PUMaC 2017 · 组合（B 组） · 第 2 题

PUMaC 2017 — Combinatorics (Division B) — Problem 2

Split a face of a regular tetrahedron into four congruent equilateral triangles. How many different ways can the seven triangles of the tetrahedron be colored using only the colors orange and black? (Two tetrahedra are considered to be colored the same way if you can rotate one so it looks like the other.)

解析

First consider unique rotations of the tetrahedral base divided into four triangles. For now, ignore the innermost triangle. Consider 2 cases: Case 1: The other three triangles on this base are the same color. There are two ways this can happen (orange or black). From here, there are four ways to color the other three sides of the tetrahedron, for 8 solutions in this case. Case 2: Two of the other three triangles of the base are the same color. There are two ways 3 this can happen, and there are 2 = 8 ways to color the other three sides of the tetrahedron, for 16 solutions in this case. Hence, there are 24 ways to color these 6 triangles. Lastly, there are 2 ways to color the innermost triangle for a total of 2 · 24 = 48 solutions. Problem written by Matt Tyler