals until he reaches a city without a portal, the probability that he ends up in city W can be expressed as \frac{x}{y}, where x and y are relatively prime positive integers. Compute x + y.