2010 PUMaC Combinatorics B6: exploding ants in pentagon
Source:
August 21, 2011
probabilityexpected value
Problem Statement
A regular pentagon is drawn in the plane, along with all its diagonals. All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded. An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of the region it's in, with an equal probability of choosing each edge, and crosses that edge into another region. If the ant enters an unbounded region, it explodes. After first leaving the central region of the pentagon, let be the expected number of times the ant re-enters the central region before it explodes. Find the closest integer to .