HMMT 二月 2005 · 冲刺赛 · 第 29 题

HMMT February 2005 — Guts Round — Problem 29

[10] Let n > 0 be an integer. Each face of a regular tetrahedron is painted in one of n colors (the faces are not necessarily painted different colors.) Suppose there are 3 n possible colorings, where rotations, but not reflections, of the same coloring are considered the same. Find all possible values of n .

解析

Let n > 0 be an integer. Each face of a regular tetrahedron is painted in one of n colors 3 (the faces are not necessarily painted different colors.) Suppose there are n possible colorings, where rotations, but not reflections, of the same coloring are considered the same. Find all possible values of n . Solution: 1 , 11 10 We count the possible number of colorings. If four colors are used, there are two ( ) n different colorings that are mirror images of each other, for a total of 2 colorings. If 4 three colors are used, we choose one color to use twice (which determines the coloring), ( ) n for a total of 3 colorings. If two colors are used, we can either choose one of those 3 colors and color three faces with it, or we can color two faces each color, for a total of ( ) ( ) n n 3 colorings. Finally, we can also use only one color, for colorings. This gives a 2 1 total of ( ) ( ) ( ) ( ) n n n n 1 2 2 2 + 3 + 3 + = n ( n + 11) 4 3 2 1 12 3 2 2 3 colorings. Setting this equal to n , we get the equation n ( n + 11) = 12 n , or equiva- 2 lently n ( n − 1)( n − 11) = 0, giving the answers 1 and 11.