HMMT 十一月 2015 · THM 赛 · 第 6 题
HMMT November 2015 — THM Round — Problem 6
题目详情
英文原题
- Consider a 6 × 6 grid of squares. Edmond chooses four of these squares uniformly at random. Whatis the probability that the centers of these four squares form a square?
解析
英文解析
- Consider a 6 × 6 grid of squares. Edmond chooses four of these squares uniformly at random. Whatis the probability that the centers of these four squares form a square?
Proposed by: Alexander Katz
1 105
Answer: OR
56136
( )
( )4
Firstly, there are possible combinations of points. Call a square proper if its sides are parallel to 36
the coordinate axes and improper otherwise. Note that every improper square can be inscribed in a 4
unique proper square. Hence, an n × n proper square represents a total of n squares: 1 proper andn − 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 ii =1 i =1 = 441 − 12(91) + 36(21) = 441 − 1092 + 756 = 105
105 1
( )
squares on the grid. Our desired probability is thus = .
56136 4