HMMT 二月 2024 · COMB 赛 · 第 1 题

HMMT February 2024 — COMB Round — Problem 1

Compute the number of ways to divide a 20 × 24 rectangle into 4 × 5 rectangles. (Rotations and reflections are considered distinct.)

解析

Compute the number of ways to divide a 20 × 24 rectangle into 4 × 5 rectangles. (Rotations and reflections are considered distinct.) Proposed by: Derek Liu Answer: 6 Solution: For convenience, say the edge of length 20 is vertical. Consider some vertical line inside the rectangle. It must pass through rectangles of some heights adding to 20 . In particular, these heights correspond to ways to add up to 20 with fours and fives, which is either 4 + 4 + 4 + 4 + 4 or 5 + 5 + 5 + 5 . These options correspond to columns of rectangles with width 5 or 4 , respectively. In particular, we need to span the width of the original 20 × 24 rectangle using these columns, meaning that we can just count the number of ways to add to 24 with fours and fives. 4 4 4 4 4 4 4 5 5 5 5 There are two ways to do this: either 4 + 4 + 4 + 4 + 4 + 4 or 4 + 5 + 5 + 5 + 5 . Considering the orders in which the second sum can be written, we get an answer of 1 + 5 = 6 .