HMMT 二月 2018 · COMB 赛 · 第 3 题
HMMT February 2018 — COMB Round — Problem 3
题目详情
- A 4 × 4 window is made out of 16 square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors? Two different windowpanes are neighbors if they share a side.
解析
- A 4 × 4 window is made out of 16 square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors? Two different windowpanes are neighbors if they share a side. Proposed by: Kevin Sun Answer: 24 For the purpose of explaining this solution, let’s label the squares as 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 Note that since the corner squares 11 , 14 , 41 , 44 each only have two neighbors, each corner square is the same color as both of its neighbors (for example, 11 , 12 , and 21 are the same color, 31 , 41 , and 42 are the same color, etc.). This corner square constraint heavily limits the possible colorings. We will now use casework. Case 1: Suppose two corner squares on the same side (without loss of generality, let them be 11 and