Number Play · Lesson 6 of 10
Generalising Magic Squares
“Transform and describe magic squares using a letter for the centre.”
• Predict how adding or multiplying every entry changes a magic sum. • Construct a magic square from any nine consecutive integers. • Express the entries relative to a centre value m. • Use the rule 3m to create squares with a required centre or magic sum. • Recognise selected historical and cultural examples of magic squares.
Change every entry, keep the balance
Start with any valid 3 × 3 magic square. If you add the same number c to every entry, each line receives c three times. Every line still has the same sum, but the magic sum rises by 3c. Multiplying every entry by a number a multiplies every line sum by a. These operations let one familiar square generate many others.
Problem
How can a square using 1–9 become one using 2–10?
- 1.Add 1 to each of its nine entries. For example, 8, 1, 6 becomes 9, 2, 7.
- 2.Each line had sum 15 and now has three entries each increased by 1, so each line totals 18.
- 3.All nine consecutive values 2–10 appear once, so the new square meets both rules.
Problem
What happens when every entry of a magic square with sum 15 is doubled?
- 1.Each line is the sum of three entries. Doubling each entry doubles the entire line.
- 2.The new magic sum is 2 × 15 = 30. The rows, columns, and diagonals still agree.
Write the structure around its centre
Suppose the centre is m and the entries are nine consecutive integers. Starting from the 1–9 square and replacing 5 by m means adding m − 5 to every cell. The resulting entries are the four integers below m, m itself, and the four above m. The table exposes the relationships; it is not nine unrelated formulas to memorise.
| Left | Middle | Right |
|---|---|---|
| m + 3 | m − 4 | m + 1 |
| m − 2 | m | m + 2 |
| m − 1 | m + 4 | m − 3 |
For instance, the first row totals (m + 3) + (m − 4) + (m + 1) = 3m. Because this table comes from adding the same amount to every cell of a valid magic square, every other row, column, and diagonal also totals 3m.
Problem
Construct a square whose centre is 25 from the letter pattern.
- 1.Replace m with 25: the top row becomes 28, 21, 26.
- 2.The middle row becomes 23, 25, 27; the bottom becomes 24, 29, 22.
- 3.Every line totals 3 × 25 = 75.
Problem
Construct a square with magic sum 60 using the consecutive-number pattern.
- 1.The sum is 3m, so 3m = 60 and m = 20.
- 2.Substitute 20 into the pattern: 23, 16, 21 / 18, 20, 22 / 19, 24, 17.
- 3.A row check gives 23 + 16 + 21 = 60; the other lines match.
Beyond one set of nine numbers
Nine consecutive numbers are a convenient starting set, but a magic square can also use nonconsecutive values. For example, multiplying the 1–9 square by 3 gives 24, 3, 18 / 9, 15, 21 / 12, 27, 6, with magic sum 45. Subtracting 5 from each cell of the 1–9 square instead gives −4 through 4 and magic sum 0. The same balancing principle works in both cases.
Magic squares appear in several mathematical traditions. The chapter shows the 4 × 4 Chautīsā Yantra at Khajuraho, whose rows, columns, and diagonals total 34, and discusses the Lo Shu square, Indian temple examples, and Navagraha and Kubera Yantras. These examples are contexts for looking for patterns; the 3 × 3 construction remains our main mathematical task.
In the Chautīsā Yantra, check a displayed row such as 7 + 12 + 1 + 14 = 34. Search for other groups of four adding to 34, while keeping the 3 × 3 rule separate from this larger example.
Changing one entry, or adding different amounts to different cells, usually destroys the equal line sums. The simple transformation rules require the same operation on every entry.
Quiz
Adding 4 to every entry of a 3 × 3 magic square changes its magic sum by how much?
Doubling every entry changes a magic sum of 15 to what?
In the letter pattern with centre m, what does every magic line total?
What must m equal when the letter-pattern magic sum is 60?
Which operation definitely preserves a valid magic square?
Practice Problems
- Create a magic square using 9–17 by changing every entry of a 1–9 square. State its sum.
- Find the magic sum when the centre is 40 and construct the full 3 × 3 square.
- Explain why adding c to every entry increases the line sum by 3c.
- Subtract 5 from every entry of a 1–9 magic square. Verify that the new sum is 0 and not every entry is zero.
- Multiply every entry of a 1–9 magic square by 3. List its entries and explain why they are nonconsecutive.
- Check a row, a column, and a diagonal in the chapter’s 4 × 4 Chautīsā Yantra. What is its magic sum?
Key Takeaways
• Adding c to each entry increases a 3 × 3 magic sum by 3c. • Multiplying every entry by a multiplies the magic sum by a. • For the generalised consecutive-number pattern, centre m gives magic sum 3m. • Uniform transformations can produce magic squares with nonconsecutive or negative entries. • Magic squares also appear in historical and cultural settings beyond the 3 × 3 example.