1990 AMC 8 第 25 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
25.
在一个 方格中，恰好给九个单位正方形中的两个涂色，可以形成多少种不同图案？通过翻转或旋转可以重合的图案不算不同。
How many different patterns can be made by shading exactly two of the nine unit squares in a grid? Patterns that can be matched by flips and/or turns are not considered different.
小提示：
九个小方格可分成三类： 个角格、 个边中格和 个中心格；用两格的类型和距离描述图案
The nine squares come in three types: corners, edge-middles, and center; describe each pattern by which types its two shaded squares use and how far apart they are
大提示：
按是否有角格被涂色分类，注意翻转和旋转会把许多位置视为同一种图案
Split into cases by whether a corner is shaded, being careful that flips and turns merge many placements into one pattern
解答：
先分类含有角格的图案：两个相邻角格、两个对角角格、角格与中心格、角格与相邻边中格、角格与较远边中格，共 种。
不含角格的图案有：两个相邻边中格、两个相对边中格、一个边中格与中心格，共 种。
总共有 种不同图案。
所以正确答案是 C。
Classify the two shaded squares up to rotations and reflections. Patterns that include a corner: two adjacent corners (same edge), two diagonally opposite corners, corner with the center, corner with an adjacent edge-middle, and corner with a far edge-middle. That is patterns.
Patterns with no corner: two adjacent edge-middles, two opposite edge-middles, and an edge-middle with the center. That is more.
In total there are distinct patterns.
Thus, the correct answer is C .
其他年份的第 25 题
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