HMMT 二月 2005 · 冲刺赛 · 第 29 题

英文解析

29. 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 n possible 3
    colorings, where rotations, but not reflections, of the same coloring are considered the same. Find all possible values of n .
    Solution: 1 , 11
    We count the possible number of colorings. If four colors are used, there are two 10
    ( )
    different colorings that are mirror images of each other, for a total of 2 colorings. Ifnthree colors are used, we choose one color to use twice (which determines the coloring),4
    ( )
    for a total of 3 colorings. If two colors are used, we can either choose one of thosencolors and color three faces with it, or we can color two faces each color, for a total of 3
    ( ) ( )
    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-
    lently n ( n − 1)( n − 11) = 0, giving the answers 1 and 11.2