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HMMT 二月 2015 · 冲刺赛 · 第 7 题

HMMT February 2015 — Guts Round — Problem 7

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 5 ] Let C be a cube of side length 2. We color each of the faces of C blue, then subdivide it into
    2 = 8 unit cubes. We then randomly rearrange these cubes (possibly with rotation) to form a new 3
    3-dimensional cube.
    What is the probability that its exterior is still completely blue?
解析

英文解析

  1. [ 5 ] Let C be a cube of side length 2. We color each of the faces of C blue, then subdivide it into
    2 = 8 unit cubes. We then randomly rearrange these cubes (possibly with rotation) to form a new 3
    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 toone unit cube, and each has a probability of being oriented with the old outer vertex as a vertex of 1
    the new length-2 cube. Multiplying gives the answer.8