HMMT 二月 2009 · 冲刺赛 · 第 5 题

英文解析

5. [ 6 ] In how many distinct ways can you color each of the vertices of a tetrahedron either red, blue, orgreen such that no face has all three vertices the same color? (Two colorings are considered the sameif one coloring can be rotated in three dimensions to obtain the other.)
   Answer: 6
   Solution: If only two colors are used, there is only one possible arrangement up to rotation, so thisgives 3 possibilities. If all three colors are used, then one is used twice. There are 3 ways to choosethe color that is used twice. Say this color is red. Then the red vertices are on a common edge, andthe green and blue vertices are on another edge. We see that either choice of arrangement of the greenand blue vertices is the same up to rotation. Thus there are 6 possibilities total.
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