HMMT 二月 2006 · 冲刺赛 · 第 8 题
HMMT February 2006 — Guts Round — Problem 8
题目详情
英文原题
- [6] How many ways are there to label the faces of a regular octahedron with the integers
1–8, using each exactly once, so that any two faces that share an edge have numbers thatare relatively prime? Physically realizable rotations are considered indistinguishable, butphysically unrealizable reflections are considered different.
解析
英文解析
- How many ways are there to label the faces of a regular octahedron with the integers 1–
8, using each exactly once, so that any two faces that share an edge have numbers thatare relatively prime? Physically realizable rotations are considered indistinguishable,
but physically unrealizable reflections are considered different.
Answer: 12
Solution: Well, instead of labeling the faces of a regular octahedron, we may labelthe vertices of a cube. Then, as no two even numbers may be adjacent, the evennumbers better form a regular tetrahedron, which can be done in 2 ways (becauserotations are indistiguishable but reflections are different). Then 3 must be opposite 2
6, and the remaining numbers — 1, 5, 7 — may be filled in at will, in 3! = 6 ways.
The answer is thus 2 × 6 = 12.