HMMT 十一月 2008 · 冲刺赛 · 第 20 题

HMMT November 2008 — Guts Round — Problem 20

个人笔记

[ 11 ] You have a die with faces labelled 1 through 6. On each face, you draw an arrow to an adjacentface, such that if you start on a face and follow the arrows, after 6 steps you will have passed throughevery face once and will be back on your starting face. How many ways are there to draw the arrowsso that this is true?

解析

[ 11 ] You have a die with faces labelled 1 through 6. On each face, you draw an arrow to an adjacentface, such that if you start on a face and follow the arrows, after 6 steps you will have passed throughevery face once and will be back on your starting face. How many ways are there to draw the arrowsso that this is true?
Answer: 32 There are 4 choices for where to go from face 1. Consider the 4 faces adjacent to 1.
We can visit either 1, 2, or 3 of them before visiting the face opposite 1. If we only visit one of theseadjacent faces, we have 4 choices for which one, then we visit face 6, opposite face 1, then we visit theremaining 3 faces in one of two orders - for a total of 8 ways. If we visit 2 adjacent faces first, there is
8 choices for these two faces, then 2 choices for the path back from face 6 to face 1. Lastly, there are 8
ways to visit three of the adjacent faces before visiting the opposite face. These choices give 32 total.