The reason why some odd numbers convert to an even number immediately after applying the transformation $\frac{3N + 1}{2}$, while others require multiple steps, is rooted in the behavior of the Collatz function, also known as the 3n + 1 problem.

The Transformation Process:

For any odd number $N$, the next number in the sequence is determined by: $\text{Next number} = \frac{3N + 1}{2}$ This transformation ensures that the result is always an integer because $3N + 1$ is always even (since $3 \times \text{odd number}$ is odd, and odd + 1 = even).

Key Points:

1. Immediate Conversion to Even:

   • Some odd numbers, when multiplied by 3 and increased by 1, yield a value that is directly divisible by 2, giving an even number.
   • Example: For $N = 13$, the next number is: $\frac{3 \times 13 + 1}{2} = \frac{39 + 1}{2} = \frac{40}{2} = 20 \quad (\text{even number})$ Here, 13 directly converts to 20 in one step.

2. Multiple Steps to Convert to Even:

   • For some odd numbers, the process doesn't lead to an even number immediately after the first transformation. Instead, the sequence requires multiple iterations before reaching an even number.
   • Example: For $N = 7$, the sequence proceeds as: $\frac{3 \times 7 + 1}{2} = \frac{21 + 1}{2} = \frac{22}{2} = 11 \quad (\text{odd})$ Since 11 is still odd, we apply the process again: $\frac{3 \times 11 + 1}{2} = \frac{33 + 1}{2} = \frac{34}{2} = 17 \quad (\text{odd})$ Continuing: $\frac{3 \times 17 + 1}{2} = \frac{51 + 1}{2} = \frac{52}{2} = 26 \quad (\text{even})$ So, 7 requires 4 steps to convert to an even number.

Why the Difference?

The difference lies in the structure of the number itself and how it interacts with the multiplication by 3 and addition of 1. Specifically:

• Some odd numbers, when transformed by $3N + 1$, yield a number that is immediately even and large enough to divide cleanly by 2, resulting in an even number right away.
• Other odd numbers, due to their particular value, lead to another odd number after the transformation. This continues until a transformation finally results in an even number.

Conclusion:

The varying number of steps to reach an even number arises from the specific mathematical properties of each odd number and how they interact with the $3N + 1$ operation. Some numbers naturally align to produce even numbers quicker, while others cycle through additional odd numbers before converting.

Would you like further details on the Collatz conjecture or other related topics?

Related Questions:

1. What is the Collatz conjecture, and how does it relate to these sequences?
2. Can all odd numbers eventually reach an even number using this method?
3. How do even numbers behave differently in the Collatz sequence?
4. What is the significance of the number of steps in these sequences?
5. Are there any known patterns or formulas to predict the number of steps needed?

Tip:

When studying sequences like the Collatz sequence, it can be useful to visualize the process using a flowchart or graph to see how numbers transform step by step.