Problem:
Let be a dodecagon (-gon). Three frogs initially sit at , and . At the end of each minute, simultaneously each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is , where and are relatively prime positive integers. Find .
Solution:
Define the distance between two frogs as the number of sides between them that do not contain the third frog. At any moment before the frogs stop jumping, the three distances, in nondecreasing order, can only be the following triples: , and . Let , and be the expected number of minutes from each status to the stopping time, respectively. Then , and satisfy the following system of equations.
Solving the system yields , and . Because the frogs start in configuration , the requested sum is .