HMMT 二月 2013 · 冲刺赛 · 第 9 题

HMMT February 2013 — Guts Round — Problem 9

[ 6 ] I have 8 unit cubes of different colors, which I want to glue together into a 2 × 2 × 2 cube. How many distinct 2 × 2 × 2 cubes can I make? Rotations of the same cube are not considered distinct, but reflections are.

解析

[ 6 ] I have 8 unit cubes of different colors, which I want to glue together into a 2 × 2 × 2 cube. How many distinct 2 × 2 × 2 cubes can I make? Rotations of the same cube are not considered distinct, but reflections are. Answer: 1680 Our goal is to first pin down the cube, so it can’t rotate. Without loss of generality, suppose one of the unit cubes is purple, and let the purple cube be in the top left front position. Now, ( ) 7 look at the three positions that share a face with the purple cube. There are ways to pick the three 3 cubes that fill those positions and two ways to position them that are rotationally distinct. Now, we’ve taken care of any possible rotations, so there are simply 4! ways to position the final four cubes. Thus, ( ) 7 our answer is · 2 · 4! = 1680 ways. 3