[Ed Miller]

Ok... I got a slightly different answer from the other folks.

So, you are looking for the chance that you will return to the original square from a square that is labeled 1. I will call this P(1).

You will move to a 2 square with only one way back 1/4 of the time, to a 2 square with two ways back 1/2 of the time, and you will reach the honey pot 1/4 of the time. Thus,

P(1) = 1/4 + 1/4 * P(2_1) + 1/2 * P(2_2)

where P(2_1) is the probability of returning to S from the 2 square with only one way back, and P(2_2) is the probability of returning to S from the 2 squares with two ways back.

We also know that the chances of returning from the 2 squares are, respectively,

P(2_1) = 1/4 * P(1)
P(2_2) = 1/2 * P(1)

Thus,

P(1) = 1/4 + 1/4 * (1/4 * P(1)) + 1/2 * (1/2 * P(1))
P(1) = 1/4 + 1/16 * P(1) + 1/4 * P(1)
P(1) = 1/4 + 5/16 * P(1)
11/16 * P(1) = 1/4
P(1) = 4/11

Since you move initially to a 1 square from S every time, the probability you are looking for is just P(1). Thus, my answer is 4/11.