Collatz Conjecture Sequence Visualizer - 3n+1 Problem

How to use this tool

1. Enter a whole number from 1 to 1,000,000 and click Visualize.
2. The chart shows the full hailstone sequence to 1.
3. Switch between linear and logarithmic scales, then open the sequence table.

What is the Collatz conjecture?

Start with any positive integer n. If it is even, divide it by 2; if it is odd, multiply it by 3 and add 1. Repeat those steps. The resulting sequence is called a Collatz sequence or hailstone sequence because its values can rise and fall before reaching 1.

The conjecture says that every positive integer eventually reaches 1, then enters the cycle 1 → 4 → 2 → 1. It has not been proved for all positive integers. As of 2025, computer calculations have verified convergence for every integer below 2⁷¹. This is a finite computational verification, not a proof of the conjecture for all integers. See Barina’s 2025 paper and the project verification log.

How to read the visualization

Enter a starting number and the chart plots its sequence down to 1. The horizontal axis is the step number and the vertical axis is the value at that step; the chart offers linear and logarithmic vertical scales. A high peak does not mean a long sequence: the path from 27 reaches 9,232 at step 77, then takes 34 more steps to reach 1. The total stopping time is the number of transitions needed to reach 1, so a sequence with 111 steps contains 112 values including its start and end. In some Collatz literature, “stopping time” means the first step where a value falls below its starting value; that is a different measurement.

Notable starting numbers

Small changes to the starting value can produce sequences with very different lengths and peaks. For example, powers of 2 descend by repeated halving, while odd values can first grow under the 3n + 1 rule. These examples are useful for checking the chart and for seeing why sequence length and peak height describe different parts of a trajectory.

• 27 takes 111 steps and reaches a peak of 9,232 at step 77. It has the longest sequence among starting values below 54.
• 97 is the record below 100: it takes 118 steps to reach 1.
• 871 reaches a peak of 190,996 and takes 178 steps.
• 837,799 is the record below 1,000,000, with 524 steps.
• Powers of 2 halve directly to 1 and have no odd transitions.

Why it matters

The Collatz rule is simple to state, but its sequences can behave in ways that are hard to predict from the starting value. The problem connects elementary arithmetic with questions in number theory and dynamical systems. Paul Erdős reportedly said, “Mathematics is not yet ready for such problems.” Despite extensive computation, no proof or counterexample is known.

Comparing starting numbers

Comparing two sequences on paper or in separate runs shows how different starting values can have very different paths. For example, 27 takes 111 steps and rises to 9,232, while 32 halves monotonically to 1 in five steps. A power of 2 descends at every step because each value remains even until it reaches 1. The visualizer plots one starting value at a time, so this comparison is a mathematical explanation rather than an overlaid comparison view.

Stopping times across starting values

Total stopping time counts the transitions from a chosen starting number all the way to 1. A related stopping time is the first transition where a sequence falls below its starting value. Researchers can compare either quantity across many starting values; the resulting counts have irregular peaks and valleys rather than a simple known formula. Numbers that set a new record for total stopping time over a growing range are sometimes called record holders or champion numbers. This page currently visualizes one sequence at a time and does not plot a range or distribution.