HMMT 十一月 2015 · THM 赛 · 第 6 题

HMMT November 2015 — THM Round — Problem 6

Consider a 6 × 6 grid of squares. Edmond chooses four of these squares uniformly at random. What is the probability that the centers of these four squares form a square?

解析

Consider a 6 × 6 grid of squares. Edmond chooses four of these squares uniformly at random. What is the probability that the centers of these four squares form a square? Proposed by: Alexander Katz 1 105 Answer: OR 36 561 ( ) 4 ( ) 36 Firstly, there are possible combinations of points. Call a square proper if its sides are parallel to 4 the coordinate axes and improper otherwise. Note that every improper square can be inscribed in a unique proper square. Hence, an n × n proper square represents a total of n squares: 1 proper and n − 1 improper. There are thus a total of 6 6 ∑ ∑ 2 3 2 i (6 − i ) = ( i − 12 i + 36 i ) i =1 i =1 6 6 ∑ ∑ ∑ 3 2 6 = i − 12 i + 36 i = 1 i i =1 i =1 = 441 − 12(91) + 36(21) = 441 − 1092 + 756 = 105 105 1 ( ) squares on the grid. Our desired probability is thus = . 36 561 4