Understanding the Collatz Conjecture
The Collatz conjecture, also known as the 3n + 1 problem, is a mathematical sequence defined as follows:
- If the starting number is even, divide it by 2.
- If the starting number is odd, multiply it by 3 and add 1.
- Repeat this process until you reach the number 1.
Applying the Collatz Conjecture to 22
Starting with the number 22, we will follow the steps outlined by the conjecture.
Step-by-Step Sequence:
- 22 is even: 22 / 2 = 11
- 11 is odd: 3 * 11 + 1 = 34
- 34 is even: 34 / 2 = 17
- 17 is odd: 3 * 17 + 1 = 52
- 52 is even: 52 / 2 = 26
- 26 is even: 26 / 2 = 13
- 13 is odd: 3 * 13 + 1 = 40
- 40 is even: 40 / 2 = 20
- 20 is even: 20 / 2 = 10
- 10 is even: 10 / 2 = 5
- 5 is odd: 3 * 5 + 1 = 16
- 16 is even: 16 / 2 = 8
- 8 is even: 8 / 2 = 4
- 4 is even: 4 / 2 = 2
- 2 is even: 2 / 2 = 1
Conclusion
After following the sequence starting from 22, we eventually arrive at the number 1. This confirms that the Collatz conjecture holds true for the starting number 22, as we reached 1 after a series of calculations.
Key Takeaway
- The Collatz conjecture is intriguing because it suggests that no matter what positive integer you start with, you will always reach 1.
This pattern has been tested for many numbers, yet a general proof remains elusive, making it a captivating topic in mathematics.