HMMT 十一月 2010 · 团队赛 · 第 1 题

HMMT November 2010 — Team Round — Problem 1

[ 3 ] Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he’s currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started?

解析

[ 3 ] Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he’s currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started? 7 Answer: Let the cube have vertices all 0 or 1 in the x, y, z, coordinate system. Travis starts at 27 (0 , 0 , 0). If after 3 moves he is at (1 , 1 , 1) he cannot get back to (0 , 0 , 0). From any other vertex he has 1 2 7 a chance of getting back on the final move. There is a chance he ends up at (1 , 1 , 1), and thus a 3 9 9 7 chance he does not end up there, and thus a chance he ends up at (0 , 0 , 0). 27