HMMT 二月 2006 · TEAM1 赛 · 第 12 题

英文解析

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 remainingsolid? Prove your answer.
    Answer: 50
    Solution: Orient the cube so that its edges are parallel to the x -, y -, and z -axes. Aset of three unit cubes whose centers differ only in their x -coordinate will be termedan “ 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 ofprecisely five unit cubes).
    To show that 50 is the minimum: Note that there cannot be two x -rows that areboth completely removed, as that would imply removing six unit cubes. (Similarstatements apply for y - and z -rows, of course.) It is also impossible for there to be onex -row, one y -row, and one z -row that are all removed, as that would imply removingseven unit cubes. Every x -, y -, or z -row that is not completely removed contributesat least 2 square units to the surface area. Thus, the total surface area is at least
    9 · 2 + 8 · 2 + 8 · 2 = 50 .