2000 AMC 12 第 25 题
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25.
八个全等的等边三角形，每个颜色都不同，被用来构造一个正八面体。构造这个八面体有多少种可区分的方法？（如果两个着色八面体不能通过旋转变得完全一样，则称它们可区分。）
Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)
小提示：
在考虑旋转之前，有 种方式把八种不同颜色放到八个面上。
There are ways to place the eight distinct colors before accounting for rotations
大提示：
八面体的旋转群有 个元素，并且颜色都不同时，每种着色的稳定子都是平凡的。
The rotation group of the octahedron has elements, and distinct colors give each arrangement a trivial stabilizer
解答：
将八种不同颜色分配给八个面共有 种方式。两个分配表示同一个八面体，当且仅当其中一个可以旋转成另一个。
正八面体的旋转群有 个元素。因为八种颜色全都不同，没有非平凡旋转会固定一种着色，所以每个可区分的八面体恰好对应 种分配。
因此可区分的八面体数量为
因此，正确答案是 E。
There are ways to assign the eight distinct colors to the eight faces. Two assignments give the same octahedron exactly when one is a rotation of the other.
The rotation group of a regular octahedron has elements. Because all eight colors are different, no nontrivial rotation fixes a coloring, so each distinguishable octahedron corresponds to exactly assignments.
Therefore the number of distinguishable octahedrons is
Thus, the correct answer is E.
其他年份的第 25 题
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