HMMT 十一月 2021 · 冲刺赛 · 第 14 题

HMMT November 2021 — Guts Round — Problem 14

[9] In a k × k chessboard, a set S of 25 cells that are in a 5 × 5 square is chosen uniformly at random. The probability that there are more black squares than white squares in S is 48%. Find k . √

解析

[9] In a k × k chessboard, a set S of 25 cells that are in a 5 × 5 square is chosen uniformly at random. The probability that there are more black squares than white squares in S is 48%. Find k . Proposed by: Akash Das Answer: 9 Solution: We know that there must be fewer black squares than white squares, and k must be odd. Additionally, we know that there are k − 4 ways to pick the left column of the 5 × 5 square so that the right column can fit within the k × k grid, and k − 4 ways to pick the top row by similar logic. 2 Therefore, there are ( k − 4) of these 5 × 5 squares on this chessboard, and because there will be more black squares than white squares whenever there exists a black square in the top left corner, there are 2 ( k − 4) − 1 of them have more black squares than white squares, corresponding to the number of black 2 squares in the upper ( k − 4) × ( k − 4) grid. Thus, we have 2 ( k − 4) − 1 2 = 0 . 48 = ⇒ k = 9 2 ( k − 4) √