A good problem!!

Algebra Level 5

In a plane there are two families of lines y = x + r y=x+r and y = − x + r y=-x+r , where r ∈ { 0 , 1 , 2 , 3 , 4 } . r\in\{0,1,2,3,4\}. How many squares of diagonal length 2 can be formed by the two lines?


The answer is 9.

Parth Lohomi by Parth Lohomi , 20, India

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1 solution

All the lines intersect to form a 4 by 4 square grid.Squares of diagonal length 2 will be formed when we choose any two parallel lines, the distance between them being sqrt(2) , from the first family of lines as well as second family of lines.This can be done in 3 2 3^2 = 9 ways. Also, if r would have taken the values 0,1,2,...n ; then the answer (by the same procedure as before) would be ( n − 1 ) 2 (n-1)^2 .

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