2017 AMC 12A Problems/Problem 22
A square is drawn in the Cartesian coordinate plane with vertices at ,
,
,
. 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
?