HMMT 二月 2002 · 冲刺赛 · 第 4 题
HMMT February 2002 — Guts Round — Problem 4
题目详情
英文原题
- [4] How many ways are there of using diagonals to divide a regular 6-sided polygon intotriangles such that at least one side of each triangle is a side of the original polygon and that each vertex of each triangle is a vertex of the original polygon?
解析
英文解析
- How many ways are there of using diagonals to divide a regular 6-sided polygon intotriangles such that at least one side of each triangle is a side of the original polygon and that each vertex of each triangle is a vertex of the original polygon?
( )
2 n
Solution: The number of ways of triangulating a convex ( n +2)-sided polygon is ,1
n n +1
which is 14 in this case. However, there are two triangulations of a hexagon which produceone triangle sharing no sides with the original polygon, so the answer is 14 − 2 = 12 .