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Solutions

Overview

Step-by-step NCERT solutions for Number Play (Chapter 6, NCERT Class 7 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Number Play textbook chapter.

Solved

What these solutions cover

All 33 questions in Number Play are solved in the PDF. Here's what's inside, exercise by exercise:

Numbers Tell us Things

  1. Arrange the stick figure cutouts (or draw a height arrangement) such that when each child calls out the number of people in front of them who are taller, 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
  2. For each statement, identify if it is Always True, Only Sometimes True, or Never True. Share your reasoning.
    • (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, they cannot say '0'.
    • (e) The person who calls out the largest number is the…

Picking Parity

  1. Using your understanding of the pictorial representation of odd and even numbers, find out the parity of the following sums:
    • (a) Sum of 2 even numbers and 2 odd numbers (e.g., even + even + odd + odd)
    • (b) Sum of 2 odd numbers and 3 even numbers
    • (c) Sum of 5 even numbers
    • (d) Sum of 8 odd numbers
  2. 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? If he did, explain why. If he didn't, how many coins of each type could he have?
  3. We know that:
    • (a) even + even = even
    • (b) odd + odd = even
    • (c) even + odd = odd Similarly, find the parity for:
    • (d) even − even = ?
    • (e) odd − odd = ?
    • (f) even − odd = ?
    • (g) odd − even = ?

Small Squares in Grids and Parity of Expressions

  1. Find the parity of the number of small squares in these grids:
    • (a) 27 × 13
    • (b) 42 × 78
    • (c) 135 × 654
  2. What is the nth even number? Write a formula. What is the 100th even number?
  3. Write a formula to find the nth odd number. What is the 100th odd number?

Fill the Grids

  1. Fill the 3 × 3 grid with numbers from 1 to 9 (no repeats) so that the row and column sums match the circled numbers shown. Grid 1 — known cell: top-left = 9, bottom-right = 5 Row sums (right circles): 13, 14, 18 Column sums (bottom circles): 24, 9, 12 (Pipe-delimited layout: each row is | c1 | c2 | c3 |) Partial grid: | 9 | _ | _ | → 13 | _ | _ | _ | → 14 | _ | _ | 5 | → 18 ↓ ↓ ↓ 24 9 12
  2. Fill the 3 × 3 grid with numbers from 1 to 9 (no repeats) so that the row and column sums match the circled numbers shown. Grid 2 — known cells: middle-left = 4, bottom-right = 3 Row sums (right circles): 24, 15, 6 Column sums (bottom circles): 12, 16, 17 Partial grid: | _ | _ | _ | → 24 | 4 | _ | _ | → 15 | _ | _ | 3 | → 6 ↓ ↓ ↓ 12 16 17

Magic Squares

  1. How many different magic squares can be made using the numbers 1 to 9?
  2. Create a magic square using the numbers 2 to 10. What strategy would you use? Compare it with magic squares made using 1 to 9.
  3. Take the magic square: | 8 | 1 | 6 | | 3 | 5 | 7 | | 4 | 9 | 2 |
    • (a) Increase each number by 1. Is the resulting grid also a magic square? How does the magic sum change?
    • (b) Double each number. Is the resulting grid also a magic square? How does the magic sum change?
  4. What other operations can be performed on a magic square to yield another magic square?
  5. Discuss ways of creating a magic square using any set of 9 consecutive numbers (like 2–10, 3–11, 9–17, etc.).

Generalising a 3 × 3 Magic Square

  1. Using the generalised form of a magic square (where m is the centre number), find a magic square if the centre number is 25. Generalised form: | m+3 | m−4 | m+1 | | m−2 | m | m+2 | | m−1 | m+4 | m−3 |
  2. What is the expression obtained by adding the 3 terms of any row, column or diagonal in the generalised form?
  3. Write the result obtained by:
    • (a) Adding 1 to every term in the generalised form.
    • (b) Doubling every term in the generalised form.
  4. Create a magic square whose magic sum is 60.
  5. Is it possible to get a magic square by filling nine non-consecutive numbers?

The Virahāṅka–Fibonacci Sequence

  1. Write the next 3 numbers in the Virahāṅka sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ___, ___, ___, … Also, can you tell whether the number after 377 is odd or even (without computing the sum)?
  2. What is the parity of each number in the Virahāṅka sequence? Do you notice any pattern?

Digits in Disguise

  1. A light bulb is ON. Dorjee toggles its switch 77 times. Will the bulb be on or off? Why?
  2. Liswini has a large old encyclopaedia. When she opened it, several loose pages fell out of it. 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. Here is a 2 × 3 grid. For each row and column, the parity of the sum is written in the circle: 'e' for even and 'o' for odd. Row parities (right circles): top row = o, bottom row = e Column parities (bottom circles): col 1 = e, col 2 = e, col 3 = o Fill the 6 boxes with 3 odd numbers and 3 even numbers to satisfy the row and column sum parities.
  4. Make a 3 × 3 magic square with 0 as the magic sum. All numbers cannot be zero. Use negative numbers as needed.
  5. Fill in the following 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 ______
  6. What is the parity of the sum of the numbers from 1 to 100?
  7. 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 in the sequence?
  8. Angaan wants to climb an 8-step staircase. His rule is that he can take either 1 step or 2 steps at a time (e.g., one path is 1, 2, 2, 1, 2). In how many different ways can he reach the top?
  9. What is the parity of the 20th term of the Virahāṅka sequence?
  10. Identify the statements that are true:
    • (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.
  11. Solve this cryptarithm: UT + TA ---- TAT (Each letter stands for a unique digit 0–9. UT is a 2-digit number, TA is a 2-digit number, TAT is a 3-digit number.)
Sample

Worked examples

3 questions from Number Play, solved step by step.

Picking Parity

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? If he did, explain why. If he didn't, how many coins of each type could he have?

Find the parity of each coin type's contribution:

₹1 coins: (odd number) × ₹1 = odd × 1 = odd total value

₹5 coins: (odd number) × ₹5 = odd × 5 = odd total value (odd × odd = odd)

₹10 coins: (even number) × ₹10 = even × 10 = even total value

Total = odd + odd + even = (odd + odd) + even = even + even = even

But ₹205 is an odd number.

An even total can never equal ₹205 (odd).

Therefore, Lakpa made a mistake.

Answer: Yes, Lakpa made a mistake. The total value must be even, but ₹205 is odd — this is impossible with the given coin counts.

Small Squares in Grids and Parity of Expressions

Write a formula to find the nth odd number. What is the 100th odd number?

Compare the even and odd sequences term by term:

Even: 2, 4, 6, 8, 10, … (nth term = 2n)

Odd: 1, 3, 5, 7, 9, … (each is one less than the corresponding even)

The nth odd number = 2n − 1.

100th odd number = 2 × 100 − 1 = 200 − 1 = 199.

Answer: The nth odd number is 2n − 1. The 100th odd number is 199.

Magic Squares

What other operations can be performed on a magic square to yield another magic square?

Any operation applied uniformly to every cell preserves the magic square property, because rows, columns, and diagonals all have the same number of cells (3 each), so the operation affects all line-sums equally.

Operations that produce another magic square:

• Add the same constant to every cell (magic sum increases by 3 × constant).

• Subtract the same constant from every cell.

• Multiply every cell by the same non-zero constant (magic sum scales by that constant).

• Divide every cell by the same non-zero constant (provided results are integers, or we allow fractions).

Also, any rotation (90°, 180°, 270°) or reflection of a magic square is again a magic square.

Answer: Adding, subtracting, multiplying, or dividing every cell by the same constant all produce another magic square. Rotations and reflections also preserve the property.

All 33 questions in Number Play are solved step by step in the Solutions PDF above.

Keep solving

More solutions in Ganita Prakash

01Large Numbers Around Us02Arithmetic Expressions03A Peek Beyond the Point04Expressions using Letter Numbers05Parallel and Intersecting Lines07A Tale of Three Intersecting Lines08Working with Fractions09Geometric Twins10Operations with Integers11Finding Common Ground12Another Peek Beyond the Point13Connecting the Dots14Constructions and Tilings15Finding the Unknown
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More NCERT Solutions for Class 7

Read the Number Play textbook chapter / PDF, or browse all NCERT Class 7 Mathematics solutions.

← PreviousCh 5: Parallel and Intersecting LinesNext →Ch 7: A Tale of Three Intersecting Lines

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