HMMT 二月 2002 · 冲刺赛 · 第 5 题

HMMT February 2002 — Guts Round — Problem 5

[6] Two 4 × 4 squares are randomly placed on an 8 × 8 chessboard so that their sides lie along the grid lines of the board. What is the probability that the two squares overlap? 2 3 4 5

解析

Two 4 × 4 squares are randomly placed on an 8 × 8 chessboard so that their sides lie along the grid lines of the board. What is the probability that the two squares overlap? Solution: 529 / 625 . Each square has 5 horizontal · 5 vertical = 25 possible positions, so there are 625 possible placements of the squares. If they do not overlap, then either one square lies in the top four rows and the other square lies in the bottom four rows, or one square lies in the left four columns and the other lies in the right four columns. The first possibility can happen in 2 · 5 · 5 = 50 ways (two choices of which square goes on top, and five horizontal positions for each square); likewise, so can the second. However, 1 this double-counts the 4 cases in which the two squares are in opposite corners, so we have 2 50 + 50 − 4 = 96 possible non-overlapping arrangements ⇒ 25 − 96 = 529 overlapping arrangements. 2 3 4 5