HMMT 十一月 2010 · 团队赛 · 第 3 题

英文解析

3. [ 3 ] While Travis is having fun on cubes, Sherry is hopping in the same manner on an octahedron. Anoctahedron has six vertices and eight regular triangular faces. After five minutes, how likely is Sherryto be one edge away from where she started?
   Answer: Let the starting vertex be the ’bottom’ one. Then there is a ’top’ vertex, and 411
   ’middle’ ones. If p ( n ) is the probability that Sherry is on a middle vertex after n minutes, p (0) = 0,16
   Team Roundp ( n + 1) = (1 − p ( n )) + p ( n ) · . This recurrence gives us the following equations.1
   p ( n )2
   p ( n + 1) = 1 −
   p (0) = 02
   p (1) = 1
   p (2) =1
   p (3) =32
   p (4) =54
   p (5) =118
   16