HMMT 二月 2026 · 冲刺赛 · 第 2 题

HMMT February 2026 — Guts Round — Problem 2

[5] Compute the number of ways to fill each of the outer 8 cells of a 3 × 3 grid with exactly one of the numbers 1 , 2 , and 3 such that the top row, bottom row, left column, and right column each contain no repeated numbers.

解析

[5] Compute the number of ways to fill each of the outer 8 cells of a 3 × 3 grid with exactly one of the numbers 1 , 2 , and 3 such that the top row, bottom row, left column, and right column each contain no repeated numbers. Proposed by: Sebastian Attlan Answer: 18 Solution: Notice that given distinct numbers in two corner cells on the same row or column, there is a unique choice for the number in the middle cell on the same row or column. Furthermore, every middle cell participates in a single row or column, so the only requirement for a number in a middle ©2026 HMMT cell is to be different from the two corner cells on its row or column. Thus, it is sufficient to compute the number of ways to fill each of the 4 corner cells with exactly one of the numbers 1 , 2 , and 3 such that no two corner cells on the same row or column have the same number. Let a , b , c , and d denote the numbers in the upper-left, upper-right, lower-left, and lower-right corner cells, respectively. If a = c = x , then there are three possible choices for x , two possible choices for b ̸ = x , and two possible choices for d ̸ = x . If a ̸ = c , then there are three are three possible choices for a , two possible choices for c , and b, d ̸ ∈ { a, c } are uniquely determined. Therefore, the total number of ways to fill the corner cells without violating the problem condition is 3 · 2 · 2 + 3 · 2 = 18 .