CBSE Class 6 Mathematics Number Play MCQs Set 11

Answer: (b) greater than the numbers in all the cells next to it

Explanation:
1. A supercell is a cell whose number is greater than the numbers in every cell next to it.
2. Being smaller, being equal to a neighbour or being a palindrome has nothing to do with it.

Teacher's Note:
1. Super means bigger than all its neighbours.
2. Only the cells that touch it are compared.

Answer: (c) 4554

Explanation:
1. A palindrome reads the same from left to right and from right to left.
2. 4554 read backwards is again 4554, so it is a palindrome.
3. 1234, 5678 and 1101 change when read backwards (4321, 8765, 1011). So the answer is 4554.

Teacher's Note:
1. Read the number from the right end to check.
2. 1101 is a trap: backwards it is 1011.

Answer: (d) 6174

Explanation:
1. Make the largest and the smallest number from the digits, subtract, and repeat.
2. For 4-digit numbers these steps always end at 6174, the Kaprekar constant.
3. 495 is the number reached for 3-digit numbers, not 4-digit ones.

Teacher's Note:
1. 6174 for four digits, 495 for three digits.
2. Learn both numbers together.

Answer: (c) 900

Explanation:
1. The 3-digit numbers start at 100 and end at 999.
2. Count = 999 − 100 + 1 = 900.

Teacher's Note:
1. Add 1 when counting from one number to another, both included.
2. Similarly there are 90 two-digit numbers.

Answer: (b) 4

Explanation:
1. 7,310 is greater than 2,450, 3,560 and 1,480, so it is a supercell.
2. 6,120 is greater than 2,450, 1,480 and 4,275, so it is a supercell.
3. 5,905 is greater than 3,560, 1,480 and 2,019, so it is a supercell.
4. 8,640 is greater than 4,275, 1,480 and 2,019, so it is a supercell.
5. Every other cell has a bigger neighbour. So there are 4 supercells.

Teacher's Note:
1. Check each cell against up, down, left and right only.
2. Corner cells have just two neighbours.

Answer: (b) multiplying it by 3 and adding 1

Explanation:
1. Collatz rule: halve an even number; for an odd number, multiply by 3 and add 1.
2. For example, after 5 comes 3 × 5 + 1 = 16. So the answer is multiplying it by 3 and adding 1.

Teacher's Note:
1. Even: halve. Odd: 3 times plus 1.
2. Taking half is the rule for even numbers.

Answer: (a) Only Statement I is true

Explanation:
1. In the activity every child calls out how many of their neighbours are taller, so Statement I is true.
2. A child at an end has only one neighbour, so that child can say only 0 or 1, never 2. Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. Count the neighbours a child has before deciding.
2. An end child has just one neighbour.

Answer: (a) 10001

Explanation:
1. The first digit must be as small as possible but not 0, so it is 1; the last digit must then also be 1.
2. The middle digits should be the smallest possible, 0, 0, 0.
3. This gives 10001. (10000 is not a palindrome, since backwards it reads 00001.)

Teacher's Note:
1. A palindrome's first and last digits must match.
2. 10000 is the smallest 5-digit number, not palindrome.

Answer: (c) 8,000

Explanation:
1. The first two marks are 7,996 and 7,997, and each mark is 1 more than the one before.
2. So the marks read 7,996, 7,997, 7,998, 7,999, 8,000, …
3. The arrow is on the 5th mark, which is 8,000.

Teacher's Note:
1. Count the marks carefully from the first label.
2. After 7,999 comes 8,000.

Answer: (d) 800

Explanation:
1. 487 rounded to the nearest hundred is 500, and 311 rounded is 300.
2. Estimate = 500 + 300 = 800.

Teacher's Note:
1. Look at the tens digit to decide whether to round up or down.
2. The exact sum, 798, is close to 800.

Answer: (d) A is false, but R is true.

Explanation:
1. Every cell next to the smallest number holds a bigger number, so the smallest number can never be a supercell. A is false.
2. R correctly gives the meaning of a supercell, so R is true.

Teacher's Note:
1. The smallest number loses to every neighbour.
2. R is just the definition, and it is correct.

Answer: (a) 1, 2 or 3

Explanation:
1. In the game of 21, each player adds 1, 2 or 3 to the number just said, and whoever says 21 wins.
2. Adding any number from 1 to 10 is the rule of the game that ends at 99.

Teacher's Note:
1. 21 game: add 1, 2 or 3.
2. 99 game: add 1 to 10.

Answer: (b) (i)-(q), (ii)-(s), (iii)-(r), (iv)-(p)

Explanation:
1. 1-digit numbers: 1 to 9, so 9 of them (q). 2-digit numbers: 10 to 99, so 90 (s).
2. 3-digit numbers: 100 to 999, so 900 (r). 4-digit numbers: 1000 to 9999, so 9,000 (p).
3. The match is (i)-(q), (ii)-(s), (iii)-(r), (iv)-(p).

Teacher's Note:
1. Each time one more digit gives ten times as many numbers.
2. 9, 90, 900, 9,000 is the pattern.

Answer: (c) 6174

Explanation:
1. The digits are 3, 8, 5, 2. Largest A = 8532 and smallest B = 2358.
2. A − B = 8532 − 2358 = 6174.
3. So 3852 reaches the Kaprekar constant in just one round.

Teacher's Note:
1. Arrange digits in decreasing order for A, increasing order for B.
2. Subtract carefully with borrowing.

Answer: (d) 10, 5

Explanation:
1. 3 is odd, so the next number is 3 × 3 + 1 = 10.
2. 10 is even, so the next number is 10 ÷ 2 = 5.
3. The next two numbers are 10, 5.

Teacher's Note:
1. Decide even or odd at every step.
2. Do not halve an odd number.

Answer: (b) Only Statement II is true

Explanation:
1. 9,999 + 99 = 10,098, which has 5 digits. So the sum is not always 4-digit, and Statement I is false.
2. Even this largest possible sum, 10,098, has only 5 digits, so a 6-digit sum is impossible. Statement II is true.
3. So only Statement II is true.

Teacher's Note:
1. Test the largest numbers to find the biggest sum.
2. One example is enough to show 'always' is false.

Answer: (c) A is true, but R is false.

Explanation:
1. For every 2-digit number, reversing and adding again and again does end in a palindrome, so A is true.
2. A number like 48 is not a palindrome (backwards it is 84), so R is false.

Teacher's Note:
1. 2-digit palindromes are only 11, 22, …, 99.
2. One counter-example makes R false.

Answer: (a) 59

Explanation:
1. No 1-digit number has digit sum 14, so we need a 2-digit number.
2. To keep the number small, keep the tens digit small and the ones digit as big as possible: 9.
3. Tens digit = 14 − 9 = 5, so the number is 59.

Teacher's Note:
1. Put the 9s at the right end to make a small number.
2. Fewer digits always means a smaller number.

Answer: (c) 450

Explanation:
1. The plus-shaped figure has 5 cells in the column and 4 more cells in the row, so 9 cells in all.
2. Each cell holds 50. Instead of adding one by one, multiply: 9 × 50 = 450.

Teacher's Note:
1. Do not count the middle cell twice.
2. Count the cells, then multiply.

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. 9,998 comes just after 9,997 and just before 9,999, so A is true.
2. 9,999 is indeed the largest 4-digit number, so R is true.
3. But R does not say anything about where 9,998 lies, so R does not explain A.

Teacher's Note:
1. Both can be true and still unrelated.
2. Ask: does R give the reason for A?

Answer: (d) (i)-(q), (ii)-(s), (iii)-(r), (iv)-(p)

Explanation:
1. After 10:01 the next palindromic time is 11:11 (q). After 2:05 it is 2:12 (s).
2. After 12:30 it is 1:01 (r). After 2:00 it is 2:02 (p).
3. So the match is (i)-(q), (ii)-(s), (iii)-(r), (iv)-(p).

Teacher's Note:
1. Ignore the colon and read the digits both ways.
2. On a 12-hour clock 12:59 is followed by 1:00.

Answer: (d) A is false, but R is true.

Explanation:
1. The Collatz conjecture has not been proved; it is still an unsolved problem. A is false.
2. Every chain tried so far has reached 1, so R is true. But testing many numbers is not a proof.

Teacher's Note:
1. Conjecture means believed but not proved.
2. Many examples are not the same as a proof.

Answer: (a) 1000

Explanation:
1. 312 → 300, 489 → 500 and 198 → 200.
2. Estimate = 300 + 500 + 200 = 1000 books.

Teacher's Note:
1. Round first, then add.
2. The exact total, 999, shows the estimate is very close.

Answer: (a) Only Statement I is true

Explanation:
1. Two cells next to each other cannot both be supercells, so supercells can sit only in alternate cells, e.g. 9, 1, 8, 2, 7. That gives at most 3. Statement I is true.
2. The cell with the largest number is always a supercell, so a row with no supercell is impossible. Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. Neighbouring cells cannot both be supercells.
2. The largest number is always a supercell.

Answer: (c) 5

Explanation:
1. Round 1: 9521 − 1259 = 8262. Round 2: 8622 − 2268 = 6354.
2. Round 3: 6543 − 3456 = 3087. Round 4: 8730 − 0378 = 8352.
3. Round 5: 8532 − 2358 = 6174. So it takes 5 rounds.

Teacher's Note:
1. Keep a 0 as a digit when it appears (3087 → 8730 and 0378).
2. Write each round on a new line.

Answer: (b) 73,999

Explanation:
1. The number must be less than 75,000, so the first digit can be at most 7.
2. With 7 first, the second digit must be odd and less than 5, so the biggest choice is 3.
3. Use the largest odd digit 9 everywhere else: 73,999.

Teacher's Note:
1. 74,999 has the even digit 4.
2. 75,555 and 79,999 are more than 75,000.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. The safe numbers are 11, 22, 33, …, 99. Whatever the other player adds (1 to 10), the safe player adds the rest to reach the next multiple of 11.
2. The first player can say only 1 to 10, never 11, so the second player can say 11 and stay on the multiples of 11 up to 99.
3. Both A and R are true, and R explains A.

Teacher's Note:
1. Work backwards from 99 in steps of 11.
2. The step is 1 more than the largest number allowed.

Answer: (d) (i)-(q), (ii)-(s), (iii)-(p), (iv)-(r)

Explanation:
1. 6174 is the Kaprekar constant for 4-digit numbers (q); 495 repeats for 3-digit numbers (s).
2. Every Collatz chain is believed to reach 1 (p); 196 is the 3-digit number suspected never to give a palindrome (r).
3. So the match is (i)-(q), (ii)-(s), (iii)-(p), (iv)-(r).

Teacher's Note:
1. Make a small card of the chapter's special numbers.
2. Match the easiest one first.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. 10,000 − 99 = 9,901 is the smallest possible answer, and it has 4 digits. So R is true.
2. Any other such subtraction gives a bigger answer, so a 3-digit answer is impossible. A is true.
3. R gives exactly the reason for A.

Teacher's Note:
1. Use the extreme numbers to find the smallest answer.
2. Smallest minus largest gives the least difference.

Answer: (b) Only Statement II is true

Explanation:
1. The tallest child has no taller neighbour and always says 0, so all five can never say 1. Statement I is false.
2. End children can never say 2, and two neighbours cannot both say 2 (one of them is taller). So only the 2nd and 4th children can say 2. Statement II is true.
3. So only Statement II is true.

Teacher's Note:
1. The tallest child always says 0.
2. A child who says 2 is shorter than both neighbours.