HMMT 二月 2002 · 冲刺赛 · 第 4 题

HMMT February 2002 — Guts Round — Problem 4

[4] How many ways are there of using diagonals to divide a regular 6-sided polygon into triangles 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

How many ways are there of using diagonals to divide a regular 6-sided polygon into triangles 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 1 Solution: The number of ways of triangulating a convex ( n +2)-sided polygon is , n n +1 which is 14 in this case. However, there are two triangulations of a hexagon which produce one triangle sharing no sides with the original polygon, so the answer is 14 − 2 = 12 .