Question 5

Step 1 · Collatz Sequence for 10

Starting with $10$:

• $10$ is even $\implies 10 \div 2 = 5$
• $5$ is odd $\implies (3 \times 5) + 1 = 16$
• $16$ is even $\implies 16 \div 2 = 8$
• $8$ is even $\implies 8 \div 2 = 4$
• $4$ is even $\implies 4 \div 2 = 2$
• $2$ is even $\implies 2 \div 2 = 1$

$10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1$

Step 2 · Collatz Sequence for 15

Starting with $15$:

• $15$ is odd $\implies (3 \times 15) + 1 = 46$
• $46$ is even $\implies 46 \div 2 = 23$
• $23$ is odd $\implies (3 \times 23) + 1 = 70$
• $70$ is even $\implies 70 \div 2 = 35$
• $35$ is odd $\implies (3 \times 35) + 1 = 106$
• $106$ is even $\implies 106 \div 2 = 53$
• $53$ is odd $\implies (3 \times 53) + 1 = 160$
• $160$ is even $\implies 160 \div 2 = 80$
• $80$ is even $\implies 80 \div 2 = 40$
• $40$ is even $\implies 40 \div 2 = 20$
• $20$ is even $\implies 20 \div 2 = 10$
• $10$ is even $\implies 10 \div 2 = 5$
• $5$ is odd $\implies (3 \times 5) + 1 = 16$
• $16$ is even $\implies 16 \div 2 = 8$
• $8$ is even $\implies 8 \div 2 = 4$
• $4$ is even $\implies 4 \div 2 = 2$
• $2$ is even $\implies 2 \div 2 = 1$

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

Step 3 · Collatz Sequence for 19

Starting with $19$:

• $19$ is odd $\implies (3 \times 19) + 1 = 58$
• $58$ is even $\implies 58 \div 2 = 29$
• $29$ is odd $\implies (3 \times 29) + 1 = 88$
• $88$ is even $\implies 88 \div 2 = 44$
• $44$ is even $\implies 44 \div 2 = 22$
• $22$ is even $\implies 22 \div 2 = 11$
• $11$ is odd $\implies (3 \times 11) + 1 = 34$
• $34$ is even $\implies 34 \div 2 = 17$
• $17$ is odd $\implies (3 \times 17) + 1 = 52$
• $52$ is even $\implies 52 \div 2 = 26$
• $26$ is even $\implies 26 \div 2 = 13$
• $13$ is odd $\implies (3 \times 13) + 1 = 40$
• $40$ is even $\implies 40 \div 2 = 20$
• $20$ is even $\implies 20 \div 2 = 10$
• $10$ is even $\implies 10 \div 2 = 5$
• $5$ is odd $\implies (3 \times 5) + 1 = 16$
• $16$ is even $\implies 16 \div 2 = 8$
• $8$ is even $\implies 8 \div 2 = 4$
• $4$ is even $\implies 4 \div 2 = 2$
• $2$ is even $\implies 2 \div 2 = 1$

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

Step 1 · Evaluate the Conjecture

• Belief: Yes, it is reasonable to believe the conjecture is true based on extensive empirical evidence.
• Reasoning:
  • Empirical Support: Billions of starting numbers have been tested by computers, and every single sequence eventually reaches $1$. No counterexample has ever been found.
  • Pattern Behavior: Multiplying an odd number by $3$ and adding $1$ always yields an even number, which is then halved—often repeatedly—bringing the value back down.
• Note: While it is strongly supported by observation, it remains an unproven conjecture because a formal mathematical proof for all infinitely many numbers has not yet been discovered.