I got many answers to this puzzle, and mostly all correct. This was a relatively simple puzzle for those who know algebra and probability, I do not know any way to do this one without them, and hence the younger audience it difficult to participate in this puzzle. Ones who sent correct answer included Amritansh Raghav, Aman Singla, Alok Kuchlous, Anubhav Garg and Sirisha Gadepalli.
Several of you pointed out that the answer to the puzzle is different depending on whether or not the rat has memory…that is, once it traverses down a path that does not lead to exit, will it remember or not. Accordingly the answer is different. Aman Singla was the first one to point this out.
In the scenario there is no memory, assume that E is the escape time. Given equal probability of each of the 4 exits, one can formulate the following equation:
E = (E + 8)/4 + (E + 2)/4 + (3 + 5)/4
Solving for E gives 9.
In the event, there is memory, the following equation will be true:
Escape time =
0.25 x 5 +
0.25 x 3 +
0.25 x (8 + 5/3 + 3/3 + 1/3*(2 + 5/2 + 3/2)) +
0.25 x (2 + 5/3 + 3/3 + 1/3*(8 + 5/2 + 3/2))
Solving, one would get 7 minutes 20 seconds as the answer.
Hope you enjoyed the puzzle!