NCERT Solutions for Class 6 Maths Chapter 3 Number Play Ex 3.12

3.12 Games and Winning Strategies 

Figure it Out (Page No. 72 – 73)

Question 1. This grid has only one supercell (number greater than all its neighbours). If you exchange two digits of one of the numbers, there will be 4 supercells. Figure out which digits to swap.

Solution:
If we swap first and last digit of central number 62,871, we get the desired result.

Question 2. How many rounds does your year of birth take to recall the Kaprekar constant?
Solution: If your year of birth is 2000
Step 1: Now from digits of number 2000
Here largest number = 2000
and smallest number = 0002
Let’s subtract them = 2000 – 0002 = 1998

Step 2: Now from digits of number 1998
Here largest number = 9981
and smallest number = 1899
Let’s subtract them = 9981 – 1899 = 8082

Step 3: Now from digits of number 8082
Here largest number = 8820
and smallest number = 0288
Let’s subtract them = 8820 – 0288 = 8532

Step 4: Now from digits of number 8532
Here largest number = 8532
and smallest number = 2358

Let’s subtract them = 8532 – 2358 = 6174

which is a Kaprekar constant.
Hence it took 4 rounds to reach the Kaprekar constant from 2000.

Question 3. We are the group of 5-digit numbers between 35,000 and 75,000 such that all of our digits are odd. Who is the largest number in our group? Who is the smallest number in our group? Who among us is the closest to 50,000?
Solution: 

The largest number with all odd digits (different) = 73951
The largest number with all odd digits (repetitive) = 73999
The smallest number (non repetitive) = 35,179
The smallest number (repetitive) = 57111
Closest to 50,000 (in case of non-repetition) = 49751
Closest to 50,000 (in case of repetition) = 49999

Question 4. Estimate the number of holidays you get in a year including weekends, festivals and vacation. Then try to get an exact number and see how close your estimate is.
Solution:
Will be done by students.

Question 5. Estimate the number of litres a mug, a bucket and an overhead tank can hold.
Solution:
Will be done by students

Question 6. Write one 5-digit number and two 3-digit numbers such that their sum is 18,670. ‘
Solution:
5 digit number = 1 8 0 0 0
3 digit number = 6 7 0
Sum = 1 8 000 + 670 = 18670

Question 7. Choose a number between 210 and 390. Create a number pattern similar to those shown in Section 3.9 that will sum up to this number.

Solution:
Sum of No. = 5 × 1 = 5
+ 10 × 3 = 30
+ 15 × 5 = 75
+ 20 × 7 = 140 = 250
which lies between 210 and 390.

Question 8. Recall the sequence of Powers of 2 from Chapter 1, Table 1. Why is the Collatz conjecture correct for all the starting numbers in this sequence?
Solution: The square of power of 2 is :
1,2,4, 8, 16, 32, 64

Let’s take the number’ 64 as per Collatz Conjecture

• 64 is even, divide by 2 = 32

• 32 is even, divide by 2 = 16

• 16 is even, divide by 2 = 8

• 8 is even, divide by 2 = 4

• 4 is even, divide by 2 = 2

• 2 is even, divide by 2 = 1

Hence Collatz conjecture is correct in all numbers in the power of 2 sequence.
As it is power of 2, and in Collatz Conjecture even number is divided by 2 in each step.

Question 9. Check if the Collatz Conjecture holds for the starting number 100.
Solution:

• 100 is even, divide by 2 =50

• 50 is even, divide by 2 = 25

• 25 is odd, so multiply by 3 and add 1 → 76

• 76 is even, divide by 2 = 38

• 38 is even, divide by 2 = 19

• 19 is odd, so multiply by 3 and add 1 → 58

• 58 is even, divide by 2 = 29

• 29 is odd, so multiply by 3 and add 1 → 88

• 88 is even, divide by 2 = 44

• 44 is even, divide by 2 = 22

• 22 is even, divide by 2 = 11

• 11 is odd, so multiply by 3 and add 1 → 34

• 34 is even, divide by 2 = 17 

• 17 is odd, so multiply by 3 and add 1 → 52

• 52 is even, divide by 2 = 26

• 26 is even, divide by 2 = 13

• 13 is odd, so multiply by 3 and add 1 → 40

• 40 is even, divide by 2 = 20

• 20 is even, divide by 2 = 10

• 10 is even, divide by 2 = 5

• 5 is odd, so multiply by 3 and add 1 → 16

• 16 is even, divide by 2 = 8

• 8 is even, divide by 2 = 4

• 4 is even, divide by 2 = 2

• 2 is even, divide by 2 = 1

Yes, the Collatz conjecture holds for the starting number 100. 

Benefits of NCERT Solutions for Class 6 Maths Ex 3.12

• Enhances problem-solving skills: By learning to analyze game strategies, students improve their ability to solve problems logically.

• Improves reasoning ability: The chapter sharpens the reasoning skills of students by teaching them to recognize patterns and predict outcomes.

• Practical application of maths: Students learn how maths can be applied to real-life situations through games.

• Boosts critical thinking: The chapter encourages critical thinking as students explore various strategies to win games.

Class 6 Maths Chapter 3: Exercises Breakdown

Important Study Material Links for Maths Chapter 3 Class 6

Conclusion

Chapter 3 Excercise 3.12  Games and Winning Strategies from Class 6 Maths is an engaging and interactive way to strengthen mathematical concepts. Students not only enjoy the process of learning but also understand how to apply mathematical reasoning in real-life scenarios. This chapter is instrumental in building logical thinking and problem-solving abilities through fun and strategic games.

Chapter-wise NCERT Solutions Class 6 Maths

After familiarising yourself with the Class 6 Maths  Chapters Question Answers, you can access comprehensive NCERT Solutions from all Class 6 Maths textbook chapters.

Related Important Links for Class 6  Maths 

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