Question 5

We will follow two simple rules to make number sequences.

Step 1 — Collatz Sequence for 10 Let us start with the number 10. We check if it is even or odd. 10 is an even number. So, we divide 10 by 2. $10 \div 2$ $= 5$ Now we have 5. 5 is an odd number. So, we multiply 5 by 3. Then we add 1. $(3 \times 5) + 1$ $= 15 + 1$ $= 16$ Now we have 16. 16 is an even number. So, we divide 16 by 2. $16 \div 2$ $= 8$ Now we have 8. 8 is an even number. So, we divide 8 by 2. $8 \div 2$ $= 4$ Now we have 4. 4 is an even number. So, we divide 4 by 2. $4 \div 2$ $= 2$ Now we have 2. 2 is an even number. So, we divide 2 by 2. $2 \div 2$ $= 1$ We stop when we reach 1. The sequence for 10 is:

$\boxed{10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1}$

Step 2 — Collatz Sequence for 15 Let us start with the number 15. We follow the same rules. 15 is odd, so $(3 \times 15) + 1 = 46$. 46 is even, so $46 \div 2 = 23$. 23 is odd, so $(3 \times 23) + 1 = 70$. 70 is even, so $70 \div 2 = 35$. 35 is odd, so $(3 \times 35) + 1 = 106$. 106 is even, so $106 \div 2 = 53$. 53 is odd, so $(3 \times 53) + 1 = 160$. 160 is even, so $160 \div 2 = 80$. 80 is even, so $80 \div 2 = 40$. 40 is even, so $40 \div 2 = 20$. 20 is even, so $20 \div 2 = 10$. 10 is even, so $10 \div 2 = 5$. 5 is odd, so $(3 \times 5) + 1 = 16$. 16 is even, so $16 \div 2 = 8$. 8 is even, so $8 \div 2 = 4$. 4 is even, so $4 \div 2 = 2$. 2 is even, so $2 \div 2 = 1$. The sequence for 15 is:

$\boxed{15 \to 46 \to 23 \to 70 \to 35 \to 106 \to 53 \to 160 \to 80 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1}$

Step 3 — Collatz Sequence for 19 Let us start with the number 19. We follow the same rules. 19 is odd, so $(3 \times 19) + 1 = 58$. 58 is even, so $58 \div 2 = 29$. 29 is odd, so $(3 \times 29) + 1 = 88$. 88 is even, so $88 \div 2 = 44$. 44 is even, so $44 \div 2 = 22$. 22 is even, so $22 \div 2 = 11$. 11 is odd, so $(3 \times 11) + 1 = 34$. 34 is even, so $34 \div 2 = 17$. 17 is odd, so $(3 \times 17) + 1 = 52$. 52 is even, so $52 \div 2 = 26$. 26 is even, so $26 \div 2 = 13$. 13 is odd, so $(3 \times 13) + 1 = 40$. 40 is even, so $40 \div 2 = 20$. 20 is even, so $20 \div 2 = 10$. 10 is even, so $10 \div 2 = 5$. 5 is odd, so $(3 \times 5) + 1 = 16$. 16 is even, so $16 \div 2 = 8$. 8 is even, so $8 \div 2 = 4$. 4 is even, so $4 \div 2 = 2$. 2 is even, so $2 \div 2 = 1$. The sequence for 19 is:

$\boxed{19 \to 58 \to 29 \to 88 \to 44 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26 \to 13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1}$

Step 4 — Do we always reach 1? All our examples ended at 1. The sequence always reached 1. This happens for many, many starting numbers. No one has found such a number. It does not reach 1. So, all sequences seem to reach 1.

Step 5 — Believing the Collatz Conjecture The Collatz conjecture says this. All sequences will reach 1. I believe the Collatz conjecture could be true. Here is why I think so. Many people have checked billions of numbers. Every single number they checked eventually reached 1. No one has found such a number. It does not reach 1. This is called empirical support. The rules make numbers smaller when they are even. Odd numbers become bigger. But then they usually become even. This often leads to the number becoming smaller again. This repetitive nature suggests sequences get reduced quickly. However, it is still a mystery. No one has proven it for all numbers yet. A very big number might exist. It does not reach 1. But based on what we know. It seems very likely to be true.

Answer

(i) Collatz Sequence Examples:

• Starting with 10: 10 → 5 → 16 → 8 → 4 → 2 → 1
• Starting with 15: 15 → 46 → 23 → 70 → 35 → 106 → 53 → 160 → 80 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
• Starting with 19: 19 → 58 → 29 → 88 → 44 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 (ii) Yes, in the above examples and many others, the sequence always eventually reaches 1. This pattern has been observed for a vast range of starting numbers, though it has not been formally proven for all possible numbers. (iii) I believe the Collatz conjecture could be true, based on empirical evidence. Every tested sequence eventually reaches 1, even for very large numbers. This makes it seem likely that the conjecture holds for all numbers. Reasoning:
• Empirical support: No counterexample has been found despite extensive testing.
• Intuition: The repetitive nature of the rules (halving even numbers and making odd numbers smaller after multiplying and adding) suggests that most sequences get reduced quickly. However, since the problem is still unsolved, there is always the possibility of an exception for some very large, unknown number.