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The Collatz conjecture (also called the 3n+1 problem) asks a deceptively simple question: starting from any positive number, if you repeatedly halve even numbers and triple-plus-one odd numbers, do you always eventually reach 1? Every number ever tested says yes — but no one has ever proven it's true for all numbers.
This calculator simulates the Collatz sequence for any starting number, showing every step.
How it works
Enter a starting number. The calculator repeatedly applies the Collatz rule — divide by 2 if even, multiply by 3 and add 1 if odd — tracking every value until the sequence reaches 1, and reports the step count and maximum value reached along the way.
- Enter starting number.
- Click Calculate to see your results.
Examples
Starting from 27
The famously long Collatz sequence starting from 27 takes 111 steps to reach 1, climbing as high as 9,232 along the way before finally descending.
Who should use it
- Exploring Collatz conjecture behavior for a math enrichment project or coursework.
- Checking the step count and sequence for a specific starting number.
Industry applications
- Mathematics education
- Recreational mathematics and computer science
Advantages
- Tracks and displays the full sequence, not just the final step count.
- Reports the maximum value reached, which is often surprisingly higher than the starting number.
Limitations
- Capped at 10,000 steps as a safety limit — no counterexample has ever been found, so this limit is never expected to be reached in practice.
Common mistakes to avoid
- Assuming the sequence always decreases steadily — it can climb dramatically higher than the starting number before eventually coming back down (as with 27, which climbs to 9,232).
- Believing the conjecture has been mathematically proven — it hasn't, despite being true for every number ever tested.
Best practices
- Try a range of starting numbers to see how unpredictably the step count and maximum value can vary, even between very close starting numbers.
Tips
- The Collatz conjecture is a popular example in computer science courses of a simple-to-state algorithm whose termination (does it always finish?) is unproven — a good illustration of the difference between "true for every case we've tried" and "proven true in general."