HMMT 二月 2006 · 冲刺赛 · 第 8 题
HMMT February 2006 — Guts Round — Problem 8
题目详情
- [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 that are relatively prime? Physically realizable rotations are considered indistinguishable, but physically unrealizable reflections are considered different.
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 that are 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 label the vertices of a cube. Then, as no two even numbers may be adjacent, the even numbers better form a regular tetrahedron, which can be done in 2 ways (because 2 rotations are indistiguishable but reflections are different). Then 3 must be opposite 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.