Solutions: Sec 3.10 Collatz Conjecture

Page 69: Questions

Q. Make some more Collatz sequences. Do you always reach 1? Answer: Yes, try any number (e.g., 6, 7, 100). They eventually spiral down to 4, 2, 1.

Q9. Check for 100. $100 \to 50 \to 25 \to 76 \to 38 \to 19 \to 58 \to 29 \to 88 \to 44 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26 \to 13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1$. Verified.

Q10 (Page 73). Game Strategy change. Start with 0. Add 1, 2, or 3. Reach 22. Strategy: Target is 22. Allowed moves 1-3. Winning numbers are multiples of 4 away from target. $22 - 4 = 18$ $18 - 4 = 14$ $14 - 4 = 10$ $10 - 4 = 6$ $6 - 4 = 2$. First player strategy: Say 2. Then whatever opponent says ($x$), you say ($4-x$). You will hit 6, 10, 14, 18, 22.