HMMT 二月 2008 · COMB 赛 · 第 1 题
HMMT February 2008 — COMB Round — Problem 1
题目详情
英文原题
- [ 3 ] A 3 × 3 × 3 cube composed of 27 unit cubes rests on a horizontal plane. Determine the numberof 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
英文解析
(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
and x = 2 are made independently, we see that the total number of possibilities is 45 = 2025.2
Remark: The solution could be guided by the following diagram. Set A is above B and connected to Bif and only if A ⊂ B . Such diagrams are known as Hasse diagrams , which are used to depict partiallyordered sets .
8A
A A
4 6
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A A A A
5 P 7 2 3
P B |PBP
B |P
P |BP
|BP
P |BP
|BP
P |BP
|P
1AP