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Number Play

NCERT Class 6 · Mathematics Based on NCERT Class 6 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter explores numbers beyond counting and basic operations. It shows how numbers can convey information, create patterns, solve puzzles, and help in estimation and games. The one big idea a student must walk away with is that numbers are not just for arithmetic—they are tools for thinking, discovering patterns, and having fun while learning.
Key Concepts (10)
Supercell
A cell in a table is called a supercell if the number in it is larger than all its adjacent cells (immediately to the left, right, top, and bottom).
e.g. In the row 200, 577, 626, 345, 790, 694, 109, the number 626 is a supercell because it is larger than 577 and 345.
Palindrome (Palindromic Number)
A number that reads the same from left to right and from right to left.
e.g. 66, 848, 575, 797, 1111, 121, 313, 222.
Digit Sum
The sum obtained by adding all the digits of a number.
e.g. For 68, digit sum = 6 + 8 = 14. For 176, digit sum = 1 + 7 + 6 = 14.
Kaprekar Constant
The number 6174, which is reached by repeatedly applying the Kaprekar routine (largest number minus smallest number formed from the digits) for any 4-digit number with at least two different digits.
e.g. Start with 6382: largest 8632, smallest 2368, difference = 6264; then 6642 – 2466 = 4176; then 7641 – 1467 = 6174.
Collatz Conjecture
An unsolved mathematical conjecture stating that starting with any whole number, if it is even, take half of it; if it is odd, multiply it by 3 and add 1; repeating this process will always eventually reach the number 1.
e.g. Start with 12: 12 → 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1.
Reverse-and-Add Palindrome
A palindrome obtained by repeatedly adding a number to its reverse until a palindrome is formed.
e.g. Start with 47: 47 + 74 = 121 (palindrome).
Estimation
Finding an approximate value or count without needing an exact number, sufficient for the purpose at hand.
e.g. Estimating the number of students in a school as 'about 500' instead of counting exactly.
Winning Strategy
A plan or pattern of moves that guarantees a player will win a game if they play correctly.
e.g. In the game of 21 (adding 1, 2, or 3 each turn), the winning player should say numbers that are multiples of 4.
Number Line
A visual representation of numbers placed at appropriate positions along a line, usually with equal intervals.
e.g. Placing 2180, 2754, 1500, 3600, 9950, 9590, 1050, 3050, 5030, 5300, 8400 on a line from 1000 to 10,000.
Digit Detectives
A problem-solving activity to find how many times a particular digit occurs in a given range of numbers.
e.g. Among numbers 1–100, the digit '7' occurs 20 times.
Formulas & Equations (3)
Largest number from digits – Smallest number from digits = C
Reads as: Largest number minus smallest number equals C
Variables: Largest number: the greatest number formed by arranging digits in descending order; Smallest number: the smallest number formed by arranging digits in ascending order; C: the result of subtraction.
Use when: Applying the Kaprekar routine for 4-digit or 3-digit numbers.
⚠ Students often forget to use all digits exactly once, or they include leading zeros incorrectly when forming the smallest number.
If number is even: number ÷ 2; If number is odd: (number × 3) + 1
Reads as: If the number is even, take half of it. If the number is odd, multiply it by 3 and add 1.
Variables: number: any whole number; ÷ 2: divide by 2; × 3: multiply by 3; + 1: add 1.
Use when: Generating a Collatz sequence from any starting whole number.
⚠ Students sometimes apply the odd rule to even numbers or forget to repeat the process until reaching 1.
Number + Reverse of number = New number (repeat until palindrome)
Reads as: Number plus its reverse equals new number, repeat until you get a palindrome.
Variables: Number: any 2-digit or multi-digit number; Reverse: the number written with digits in opposite order.
Use when: Finding a reverse-and-add palindrome.
⚠ Students stop after one addition even if the result is not a palindrome, or they reverse the number incorrectly.
How It Works
Identifying Supercells in a Table
  1. Step 1: Look at each number in the table.
  2. Step 2: Identify its adjacent cells (immediately left, right, top, bottom — if they exist).
  3. Step 3: Compare the number with all its neighbours.
  4. Step 4: If the number is greater than all its neighbours, mark it as a supercell.
  5. Step 5: If a cell has only one neighbour (e.g., at an edge), compare only with that neighbour.
💡 The largest number in the table is always a supercell, but the smallest number can never be a supercell.
Kaprekar Routine (4-digit numbers)
  1. Step 1: Pick any 4-digit number with at least two different digits.
  2. Step 2: Arrange its digits in descending order to form the largest number (A).
  3. Step 3: Arrange its digits in ascending order to form the smallest number (B).
  4. Step 4: Subtract B from A to get C.
  5. Step 5: Use the digits of C as the new number and repeat steps 2–4.
  6. Step 6: Continue until you reach the Kaprekar constant 6174.
💡 The process always reaches 6174 for any 4-digit number with at least two different digits. For 3-digit numbers, the constant is 495.
Collatz Sequence Generation
  1. Step 1: Start with any whole number.
  2. Step 2: If the number is even, divide it by 2.
  3. Step 3: If the number is odd, multiply it by 3 and add 1.
  4. Step 4: Write down the new number.
  5. Step 5: Repeat steps 2–4 using the new number.
  6. Step 6: Continue until you reach the number 1.
💡 The Collatz conjecture says this always reaches 1, but it is still unproven for all numbers.
Reverse-and-Add Palindrome Method
  1. Step 1: Start with a 2-digit number.
  2. Step 2: Reverse the digits of the number.
  3. Step 3: Add the original number and its reverse.
  4. Step 4: Check if the sum is a palindrome (reads the same forwards and backwards).
  5. Step 5: If it is a palindrome, stop. If not, repeat steps 2–4 using the sum as the new number.
💡 For 2-digit numbers, this always yields a palindrome. For 3-digit numbers, it is unknown for some numbers like 196.
Winning Strategy for the Game of 21
  1. Step 1: The first player says 1, 2, or 3.
  2. Step 2: Players take turns adding 1, 2, or 3 to the previous number.
  3. Step 3: The first player to reach 21 wins.
  4. Step 4: The winning strategy is to always say numbers that are multiples of 4 (4, 8, 12, 16, 20).
  5. Step 5: The first player can win by starting with 1 and then always making the total a multiple of 4.
💡 The key is to control the multiples of the maximum addable number plus one (here, 3+1=4).
Exam Traps — Don't Make These Mistakes
⚠ Confusing 'adjacent' in Supercells
✗ Students often consider all cells in the same row or column as neighbours, not just the immediate left, right, top, and bottom.
→ The definition clearly states 'immediately to the left, right, top and bottom' — only the cells that touch the given cell directly.
✓ Only the cells that share a side with the given cell are its neighbours.
⚠ Forgetting to use all digits in Kaprekar routine
✗ Students sometimes drop a digit or use a digit more than once when forming the largest and smallest numbers.
→ The routine requires using all the digits of the current number exactly once to form the new largest and smallest numbers.
✓ Arrange all digits of the number in descending order for the largest, and ascending order for the smallest, using each digit exactly once.
⚠ Mistaking the Collatz rule for odd numbers
✗ Students write 'multiply by 3 and subtract 1' or 'add 1 without multiplying by 3'.
→ The rule is explicitly: if odd, multiply by 3 and add 1. Any variation changes the sequence completely.
✓ For an odd number, compute (number × 3) + 1.
⚠ Thinking the smallest number in a table can be a supercell
✗ Students sometimes mark the smallest number as a supercell if it is at an edge with only one neighbour.
→ A supercell must be greater than all its neighbours. The smallest number cannot be greater than any neighbour, so it can never be a supercell.
✓ The smallest number in a table can never be a supercell because all its neighbours are larger.
⚠ Confusing palindrome with symmetry in digits only
✗ Students think any number with repeated digits (like 1221) is a palindrome, but they forget to check if it reads the same backwards.
→ A palindrome must read exactly the same when the order of digits is reversed. 1221 reversed is 1221, so it is a palindrome. But 1231 reversed is 1321, so it is not.
✓ Reverse the number and compare with the original. If they are identical, it is a palindrome.
Compare & Contrast
Definition and relationship
Supercell
Largest number in a table
The largest number in a table is always a supercell, but a supercell does not have to be the largest number in the entire table — it only needs to be larger than its immediate neighbours.
Number of digits and the constant
Kaprekar Constant for 4-digit numbers (6174)
Kaprekar Constant for 3-digit numbers (495)
For 4-digit numbers with at least two different digits, the process always reaches 6174. For 3-digit numbers, the process always reaches 495. The steps are identical, but the final constant differs based on the number of digits.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 Short Answer (3 marks), 1 Long Answer (5 marks) — total 10 marks typically.
Most asked: Supercells: identifying and creating tables with maximum supercells · Palindromic numbers: writing all possible palindromes using given digits · Kaprekar routine: applying steps to reach 6174 or 495 · Digit sum: finding smallest/largest numbers with a given digit sum · Collatz conjecture: generating sequences and understanding the rule
Don't skip: Long answer on Kaprekar routine: show steps for a given 4-digit number to reach 6174; Creating a table with maximum supercells and explaining the pattern; Writing all 3-digit palindromes using given digits
One-Line Revision (read aloud before the exam)
  1. A supercell is a number greater than all its immediate neighbours (left, right, top, bottom).
  2. The largest number in a table is always a supercell; the smallest number can never be a supercell.
  3. A palindrome reads the same forwards and backwards, e.g., 121, 848, 1111.
  4. The Kaprekar constant for 4-digit numbers is 6174.
  5. The Kaprekar constant for 3-digit numbers is 495.
  6. Digit sum is the sum of all digits of a number, e.g., digit sum of 68 is 14.
  7. The Collatz rule: if even, divide by 2; if odd, multiply by 3 and add 1.
  8. The Collatz conjecture says every sequence reaches 1, but it is still unproven.
  9. Reverse-and-add: add a number to its reverse repeatedly until you get a palindrome.
  10. In the game of 21, the winning strategy is to always say multiples of 4.
  11. In the game of 99 (adding 1–10), the winning strategy is to always say multiples of 11.
  12. Estimation means finding an approximate value, not an exact count.
  13. Numbers can tell us things like height order, patterns, and information about neighbours.
  14. A 5-digit palindrome has the form abcba, e.g., 12421.
  15. The smallest 5-digit palindrome is 10001; the largest is 99999.

MCQs — 10 Questions with Answers

Q1. How many 1-digit numbers are there?

  • A. 5
  • B. 9 ✓
  • C. 10
  • D. 1

Answer: B — The 1-digit numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, which gives us exactly 9 numbers.

Q2. What is the digit sum of the number 456?

  • A. 12
  • B. 13
  • C. 14
  • D. 15 ✓

Answer: D — The digit sum is 4 + 5 + 6 = 15.

Q3. Which of the following is a palindrome?

  • A. 234
  • B. 345
  • C. 454 ✓
  • D. 456

Answer: C — 454 reads the same forwards and backwards, making it a palindrome.

Q4. In the taller neighbours game, what does a child say if neither neighbour is taller?

  • A. 0 ✓
  • B. 1
  • C. 2
  • D. 3

Answer: A — A child says 0 if neither of the children standing next to them are taller.

Q5. A supercell in a table must have a number that is ___________ than all its neighbours.

  • A. smaller
  • B. equal
  • C. greater ✓
  • D. different

Answer: C — A supercell is defined as a number that is greater than all the numbers in its neighbouring cells.

Q6. Which of these numbers has the same digit sum as 68?

  • A. 85
  • B. 77
  • C. 59 ✓
  • D. 48

Answer: C — 68 has digit sum 6+8=14, and 59 has digit sum 5+9=14, so both have the same digit sum.

Q7. How many 2-digit numbers exist in total?

  • A. 90 ✓
  • B. 99
  • C. 100
  • D. 81

Answer: A — 2-digit numbers run from 10 to 99, which gives us 99 - 10 + 1 = 90 numbers total.

Q8. On a number line marked with 0, 1000, 2000, where would the number 1500 be placed?

  • A. At the start
  • B. Halfway between 1000 and 2000 ✓
  • C. At 2000
  • D. Beyond 2000

Answer: B — 1500 is exactly halfway between 1000 and 2000 on a number line.

Q9. Which digit appears most frequently when writing numbers from 1 to 100?

  • A. 1 ✓
  • B. 5
  • C. 7
  • D. 9

Answer: A — The digit 1 appears many times: in 1, 10, 11 (twice), 12-19, 21, 31, 41, 51, 61, 71, 81, 91, and 100, making it the most frequent.

Q10. If you start with the number 19 and apply the reverse-and-add method, what is the result after the first step?

  • A. 90
  • B. 100
  • C. 110 ✓
  • D. 109

Answer: C — Reversing 19 gives 91, and 19 + 91 = 110.

Flashcards

What is a supercell in the table activity?

A supercell is a number that is greater than all the numbers in the cells next to it (left, right, top, or bottom neighbours).

What does digit sum mean?

Digit sum is the total you get when you add all the digits of a number together.

What is a palindromic number?

A palindromic number reads the same from left to right and from right to left, like 121, 343, or 5005.

How many 1-digit numbers exist from 1 to 9?

There are exactly 9 one-digit numbers: 1, 2, 3, 4, 5, 6, 7, 8, and 9.

How many 2-digit numbers exist?

There are 90 two-digit numbers, from 10 to 99.

In the taller neighbours game, what does a child say if both neighbours are taller?

A child says 2 if both children standing next to them are taller.

Can the largest number in a table always be a supercell?

Yes, the largest number in a table will always be a supercell because it is greater than all its neighbouring cells.

What is the digit sum of 68?

The digit sum of 68 is 14 because 6 + 8 = 14.

What are three examples of 3-digit palindromes using digits 1, 2, and 3?

Three examples are 121, 222, and 313.

In the reverse-and-add activity, what do you do first?

First, you take any number and reverse its digits, then add the original number to its reverse.

Important Board Questions

What is a supercell? Give one example from the chapter. [1 mark]

A supercell is a number greater than ALL its neighbouring cells. Look at the table examples where numbers are coloured — those are supercells.

Find the digit sum of 345. Is it the same as the digit sum of 543? Why or why not? [2 marks]

Add the digits: 3+4+5. Then add the digits of 543 and compare. The answer shows that digit order doesn't matter for digit sum.

In the taller neighbours game, explain why a child standing at the very end of the line can never say 2. Use a reason to support your answer. [3 marks]

A child at the end has only ONE neighbour (the child next to them), not two. So they can say at most 1 (if that neighbour is taller) or 0 (if not taller). A child needs TWO neighbours to say 2.

Create a 3×3 table of 4-digit numbers where exactly 2 cells are supercells. Explain why those two cells are supercells and the others are not. [5 marks]

Choose numbers so that exactly 2 are greater than ALL their neighbouring cells. The corner and edge numbers have fewer neighbours, while the centre number has 4 neighbours. Show your table and list each number with its neighbours.

True or False: The number 12321 is a palindrome. Give a reason for your answer. [2 marks]

Read the number forwards: 1-2-3-2-1. Now read it backwards. If they are the same, it is a palindrome.

Assertion-Reason Questions for Number Play

Q1. Assertion (A):

A palindromic number reads the same from left to right and from right to left.

Reason (R):

All numbers with repeated digits are palindromes.

Answer: C — A is true but R is false — The assertion is true—palindromes like 121 and 848 read the same both ways. However, the reason is false because not all numbers with repeated digits are palindromes. For example, 1221 has repeated digits and is a palindrome, but the reason is too narrow and doesn't explain why palindromes work.

Q2. Assertion (A):

In a table, the largest number will always be a supercell.

Reason (R):

Because the largest number is greater than all other numbers, it must be greater than all its adjacent cells.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The largest number in the entire table must be greater than all its neighbouring cells, making it a supercell by definition. The reason correctly explains why this is always true.

Q3. Assertion (A):

A child standing at the end of the line can say the number '2' in the height-based number game.

Reason (R):

Children at the ends of the line have only one neighbour, so they can say at most '1'.

Answer: D — A is false but R is true — The assertion is false, and the reason is true. Since children at the ends have only one neighbour, they can say only 0 or 1, never 2. The reason correctly explains why this is impossible.

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Case-Based Questions (CBQ) for Number Play

Case Study:

Rajesh is organizing his collection of trading cards. He arranges them in a display table with 9 cells arranged in a 3×3 grid. He fills the cells with card numbers: some are 4-digit numbers between 1000 and 9999. Rajesh notices that a cell becomes special (supercell) when the number in it is larger than all the numbers in the cells next to it (above, below, left, right). He wants to place cards numbered 5420, 7850, 3210, 6540, 4320, 8900, 2100, 5890, and 3450 in this grid to make as many supercells as possible.

Q1. Which card number should be placed in the centre cell to maximize the number of supercells?

  • A. 8900 ✓
  • B. 2100
  • C. 3210
  • D. 4320

Answer: A

Q2. If 5420 and 7850 are both supercells, what can we say about their neighbours?

  • A. Their neighbours are all smaller numbers ✓
  • B. Their neighbours are all larger numbers
  • C. They must be in the corner cells
  • D. They must be in the middle row

Answer: A

Q3. What is the digit sum of the largest card number in Rajesh's collection?

  • A. 24
  • B. 25
  • C. 26
  • D. 27 ✓

Answer: D

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