Q1.
Choose any magic square that you have made so far using consecutive numbers. If m is the letter-number of the number in the centre, express how other numbers are related to m, how much more or less than m. [Hint: Remember, how we described a 2 × 2 grid of a calendar month in the Algebraic Expressions chapter].
Answer
Take the magic square 8 1 6 / 3 5 7 / 4 9 2. Its centre is 5, so write every entry as “5 and something”.
8 = 5 + 3 • 1 = 5 – 4 • 6 = 5 + 1
3 = 5 – 2 • 5 = 5 • 7 = 5 + 2
4 = 5 – 1 • 9 = 5 + 4 • 2 = 5 – 3
3 = 5 – 2 • 5 = 5 • 7 = 5 + 2
4 = 5 – 1 • 9 = 5 + 4 • 2 = 5 – 3
Replacing 5 by the letter-number m gives the generalised form:
| m + 3 | m – 4 | m + 1 |
| m – 2 | m | m + 2 |
| m – 1 | m + 4 | m – 3 |
Row 1: (m + 3) + (m – 4) + (m + 1) = 3m
Row 2: (m – 2) + m + (m + 2) = 3m
Row 3: (m – 1) + (m + 4) + (m – 3) = 3m
Column 1: (m + 3) + (m – 2) + (m – 1) = 3m
Diagonal: (m + 3) + m + (m – 3) = 3m ✓
Row 2: (m – 2) + m + (m + 2) = 3m
Row 3: (m – 1) + (m + 4) + (m – 3) = 3m
Column 1: (m + 3) + (m – 2) + (m – 1) = 3m
Diagonal: (m + 3) + m + (m – 3) = 3m ✓
Why it happens: the nine numbers m – 4, m – 3, …, m + 4 are just nine consecutive numbers with m in the middle. In every line the two numbers on either side of m cancel each other — one is as much above m as the other is below.