HMMT 十一月 2015 · 冲刺赛 · 第 21 题

英文解析

21. [ 11 ] Consider a 2 × 2 grid of squares. Each of the squares will be colored with one of 10 colors, andtwo colorings are considered equivalent if one can be rotated to form the other. How many distinctcolorings are there?
    Proposed by: Sam Korsky
    Answer: 2530
    This solution will be presented in the general case with n colors. Our problem asks for n = 10 .
    We isolate three cases:
    Case 1: Every unit square has the same color
    In this case there are clearly n ways to color the square.
    Case 2: Two non-adjacent squares are the same color, and the other two squares are also the samecolor (but not all four squares are the same color).
    ( )
    n ( n − 1)
    In this case there are clearly = ways to color the square.n
    2 2
    Case 3: Every other case
    Since without the ”rotation” condition there would be n colorings, we have that in this case by 4
    n − n ( n − 1) − n 4
    complementary counting there are ways to color the square.
    Therefore the answer is 4
    2 4 2 4 2
    n − n n − n n + n + 2 nn + + = = 2530
    2 4 4
    5 4 3 2