AMC 8 1990 Problem 25
AMC8 1990 年第 25 题
排列与组合★★★★☆
How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different.
通过恰好给九个正方形中的两个着色，可以形成多少种不同的图案？通过翻转和/或旋转能重合的图案不算不同。例如，下面显示的图案不算不同。
- A.3
- B.6
- C.8
- D.12
- E.18
正确答案 · Correct Answer
C
解析 · Solution
解法1
常规解法 · 分类计数
在翻转/旋转下分类。若有阴影格在角上：由对称固定左上角，另一格给出 5 种不同位置。若有阴影格在边上、但都不在角：边-边 2 种、边-中 1 种，共 3 种。合计 5+3=8。
【进阶】Burnside 引理（大学群论工具，超出 AMC8 范围，了解即可）
对正方形的 8 个对称操作各数被固定的「两格涂色」：恒等固定 ；两个 旋转各 0； 旋转 4；4 条对称轴反射各 6。模式数
故共 8 种模式。
Solution 1
Elementary · count by cases
Count by cases up to flips and turns. If a shaded square is a corner, fix the top-left corner by symmetry; the other shaded square gives 5 distinct positions. If a shaded square is an edge but none is a corner: 2 edge-edge plus 1 edge-center, giving 3. In total 5+3=8.
[Advanced] Burnside's lemma (a college-level group-theory tool, beyond AMC8)
By Burnside over the 8 symmetries of the square, count the two-cell shadings fixed by each: identity fixes ; the two rotations fix 0; the rotation fixes 4; each of the 4 reflections fixes 6. The number of patterns is
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