Recall the sequence of Powers of 2 from Chapter 1, Table 1. Why is the Collatz conjecture correct Class 6

Recall the sequence of Powers of 2 from Chapter 1, Table 1. Why is the Collatz conjecture correct Class 6

Question 1.

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? Class 6

Solution:

Sequence of powers of 2 is 1, 2, 4, 8, 16, 32, .......... Here, 1 itself ends in 1. Regarding other powers of 2 say 64, we will get

[latex]\frac{64}{2}[/latex] = 32, [latex]\frac{32}{2}[/latex] = 16, [latex]\frac{16}{2}[/latex] = 8, [latex]\frac{8}{2}[/latex] = 4, [latex]\frac{4}{2}[/latex] = 2, [latex]\frac{2}{2}[/latex] = 1.

Thus, it ends in 1. Similarly, other powers like 128 etc., they also ends in 1, such as

[latex]\frac{128}{2}[/latex] = 64, [latex]\frac{64}{2}[/latex] = 32, [latex]\frac{32}{2}[/latex] = 16, ......., [latex]\frac{4}{2}[/latex] = 2, [latex]\frac{2}{2}[/latex] = 1.

ends in 1. Hence, Collatz conjecture is correct for all the starting numbers of the sequence of powers of 2.

Question 2.

Check if the Collatz Conjecture holds for the starting number 100. Class 6

Solution:

[latex]\frac{100}{2}[/latex] = 50, [latex]\frac{50}{2}[/latex] = 25, 25 × 3 + 1 = 76,

[latex]\frac{76}{2}[/latex] = 38, [latex]\frac{38}{2}[/latex] = 19, 19 × 3 + 1 = 58, [latex]\frac{58}{2}[/latex] = 29,

29 × 3 + 1 = 88, [latex]\frac{88}{2}[/latex] = 44, [latex]\frac{44}{2}[/latex] = 22, [latex]\frac{22}{2}[/latex] = 11,

11 × 3 + 1 = 34, [latex]\frac{34}{2}[/latex] = 17, 17 × 3 + 1 = 52,

[latex]\frac{52}{2}[/latex] = 26, [latex]\frac{26}{2}[/latex] = 13, 13 × 3 + 1 = [latex]\frac{40}{2}[/latex] = 20,

[latex]\frac{20}{2}[/latex] = 10, [latex]\frac{10}{2}[/latex] = 5, 5 × 3 + 1 = 16, [latex]\frac{16}{2}[/latex] = 8,

[latex]\frac{8}{2}[/latex] = 4, [latex]\frac{4}{2}[/latex] = 2, [latex]\frac{2}{1}[/latex] = 1.

Thus, Collatz Conjecture holds.

Question 3.

Starting with 0, players alternate adding numbers from 1 to 3. The first person to reach 22 wins. What is the winning strategy now? Class 6

Solution:

In this game, the winning strategy for the winner is to say such a number which on addition give a multiple of 2 (except 20) because 2 × 11 = 22 and 22 is the next multiple of 2 after 20.