Notes Maths Class 7 Chapter 6 Ganita Prakash

Notes For All Chapters – Ganita Prakash Class 7th

Number Play

1. Numbers Tell Us Things

A sequence of numbers can convey information such as height order.

Rule: Each child says how many children in front of them are taller.

Example sequence:

If sequence is: 0, 1, 2, 3, it means each person is shorter than the ones ahead.

Think logically about statements like:

“If a person says 0, they are tallest?” → Not always true.
“If someone is the tallest, their number must be 0?” → Always true.

2. Picking Parity

Parity:

Even numbers: Can be grouped in pairs; examples: 2, 4, 6.

Odd numbers: Cannot be grouped in pairs; examples: 1, 3, 5, 7,….

Rules of Addition:

Type of Numbers Sum Is
Even + Even Even
Odd + Odd Even
Even + Odd Odd

Key Concepts:

Sum of 5 odd numbers can’t be even → So you cannot make 30 using 5 odd numbers.

Two consecutive numbers always include one even and one odd → Sum will be odd.

Example: 51 + 52 = 103; Can never add to an even like 112.

Formulas:

nth even number = 2n
nth odd number = 2n – 1

3. Explorations in Grids

A grid is a table of rows and columns where we place numbers.

Example:

A 3 × 3 grid has 3 rows and 3 columns and contains 9 squares (or boxes).

What is a Magic Square?

A magic square is a special kind of grid where:
All rows, all columns, and both diagonals add up to the same number(15).
This common number is called the magic sum.

Tips to Create a Magic Square:

Total sum of all numbers = 45
Place 5 in the center.
Place 1 and 9 in side-center positions.
Carefully fill other boxes so all rows, columns, diagonals add to 15.

4. Nature’s Favourite Sequence – Virahāṅka–Fibonacci Numbers

Sequence:

1, 2, 3, 5, 8, 13, 21, 34, 55, …
Each number is the sum of the previous two.
Originated in Indian poetry to count rhythms of syllables.
Short syllable = 1 beat, Long syllable = 2 beats.

Application:

Ways to make 8 using 1s and 2s:

1+1+1+1+1+1+1+1
2+2+2+2
… total 34 ways → 8th term of the sequence.

Famous Scholars:

Virahāṅka, Piṅgala, Gopala, Hemachandra (all Indian)

Known in the West as Fibonacci numbers.

In Nature:

Number of petals in daisies: 13, 21, 34 – all Virahāṅka numbers!

5. Digits in Disguise – Cryptarithms

What is a Cryptarithm?

A cryptarithm is a math puzzle where numbers are replaced by letters, and each letter stands for a unique digit (0–9).
The goal is to find which digit each letter represents to make the math equation (like addition or multiplication) correct.

Rules:

Each letter represents one digit (0–9).
Different letters represent different digits (e.g., if U = 5, no other letter can be 5).
Numbers cannot start with 0 (e.g., a two-digit number UT cannot have U = 0).
The equation must be true when digits are substituted.

Examples from the Textbook

1. Cryptarithm 1: U + U + U = UT

Meaning: A one-digit number (U) is added to itself three times (3U), resulting in a two-digit number (UT). The units digit of the sum (T) is the same as U.

Solution:

Equation: 3U = 10U + T (UT is a two-digit number with tens digit U, units digit T).
Since T = U, we have 3U = 10U + U = 11U (but this leads to 3U = 11U, which is impossible unless U = 0, and U ≠ 0 for a two-digit UT).
Consider carry-over: 3U may produce a two-digit number.
Try U = 5: 3 × 5 = 15. UT = 15 (U = 1, T = 5). Since T = U, check if T = 5, U = 5: 3 × 5 = 15, which is 10 × 1 + 5. This works!
Test other digits:
- U = 3: 3 × 3 = 9 (single digit, not UT).
- U = 6: 3 × 6 = 18 (U = 1, T = 8, but T ≠ U).
Answer: U = 5, T = 5, so UT = 15.
Check: 5 + 5 + 5 = 15, units digit = 5 (matches U).

2. Cryptarithm 2: K2 + K2 = HMM

Meaning: K2 is a two-digit number with tens digit K and units digit 2 (e.g., K = 3 means K2 = 32). Adding K2 to itself gives a three-digit number HMM, where the tens and units digits are the same (M).