PUMaC 2016 · 组合（B 组） · 第 3 题

PUMaC 2016 — Combinatorics (Division B) — Problem 3

Chitoge is painting a cube; she can paint each face either black or white, but she wants no vertex of the cube to be touching three faces of the same color. In how many ways can Chitoge paint the cube? Two paintings of a cube are considered to be the same if you can rotate one cube so that it looks like the other cube.

解析

(ID 070) Pick a vertex of the cube. Suppose two faces that meet at that vertex are black and the other is white. The face opposite the white face is also white. Among the other two faces, not both are white. We thus get two possibilities: either one face of the other two is black, in which case the black faces form a “strip” of length 3, as do the white faces, or we have two opposite white faces and the other faces are black. The other possibility is that the two faces that meet at that vertex are white and the other is black. This gives the “strip” case again, as well as the case where we have two opposite black faces and the other faces are white. Thus, Chitoge can paint the cube in 3 ways. Problem written by Bill Huang.