Multiplication Identities - STD 9 - Page 73.1 EM

5×5 Grid Square Difference

5×5 Grid Square Difference

Why is the difference always the same?

Problem

In the 5×5 grid below, mark any square of four numbers. Find the difference of the products of the numbers on the two diagonals. Is the difference always the same?

Selected 2×2 Square

Top-Left
7
Top-Right
8
Bottom-Left
12
Bottom-Right
13
Main Diagonal
7 × 13 = 91
Other Diagonal
8 × 12 = 96
Difference
|91 - 96| = 5

Algebraic Proof

Let the top-left number be n.

In a 5×5 grid, the four numbers are:

• Top-Left: n

• Top-Right: n + 1

• Bottom-Left: n + 5

• Bottom-Right: n + 6

Main Diagonal Product = n × (n + 6) = n² + 6n

Other Diagonal Product = (n + 1) × (n + 5) = n² + 6n + 5

Difference = (n² + 6n) − (n² + 6n + 5) = -5

Absolute Difference = 5 (always!)