Number Play – Class 7 Maths Chapter 6 Notes & NCERT PDF

Introduction

"Numbers tell us things." The chapter opens with the children from the Grade 6 book standing in a line and calling out numbers. The rule: each child calls out how many children in front of them are taller than them. So the first child always says 0, and so does the tallest child. In a group of 8, the largest number anyone can say is 7. This shows how numbers can describe an arrangement without our knowing the actual heights.

Key Ideas

6.1 Numbers Tell us Things

Useful facts about the height game:

6.2 Picking Parity

Even numbers can be arranged in pairs with no leftover; odd numbers are one more (or one less) than a collection of pairs. Parity means the property of being even or odd.

The nth even number is 2n; the nth odd number is 2n – 1. So the 100th odd number is 199.

6.3 Some Explorations in Grids

Fill a 3 × 3 grid with 1–9 (no repeats) and write the row and column sums outside.

A magic square has every row, column and diagonal adding to the same number, the magic sum.

Reasoning step by step (instead of trying all 3,62,880 ways of filling the grid):

One magic square (all others are its rotations and reflections):

Generalised form with centre m:

Every row, column and diagonal adds to 3m. So a magic sum of 60 needs centre 20, and centre 25 gives magic sum 75. Adding 1 to every number, or doubling every number, still gives a magic square.

History: the first recorded 4 × 4 magic square, the Chautīsā Yantra, is in a 10th-century inscription at the Pārśhvanath Jain temple, Khajuraho; its rows, columns and diagonals add to 34 (chautīs). China's Lo Shu square is over 2000 years old. A 3 × 3 magic square is carved on a pillar of an 8th-century temple in Palani, Tamil Nadu, and Navagraha and Kubera Yantras are found in homes and shops.

6.4 Nature's Favourite Sequence: The Virahāṅka–Fibonacci Numbers

In Indian poetry, a short syllable lasts 1 beat and a long syllable 2 beats. How many rhythms fill n beats? Equivalently: in how many ways can n be written as a sum of 1s and 2s?

Virahāṅka sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …, where each number is the sum of the two before it.

Virahāṅka gave this rule around 700 CE, building on Piṅgala (about 300 BCE). Gopala (c. 1135 CE) and Hemachandra (c. 1150 CE) wrote about it before Fibonacci (1202 CE). There are 34 rhythms with 8 beats. Daisies often have 13, 21 or 34 petals. The parities follow the pattern odd, even, odd, odd, even, odd, …, so every third number is even.

6.5 Digits in Disguise (Cryptarithms)

Letters stand for digits.

Worked Examples

Figure it Out Hints

Maths Words

Common Mistakes

Maths in Daily Life

Parity checks catch counting errors: "14,70,369 people got married" must be wrong, because marriages pair people, so the count must be even. Magic squares appear in temple carvings and yantras, and Virahāṅka numbers appear in music, poetry, drumming and flower petals.