Write number like this:

Instead of a square of four numbers, take a square of nine numbers and mark only the numbers at the four corners.

What is the difference of diagonal products? Explain using algebra.

Consider the green square and consider the four corner elements ( red circles)

Diagonal products: 1× 13 = 13

3× 11 = 33

Difference between diagonal products = 33 - 13 = 20

Now, consider this green square and consider the corner elements (red circles)

Diagonal products: 8× 20 = 160

18× 10 = 180

Difference between diagonal products = 180 – 160 = 20

So, we observe that the difference between diagonal products in the above cases is also coming out to be same and is equal to 20.

Explanation using Algebra,

Let the first number in the square be x, the others can be filled in as below,

Diagonal products: x (x+12) = x² + 12x

(x+10)(x+2) = x(x+2) + 10(x+2) = (x² + 2x) + (10x+ 20)

= x² + 12x + 20

Clearly, we can observe that the difference between diagonal products is always equal to 20.