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NCERT Solutions Ganita Prakash Chapter 3 In-text Questions — An Unsolved Mystery

Book page 68 & 69 Updated on2026-09-05

Q1.
Look at the sequences below—the same rule is applied in all the sequences: a. 12, 6, 3, 10, 5, 16, 8, 4, 2, 1 b. 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 c. 21, 64, 32, 16, 8, 4, 2, 1 d. 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Do you see how these sequences were formed?
Answer

Yes. There is one rule with two parts, applied again and again:

If the number is even → take half of it
If the number is odd → multiply by 3 and add 1

Check sequence (a) step by step:

12 is even → 12 ÷ 2 = 6
6 is even → 6 ÷ 2 = 3
3 is odd → 3 × 3 + 1 = 10
10 is even → 5
5 is odd → 5 × 3 + 1 = 16
16 → 8 → 4 → 2 → 1

Sequence (c) is the shortest because 21 → 64, and 64 is a power of 2, so it just keeps halving: 64, 32, 16, 8, 4, 2, 1.

Did you notice? Every sequence ends with the same tail … 16, 8, 4, 2, 1. Once a sequence reaches a power of 2, the rest is only halving.
Q2.
Make some more Collatz sequences like those above, starting with your favourite whole numbers. Do you always reach 1?
Answer

Yes — every number tried so far reaches 1.

StartSequenceSteps
66, 3, 10, 5, 16, 8, 4, 2, 18
77, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 116
99, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 119
1515, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 117
3232, 16, 8, 4, 2, 15
Why the sequence goes up and down: an odd number is nearly tripled, so the sequence jumps upward; but the answer 3n + 1 is always even, so the very next step halves it. Over many steps the halving wins and the numbers come down to 1.
Q3.
Do you believe the conjecture of Collatz that all such sequences will eventually reach 1? Why or why not?
Answer

Yes, it certainly looks true — but nobody has been able to prove it, and that is exactly why it is famous.

  • Reasons to believe it: computers have checked every starting number up to more than 2,00,00,00,00,00,00,00,00,000 and each one comes down to 1. Also, whenever a number is odd, the next number 3n + 1 is even and gets halved at once — so halving happens at least half the time, and halving pulls the numbers down fast.
  • Why that is not a proof: checking crores of numbers is still not all numbers. There are infinitely many whole numbers, and mathematics demands a reason that covers every single one of them. Perhaps some huge number rises for ever, or gets stuck in a loop that never touches 1.
Did you know? Lothar Collatz proposed this in 1937. Almost ninety years later it is still open — a problem a Class 6 student can understand but no mathematician in the world has solved.
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