Collatz Conjecture Calculator

About the Collatz Conjecture Calculator

The **Collatz Conjecture Calculator** lets you explore one of the most famous unsolved problems in mathematics — the 3n+1 conjecture, also known as the hailstone sequence or Syracuse problem. Enter any positive integer and watch the algorithm iterate: if the number is even, divide by 2; if odd, multiply by 3 and add 1. The conjecture states that every starting number eventually reaches 1, but no one has been able to prove it.

This calculator computes the full Collatz sequence for your chosen starting number, displays the total number of steps to reach 1, identifies the maximum value the sequence hits along the way, and calculates the stopping time (the first step where the value drops below the starting number). You can choose how many terms to display, toggle between showing all steps or just key milestones, and set a maximum iteration limit for safety.

A detailed sequence table shows every step with the value at that step, the operation applied (n/2 or 3n+1), and whether the value is even or odd. The step chart visual plots the sequence as a bar graph so you can see the dramatic rises and falls that give hailstone sequences their name. Preset buttons load famous examples like 27 (which reaches 9,232 before returning to 1 in 111 steps) and other interesting starting values.

Whether you are a student learning about iterative sequences, a math enthusiast exploring open problems, or a teacher looking for an engaging classroom demonstration, the page makes the Collatz conjecture tangible and interactive.

When This Page Helps

Collatz sequences can run for hundreds of steps and spike to enormous values — manually iterating even a modest starting number like 27 (111 steps, peak 9,232) is impractical. This calculator computes the full sequence, charts the dramatic rises and falls, and highlights milestones so you can focus on the mathematical patterns. Number-theory students explore open conjectures interactively, teachers demonstrate how simple rules produce complex behaviour, and recreational mathematicians hunt for extreme stopping times.

How to Use the Inputs

1. Enter a positive starting integer.
2. Optionally set a maximum iteration limit.
3. Select a display mode to show all steps or key milestones only.
4. Click a preset to explore famous starting values.
5. Review the output cards for total steps, max value, and stopping time.
6. Scroll the sequence table to follow each step of the algorithm.
7. Examine the bar chart visual to see the rise-and-fall pattern.

Example Calculation

Tips & Best Practices

• Check that all inputs use the same scale and assumptions before trusting the result.
• Compare the answer with the worked example or a rough estimate to catch entry mistakes.

The 3n+1 Problem Explained

The rule is deceptively simple. Take any positive integer n. If it is even, divide by 2; if it is odd, compute 3n + 1. Repeat. The Collatz conjecture asserts that no matter what starting value you choose, the sequence will always eventually reach the cycle 4 → 2 → 1. Despite being easy to state, the problem has resisted proof since Lothar Collatz first posed it in 1937. Paul Erdős famously commented that "mathematics may not be ready for such problems." The conjecture has been computationally verified for all integers up to roughly 2.95 × 10²⁰, yet no one knows whether a counterexample — a starting value whose sequence diverges or enters a different cycle — exists.

Surprising Behaviour and Record-Holders

What makes the Collatz conjecture captivating is the gap between the simplicity of the rule and the wildness of the resulting sequences. The number 27 is the classic example: its sequence climbs to a peak of 9,232 before finally descending to 1 after 111 steps. Larger starting values can produce even more extreme behaviour — 77,031 takes 350 steps and peaks above 21 million. Powers of 2 reach 1 in exactly log₂(n) steps (trivially, since every step halves), while odd numbers tend to spike via the 3n + 1 rule before collapsing. There is no known formula to predict total stopping time from the starting value; the relationship appears chaotic.

Open Questions and Partial Results

Mathematicians have proven several partial results. In the late 2010s, Terrence Tao showed that "almost all" Collatz orbits attain values as small as desired — loosely, the set of potential counterexamples has logarithmic density 0. However, this falls short of the full conjecture, which requires *every* orbit to reach 1. Other research focuses on the statistical distribution of stopping times and the density of even/odd steps. The 3n + 1 map can be extended to negative integers, where cycles other than 4 → 2 → 1 are known (e.g., −1 → −2 → −1). Exploring these related maps and generalisations (such as 5n + 1 or qx + 1) is an active area of recreational and professional mathematics alike.

Frequently Asked Questions

• What is the Collatz conjecture?

  It states that for any positive integer, repeatedly applying the rule (n/2 if even, 3n+1 if odd) will eventually reach 1. It remains unproven.

• What is a hailstone sequence?

  Another name for the Collatz sequence, because the values rise and fall like hailstones in a cloud before eventually landing (reaching 1).

• What is stopping time?

  The number of steps before the sequence first drops below the starting value. For n = 6 the sequence hits 3 at step 2, so stopping time is 2.

• Has the conjecture been verified computationally?

  Yes, it has been verified for all starting values up to about 2.95 × 10^20, but no general proof exists.

• Why is 27 famous in Collatz?

  Starting from 27, the sequence takes 111 steps and reaches a peak of 9,232 — surprisingly high for such a small starting number.

• Can the sequence go on forever?

  The conjecture says no, but it is unproven. This calculator uses a configurable iteration limit to prevent infinite loops in case of very long sequences.

• What patterns appear in Collatz sequences?

  Powers of 2 reach 1 very quickly. Odd numbers tend to spike up before falling. The sequence length is unpredictable from the starting value.