A spider has its web in the shape of a regular hexagon. A fly is stuck to the web at a vertex that is diametrically opposite from the vertex at which the spider is. The spider can walk freely along the edges of the hexagon. At each vertex, it randomly chooses between walking on one of the two adjacent edges or staying at the vertex, all three choices with equal probability. If the time it takes to travel an edge is 5 seconds, while the waiting time at a vertex is 2 seconds, find the expected time it will take the spider to get to the fly?

Note: At the end of a waiting period at a vertex, a new random decision to stay at the vertex, or move along one of the two edges is made, with equal probability for the three choices.

(In reply to re: my solution by Charlie)

The expected values of the different starting points can be computed once one is known:
E3 = 6 + E2
E2 = 6 + (1/2)(E3 + E1)
E1 = 6 + (1/2)(E2 + E0)
where E0 = 0 (unless the spider has to decide whether or not to eat the fly)

So if E3 = 54, then E2 = 48 and E1 = 30
in agreement with the Charlie's results.

Posted by Larry on 2019-09-13 07:00:10