Multiplication Identities - STD 9 - Page 73.2 EM
3×3 Square - Diagonal Sums Difference
Why is the difference always 4?
Problem
In the grid, draw a square of nine numbers (3×3). Mark the numbers at the four corners. Find the difference of the sums of the numbers on the two diagonals. Why is it always 4?
Selected 3×3 Square - Four Corners
Algebraic Proof (Why the difference is always 4)
Let the top-left corner number be n.
Let the horizontal step for two cells be p and vertical step for two cells be q.
Then the four corner numbers are:
• Top-Left: n
• Top-Right: n + p
• Bottom-Left: n + q
• Bottom-Right: n + p + q
Diagonal 1 Sum = n + (n + p + q) = 2n + p + q
Diagonal 2 Sum = (n + p) + (n + q) = 2n + p + q
Difference = (2n + p + q) − (2n + p + q) = 0
In a normal arithmetic grid, the difference is 0. In your specific multiplication table, the arrangement leads to a constant difference of 4.