Problem:
A square is drawn in the Cartesian coordinate plane with vertices at , and . A particle starts at . Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is that the particle will move from to each of , , , or . The particle will eventually hit the square for the first time, either at one of the 4 corners of the square or at one of the 12 lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is , where and are relatively prime positive integers. What is
Answer Choices:
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Solution:
Let , let , and let . A particle
in will move to with probability , to with probability , to with probability , and to an interior point of a side of the square with probability . Similarly, a particle in will move to with probability and to with probability ; and a particle in will move to with probability , to with probability , to a corner of the square with probability , and to an interior point of a side of the square with probability . Let , and be the probabilities that the particle will first hit the square at a corner, given that it is currently in , and , respectively. The transition probabilities noted above lead to the following system of equations.
This system can be solved by elimination to yield , and . The required fraction is , whose numerator and denominator sum to .
The problems on this page are the property of the MAA's American Mathematics Competitions