HMMT 二月 2006 · TEAM1 赛 · 第 12 题

HMMT February 2006 — TEAM1 Round — Problem 12

[25] A 3 × 3 × 3 cube is built from 27 unit cubes. Suddenly five of those cubes mysteriously teleport away. What is the minimum possible surface area of the remaining solid? Prove your answer.

解析

[25] A 3 × 3 × 3 cube is built from 27 unit cubes. Suddenly five of those cubes mys- teriously teleport away. What is the minimum possible surface area of the remaining solid? Prove your answer. Answer: 50 Solution: Orient the cube so that its edges are parallel to the x -, y -, and z -axes. A set of three unit cubes whose centers differ only in their x -coordinate will be termed an “ x -row”; there are thus nine x -rows. Define “ y -row” and “ z -row” similarly. To achieve 50, simply take away one x -row and one y -row (their union consists of precisely five unit cubes). To show that 50 is the minimum: Note that there cannot be two x -rows that are both completely removed, as that would imply removing six unit cubes. (Similar statements apply for y - and z -rows, of course.) It is also impossible for there to be one x -row, one y -row, and one z -row that are all removed, as that would imply removing seven unit cubes. Every x -, y -, or z -row that is not completely removed contributes at least 2 square units to the surface area. Thus, the total surface area is at least 9 · 2 + 8 · 2 + 8 · 2 = 50 .