Red Perimeter 2 – Express With Variables

Probability Level 1

Which of the following expresses the correct formula for the number of unit squares with exactly 1 red side in an $n \times n$ square with $n \geq 2$ ?

4(n-2)

4n - 4

(n-2)^2

n^2 - 8

5 solutions

Chandrachur Banerjee
May 15, 2014

Each side has N red suares among whom 2 are red on 2 sides.As there are 4 sides Number reqd = 4(N-2)

Number ready?

Schuyler Sarzaba - 4 years, 5 months ago

I meant reqd

Schuyler Sarzaba - 4 years, 5 months ago

Or subtract the inner $(n-2) \times (n-2)$ square and the $4$ squares at the corners to get $n^2 - (n-2)^2 - 4 = (n - (n-2))(n + (n-2)) - 4 = 2(2n - 2) - 4 = 4n - 8$ , by difference of squares.

Vikram Sarkar - 8 months, 1 week ago

Mona Mazen
May 18, 2014

N is number of squares on each side. Subtracting the corners which are 2 on each side give the one side red squares on each side . Multiplying by 4 gives the one side red square in the whole shape.

Mikhaella Layos
May 16, 2014

We can observe from the figure that each side has two squares which have red on two sides, and the rest of the squares have only one. The number of squares with red on only one side can be expressed as $s-2$ . Since a square has four sides, we multiply the earlier equation by 4, giving us $\boxed{4(s-2)}$

Mostakim Shakil
Feb 20, 2016

From the above diagram we see that =>

when n =2 ,then number of one red side square = 0 ;

n =3, then number of one red side square = 4 ;

n =4, then number of one red side square = 8 ;

and from the mentioned expression only $4(n-2)$ satisfy this.

I used the same reasoning, but I wish I used 'intrinsic' reasons to reach the answer, rather than working backward from the diagrams.

A Former Brilliant Member - 4 years, 7 months ago

Shohag Hossen
May 15, 2014

its easily say that , 4(n -2)

brother can u explain

Nazmus Sakib - 6 years, 10 months ago