HMMT 二月 2008 · COMB 赛 · 第 1 题
HMMT February 2008 — COMB Round — Problem 1
题目详情
- [ 3 ] A 3 × 3 × 3 cube composed of 27 unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a 3 × 3 × 1 block (the order is irrelevant) with the ◦ property that the line joining the centers of the two cubes makes a 45 angle with the horizontal plane.
解析
(1) whether x ∈ A ; (2) whether x ∈ A ; (3) whether x ∈ A , and if yes, whether x ∈ A ; (4) 5 7 3 6 whether x ∈ A , and if yes, whether x ∈ A . There are 2 × 2 × 3 × 3 = 36 choices here. 4 8 Therefore, there are 1 + 8 + 36 = 45 ways to place x into some of the sets. Since the choices for x = 1 2 and x = 2 are made independently, we see that the total number of possibilities is 45 = 2025. Remark: The solution could be guided by the following diagram. Set A is above B and connected to B if and only if A ⊂ B . Such diagrams are known as Hasse diagrams , which are used to depict partially ordered sets . A 8 A A 4 6 | | | | | | | | A A A A 5 P 7 2 3 P B P P B | P B | P B P | P B | P B P | P B | P B P | P | P A 1