Calendar Square Difference - Multiplication Identities - STD 9 - EM
Calendar Square Difference
Why is the difference always 7?
Problem
In a calendar, mark any four dates that form a 2×2 square. Multiply the numbers on each diagonal and find the difference. Why is it always 7?
Sample Calendar (One Week View)
Selected 2×2 Square
Top-Left
15
Top-Right
16
Bottom-Left
22
Bottom-Right
23
Main Diagonal Product
15 × 23 = 345
Other Diagonal Product
16 × 22 = 352
Difference
|345 - 352| = 7
Algebraic Proof (Why it is always 7)
Let the top-left date be n.
Then the four dates are:
• Top-Left: n
• Top-Right: n + 1
• Bottom-Left: n + 7
• Bottom-Right: n + 8
Main Diagonal Product = n × (n + 8) = n² + 8n
Other Diagonal Product = (n + 1) × (n + 7) = n² + 8n + 7
Difference = (n² + 8n) − (n² + 8n + 7) = -7
Absolute Difference = 7