HMMT 十一月 2010 · 团队赛 · 第 3 题

HMMT November 2010 — Team Round — Problem 3

[ 3 ] While Travis is having fun on cubes, Sherry is hopping in the same manner on an octahedron. An octahedron has six vertices and eight regular triangular faces. After five minutes, how likely is Sherry to be one edge away from where she started?

解析

[ 3 ] While Travis is having fun on cubes, Sherry is hopping in the same manner on an octahedron. An octahedron has six vertices and eight regular triangular faces. After five minutes, how likely is Sherry to be one edge away from where she started? 11 Answer: Let the starting vertex be the ’bottom’ one. Then there is a ’top’ vertex, and 4 16 ’middle’ ones. If p ( n ) is the probability that Sherry is on a middle vertex after n minutes, p (0) = 0, Team Round 1 p ( n + 1) = (1 − p ( n )) + p ( n ) · . This recurrence gives us the following equations. 2 p ( n ) p ( n + 1) = 1 − 2 p (0) = 0 p (1) = 1 1 p (2) = 2 3 p (3) = 4 5 p (4) = 8 11 p (5) = 16