Some Explorations in Grids

Class 7 Maths · Chapter 6: Number Play · NCERT Ganita Prakash section 6.3

Learn

Fill a 3 × 3 grid with 1 to 9, each once, and write the sum of each row and column outside. Zoya notices: the three row sums always add to 45, and so do the three column sums. Why? The rows together contain every number once: 1 + 2 + … + 9 = 45.

1 + 2 + … + 9 = 45

Some puzzles are impossible: a row of three different numbers from 1–9 adds to at least 1 + 2 + 3 = 6 and at most 7 + 8 + 9 = 24. A grid asking for a row sum of 5 or 26 can never be filled.

6 ≤ row sum ≤ 24

A magic square has every row, column and diagonal adding to the same magic sum. With 1–9 the rows add to 45 in total, so each row is 45 ÷ 3 = 15. The centre must be 5 (with 9 there, 8 + 9 is already too big; with 1, 2 + 1 would need 12). And 1 and 9 cannot sit in corners.

8 1 6 / 3 5 7 / 4 9 2

General form: with centre m, a magic square of consecutive numbers looks like m + 3, m − 4, m + 1 / m − 2, m, m + 2 / m − 1, m + 4, m − 3. Every line adds to 3m. Add the same number to every cell, or multiply every cell by the same number, and it stays magic. India's 10th-century Chautīsā Yantra at Khajuraho is a 4 × 4 magic square with sum 34.

each line = 3m

Explore: magic numbers.

1. Set the magic sum of a magic square made from 1–9.
2. Set the number that must be in its centre.
3. A magic square has centre 25. Set its magic sum.

Remember

• In a 3 × 3 grid of 1–9, the row sums (and the column sums) add up to 45.
• A 1–9 magic square has sum 15 and 5 in the centre; in general every line is 3 × the centre.

Quick check

A magic square uses 11 to 19. What is in its centre?

1. 15
2. 11
3. 45
4. 19

Think about it

Make a magic square whose magic sum is 60.

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.