GRADE 7 • CHAPTER 6
Number Play
Revision Notes + Solved Examples + Practice Problems
Use this page for quick revision. Try each problem first, then tap Show Answer.
🔢 Parity✨ Magic Squares🔁 Virahāṅka–Fibonacci🔐 Cryptarithms
1. Numbers Tell Us Things
Main idea: In the height-line activity, each person says the number of people standing in front of them who are taller than them.
- The first person's number is always 0.
- The tallest person's number is 0, wherever that person stands.
- The largest possible number for a group of n people is n − 1.
🧠 Remember: Count taller people before me.
2. Parity — Even & Odd
| Operation | Result |
|---|---|
| even + even | even |
| odd + odd | even |
| even + odd | odd |
| even − even | even |
| odd − odd | even |
| even − odd | odd |
| odd − even | odd |
Fast rule for a sum: A sum is odd exactly when it contains an odd number of odd terms. Otherwise it is even.
🧠 Memory trick: Count the ODDs. Odd count → Odd sum; Even count → Even sum.
3. Parity of Products & Grids
- A product is even if at least one factor is even.
- A product is odd only when every factor is odd.
- For an m × n grid, the number of small squares is mn. Its parity is odd only when both m and n are odd.
Example: 27 × 13 → odd × odd → odd.
42 × 78 → even × even → even.
135 × 654 → odd × even → even.
42 × 78 → even × even → even.
135 × 654 → odd × even → even.
4. Parity of Algebraic Expressions
nth even number = 2n
nth odd number = 2n − 1
nth odd number = 2n − 1
- Expressions of the form 2n always give even numbers.
- Expressions of the form 2n + 1 or 2n − 1 always give odd numbers.
- An expression such as 3n + 4 can be even or odd depending on whether n is even or odd.
🧠 If the expression has a factor 2 throughout → even. If it is 2n ± 1 → odd.
5. Magic Squares
Magic square: A square grid in which every row, every column and every diagonal has the same sum. That common value is the magic sum.
3 × 3 magic square using 1–9
8
1
6
3
5
7
4
9
2
- 1 + 2 + … + 9 = 45.
- There are 3 equal row sums, so magic sum = 45 ÷ 3 = 15.
- The centre must be 5.
- 1 and 9 cannot occupy corners in this 1–9 magic square.
🧠 15–5–1/9: magic sum 15, centre 5, and 1 & 9 go in middle-edge positions.
6. Generalising a 3 × 3 Magic Square
If the centre number is m, the corresponding 3 × 3 pattern using nine consecutive numbers can be written as:
m+3
m−4
m+1
m−2
m
m+2
m−1
m+4
m−3
Each row, column and diagonal has sum 3m.
For example, if m = 5, this gives the standard 1–9 magic-square pattern.
7. Virahāṅka–Fibonacci Numbers
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
Rule: Every term after the first two is the sum of the previous two terms.
an = an−1 + an−2
- 5th term = 8
- 8th term = 34
- 10th term = 89
- Next after 55 = 89
The chapter connects the sequence with counting rhythms made from 1-beat and 2-beat syllables.
🧠 Next = Previous + Previous.
8. Cryptarithms / Digits in Disguise
In a cryptarithm, each letter represents a particular digit from 0 to 9. The same letter always represents the same digit.
- Work column by column from the units place.
- Track carries carefully.
- Different letters normally represent different digits.
- A leading letter of a multi-digit number cannot be 0.
Example:
T + T + T = UT
The units digit must be T. Testing digits gives T = 5, so UT = 15.
T + T + T = UT
The units digit must be T. Testing digits gives T = 5, so UT = 15.
9. Revision Practice — Try First!
There are 24 mixed problems. Answers are hidden so this can be used as a self-test.
P1 • Height arrangement
Seven people stand in a line. A person reports the number of taller people in front of them. What is the largest possible number anyone can report?
6
P2 • Parity
Find the parity of: even + odd + odd + even + odd.
Odd — there are 3 odd terms.
P3 • Parity
Find the parity of the sum of 8 odd numbers.
Even — an even number of odd terms has an even sum.
P4 • Difference
If a is odd and b is even, what is the parity of a − b?
Odd.
P5 • Grid
Without multiplying, determine the parity of 135 × 654.
Even, because one factor (654) is even.
P6 • Grid
A grid has 27 rows and 13 columns. Is its number of small squares odd or even?
Odd — both dimensions are odd.
P7 • Expression
Is 4n − 1 always odd?
Yes. 4n is even, and even − 1 is odd.
P8 • Expression
Write a formula for the 25th odd number.
2(25) − 1 = 49.
P9 • Expression
What is the 40th even number?
2(40) = 80.
P10 • Consecutive numbers
Can two consecutive integers have an even sum? Explain briefly.
No. One is even and the other is odd, so their sum is odd.
P11 • Magic square
What is the magic sum of a 3 × 3 magic square using 1–9?
15, because 1+2+…+9 = 45 and 45 ÷ 3 = 15.
P12 • Magic square
What number must be at the centre of a 1–9 3 × 3 magic square?
5.
P13 • Magic square
Using the standard 1–9 magic square, what is the sum of the main diagonal 8, 5, 2?
15.
P14 • Magic square
If the centre of the generalised 3 × 3 square is m, what is its magic sum?
3m.
P15 • Magic square
Create a 3 × 3 magic square using 2–10 by adding 1 to every entry of the standard 1–9 square.
9 2 7 / 4 6 8 / 5 10 3. Magic sum = 18.
P16 • Virahāṅka
Find the next three terms: 1, 2, 3, 5, 8, 13, …
21, 34, 55.
P17 • Virahāṅka
Two consecutive terms are 987 and 1597. Find the next two terms.
2584 and 4181.
P18 • Virahāṅka
Two consecutive terms are 987 and 1597. Find the previous two terms.
610 and 377.
P19 • Staircase
A staircase has 8 steps. You may take 1 or 2 steps at a time. How many different ways can you reach the top?
34 ways.
P20 • Cryptarithm
Solve: UT + TA = TAT.
U = 9, T = 1, A = 0, giving 19 + 10 = 101.
P21 • Toggle
A bulb is ON. Its switch is toggled 77 times. Is it ON or OFF?
OFF. An odd number of toggles changes ON to OFF.
P22 • Sum 1 to 100
Is the sum 1 + 2 + … + 100 odd or even?
Even. Pair 1+100, 2+99, …, 50+51; each pair is 101 (odd), and there are 50 pairs, so the total is even.
P23 • Reasoning
Can five odd numbers have an even sum?
No. An odd number of odd terms gives an odd sum.
P24 • Challenge
Give one expression that always gives odd numbers and one that always gives even numbers.
Odd: 2n+1. Even: 2n.
10. 60-Second Final Recall
Parity
Even = pairs; Odd = one left over.
Even = pairs; Odd = one left over.
nth even
2n
2n
nth odd
2n − 1
2n − 1
1–9 magic square
Magic sum 15; centre 5.
Magic sum 15; centre 5.
Virahāṅka
Next = sum of previous two.
Next = sum of previous two.
Cryptarithm
Same letter = same digit; watch carries.
Same letter = same digit; watch carries.