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NCERT Solutions · Maths Ch 6
Number Play
28 questions · 4 exercises
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Ex 6.1
Ex 6.2
Ex 6.3
Ex 6.4
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
EXERCISE 6.1 · 3 QUESTIONS
Numbers Tell Us Things
Q1
Each child calls out the number of children standing in front of them who are taller than they are. Write the number each child should say for the arrangement of 7 children shown (heights from left to right: 4th tallest to 1st as in the picture).
(3 marks)
Q2
Arrange the stick-figure cutouts (or draw heights, 1 = shortest, 7 = tallest) such that the sequence reads: (a) 0, 1, 1, 2, 4, 1, 5 (b) 0, 0, 0, 0, 0, 0, 0 (c) 0, 1, 2, 3, 4, 5, 6 (d) 0, 1, 0, 1, 0, 1, 0 (e) 0, 1, 1, 1, 1, 1, 1 (f) 0, 0, 0, 3, 3, 3, 3
(5 marks)
Q3
For each statement, identify if it is Always True, Only Sometimes True, or Never True. (a) If a person says '0', then they are the tallest in the group. (b) If a person is the tallest, then their number is '0'. (c) The first person's number is '0'. (d) If a person is not first or last in line, then they cannot say '0'. (e) The person who calls out the largest number is the shortest. (f) What is the largest number possible in a group of 8 people?
(3 marks)
2
EXERCISE 6.2 · 5 QUESTIONS
Parity
Q4
Using your understanding of the pictorial representation of odd and even numbers, find the parity of the following sums: (a) 2 even numbers and 2 odd numbers (b) 2 odd numbers and 3 even numbers (c) 5 even numbers (d) 8 odd numbers
(3 marks)
Q5
Lakpa has an odd number of ₹1 coins, an odd number of ₹5 coins and an even number of ₹10 coins in his piggy bank. He calculated the total and got ₹205. Did he make a mistake?
(3 marks)
Q6
We know even + even = even, odd + odd = even, even + odd = odd. Find the parity: (d) even − even (e) odd − odd (f) even − odd (g) odd − even
(3 marks)
Q7
Given the dimensions of a grid, can you tell the parity of the number of small squares without calculating the product? Find the parity for grids of size (a) 27 × 13 (b) 42 × 78 (c) 135 × 654.
(3 marks)
Q8
Come up with an expression that always has odd parity. Come up with other expressions, like 3n + 4, which could have either odd or even parity. Are there expressions using which we can list all odd numbers? What is the nth even number?
(3 marks)
3
EXERCISE 6.3 · 8 QUESTIONS
Magic Squares
Q9
How many different magic squares can be made using the numbers 1–9?
(3 marks)
Q10
Create a magic square using numbers 2–10. What strategy would you use? Compare it with the magic square made using 1–9.
(3 marks)
Q11
Take a magic square and (a) increase each number by 1 (b) double each number. In each case is the resulting grid also a magic square? How do the magic sums change?
(3 marks)
Q12
What other operations can be performed on a magic square to yield another magic square? Discuss ways of creating a magic square using any set of 9 consecutive numbers (like 2–10, 3–11, 9–17).
(2 marks)
Q13
Take the magic square with consecutive numbers and let m be the number in the centre. Express how the other numbers are related to m (how much more or less than m).
(5 marks)
Q14
Using the generalised form, find a magic square if the centre number is 25. What is the expression obtained by adding the 3 terms of any row, column or diagonal?
(5 marks)
Q15
Write the result obtained by (a) adding 1 to every term in the generalised form (b) doubling every term in the generalised form.
(3 marks)
Q16
Create a magic square whose magic sum is 60. Is it possible to get a magic square by filling nine non-consecutive numbers?
(5 marks)
4
EXERCISE 6.4 · 12 QUESTIONS
Virahāṅka Numbers and Cryptarithms
Q17
Write the next 3 numbers in the sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ___, ___, ___. If you write one more number, can you tell whether it will be odd or even without adding? What is the parity pattern?
(3 marks)
Q18
A light bulb is ON. Dorjee toggles its switch 77 times. Will the bulb be on or off? Why?
(2 marks)
Q19
Liswini has a large old encyclopaedia. Several loose pages fell out. She counted 50 sheets in total, each printed on both sides. Can the sum of the page numbers of the loose sheets be 6000? Why or why not?
(3 marks)
Q20
In a 2 × 3 grid, the parity of the sum of each row and column is written in the circles: rows 'o' (odd) and 'e' (even); columns 'e', 'e', 'o'. Fill the 6 boxes with 3 odd numbers ('o') and 3 even numbers ('e') to satisfy these parities.
(3 marks)
Q21
Make a 3 × 3 magic square with 0 as the magic sum. All numbers cannot be zero. Use negative numbers as needed.
(3 marks)
Q22
Fill in the blanks with 'odd' or 'even': (a) Sum of an odd number of even numbers is ___ (b) Sum of an even number of odd numbers is ___ (c) Sum of an even number of even numbers is ___ (d) Sum of an odd number of odd numbers is ___
(3 marks)
Q23
What is the parity of the sum of the numbers from 1 to 100?
(2 marks)
Q24
Two consecutive numbers in the Virahāṅka sequence are 987 and 1597. What are the next 2 numbers in the sequence? What are the previous 2 numbers?
(3 marks)
Q25
Angaan wants to climb an 8-step staircase. He can take either 1 step or 2 steps at a time. For example, one of his paths is 1, 2, 2, 1, 2. In how many different ways can he reach the top?
(5 marks)
Q26
What is the parity of the 20th term of the Virahāṅka sequence?
(2 marks)
Q27
Identify the true statements. (a) The expression 4m − 1 always gives odd numbers. (b) All even numbers can be expressed as 6j − 4. (c) Both expressions 2p + 1 and 2q − 1 describe all odd numbers. (d) The expression 2f + 3 gives both even and odd numbers.
(3 marks)
Q28
Solve this cryptarithm: UT + TA = TAT (each letter stands for a different digit).
(2 marks)