HMMT 十一月 2025 · 冲刺赛 · 第 6 题

HMMT November 2025 — Guts Round — Problem 6

[6] Compute the number of ways to color each cell of an 8 × 8 grid either red, green, or blue such that every 1 × 3 and 3 × 1 rectangle with edges on the grid lines contains exactly one cell of each color. © 2025 HMMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2025, November 08, 2025 — GUTS ROUND Organization Team Team ID#

解析

[6] Compute the number of ways to color each cell of an 8 × 8 grid either red, green, or blue such that every 1 × 3 and 3 × 1 rectangle with edges on the grid lines contains exactly one cell of each color. Proposed by: Jackson Dryg Answer: 12 Solution: Notice that two squares 3 units apart must be the same color. For example, the cell marked ? must be red. ? If we color only the top-left 3 × 3 square, by using this fact repeatedly, there’s exactly one way to color the rest of the grid: © 2025 HMMT → So we only need to count the ways to color the top-left 3 × 3 square. There are 3 × 2 × 1 = 6 ways to choose the colors on the top row. Consider one of these options, say red-green-blue. Each row and column must contain exactly one red cell, so there are exactly two ways to place the other two red cells. Both of these lead to exactly one way to color the entire 3 × 3 square: Similarly, for each of the 6 ways to color the top row, there are 2 ways to color the rest of the 3 × 3 square. Therefore, the answer is 6 × 2 = 12 .