HMMT 十一月 2017 · GEN 赛 · 第 7 题

HMMT November 2017 — GEN Round — Problem 7

Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces and eight equilateral triangular faces. How many faces are on the polyhedron after the fifth step? 8

解析

Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces and eight equilateral triangular faces. How many faces are on the polyhedron after the fifth step? Proposed by: Qi Qi Answer: 974 Notice that the number of vertices and edges triple with each step. We always have 3 edges meeting at one vertex, and slicing off a pyramid doesn’t change this (we make new vertices from which one edge from the previous step and two of the pyramid edges emanate). So at each step we replace the sliced-off vertex with three new vertices, and to each edge we create four new “half-edges” (two from the pyramid at each endpoint), which is equivalent to tripling the number of vertices and edges. Then, 5 5 by Euler’s Theorem the number of faces is E − V + 2 = 12 · 3 − 8 · 3 + 2 = 974. 8