HMMT 二月 2015 · 冲刺赛 · 第 7 题

HMMT February 2015 — Guts Round — Problem 7

[ 5 ] Let C be a cube of side length 2. We color each of the faces of C blue, then subdivide it into 3 2 = 8 unit cubes. We then randomly rearrange these cubes (possibly with rotation) to form a new 3-dimensional cube. What is the probability that its exterior is still completely blue?

解析

[ 5 ] Let C be a cube of side length 2. We color each of the faces of C blue, then subdivide it into 3 2 = 8 unit cubes. We then randomly rearrange these cubes (possibly with rotation) to form a new 3-dimensional cube. What is the probability that its exterior is still completely blue? 1 1 1 Answer: or or Each vertex of the original cube must end up as a vertex of the 24 8 2 8 16777216 new cube in order for all the old blue faces to show. There are 8 such vertices, each corresponding to 1 one unit cube, and each has a probability of being oriented with the old outer vertex as a vertex of 8 the new length-2 cube. Multiplying gives the answer.