HMMT 十一月 2015 · 冲刺赛 · 第 21 题

HMMT November 2015 — Guts Round — Problem 21

[ 11 ] Consider a 2 × 2 grid of squares. Each of the squares will be colored with one of 10 colors, and two colorings are considered equivalent if one can be rotated to form the other. How many distinct colorings are there? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2015, November 14, 2015 — GUTS ROUND Organization Team Team ID# 5 4 3 2

解析

[ 11 ] Consider a 2 × 2 grid of squares. Each of the squares will be colored with one of 10 colors, and two colorings are considered equivalent if one can be rotated to form the other. How many distinct colorings 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 same color (but not all four squares are the same color). ( ) n ( n − 1) n In this case there are clearly = ways to color the square. 2 2 Case 3: Every other case 4 Since without the ”rotation” condition there would be n colorings, we have that in this case by 4 n − n ( n − 1) − n complementary counting there are ways to color the square. 4 Therefore the answer is 2 4 2 4 2 n − n n − n n + n + 2 n n + + = = 2530 2 4 4 5 4 3 2