Collatz Step
The rule of the famous unsolved Collatz conjecture.
Formula
Variables
Example
Always seems to reach 1
How it works
This rule defines one step of the Collatz process: if the current positive integer is even, divide it by 2; if it is odd, multiply by 3 and add 1. The reason it is interesting is that it creates a simple repeated transformation with surprisingly chaotic behavior, yet many numbers quickly fall into a path that eventually reaches 1. It comes from studying iteration in number theory, where the main question is whether every positive integer will ultimately reach 1 under repeated use of this step.
When to use it
Use this when exploring iterative number patterns, recursive sequences, or classic open problems in number theory. It is also useful in algorithmic demonstrations of how a simple rule can generate complex behavior.
Worked example, step by step
Start with n = 7 7 is odd, so apply 3n + 1: 3(7) + 1 = 22 22 is even, so divide by 2: 22 / 2 = 11 11 is odd, so apply 3n + 1: 3(11) + 1 = 34 34 is even, so divide by 2: 34 / 2 = 17 17 is odd, so apply 3n + 1: 3(17) + 1 = 52 52 is even, so divide by 2: 52 / 2 = 26 26 is even, so divide by 2: 26 / 2 = 13 So the Collatz steps begin: 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13
Common mistakes to avoid
- Applying 3n + 1 to even numbers instead of odd numbers
- Forgetting to divide by 2 only one step at a time
- Assuming the rule proves every number reaches 1, which is still unproven
Did You Know?
The Collatz conjecture is so simple a child grasps it, yet it has defied every mathematician for 90 years.
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