HMMT 十一月 2025 · 团队赛 · 第 1 题

HMMT November 2025 — Team Round — Problem 1

[20] Compute the number of ways to divide an 8 × 8 square into 3 rectangles, each with (positive) integer side lengths.

解析

[20] Compute the number of ways to divide an 8 × 8 square into 3 rectangles, each with (positive) integer side lengths. Proposed by: Matthew Qian Answer: 238 Solution: Call a cut full-length if it connects two opposite sides of the 8 × 8 square. We consider two cases: • If both cuts are full-length and parallel, there are two possible directions (vertical or horizontal) 7 7 and ways to choose their positions, giving us 2 · = 42 ways. 2 2 • If one cut is full-length and the other is a shorter perpendicular cut, there are – 2 choices for the full-length cut’s direction, – 7 choices for its position, – 2 choices for the side of the full-length cut that the shorter cut lies on, and – 7 choices for its position, giving us 2 · 7 · 2 · 7 = 196 ways. Thus, the total number of ways to divide the square into 3 rectangles is 42 + 196 = 238 .