Collatz Conjecture Visualizer

The Collatz conjecture (also called the 3n+1 problem) is a mathematical conjecture that states: starting from any positive integer n, repeatedly applying the rule “if even, divide by 2; if odd, multiply by 3 and add 1” will eventually reach the number 1. Despite its simple rules, it has not been proven for all numbers, making it one of the most famous unsolved problems in mathematics.

Hailstone numbers refer to the sequences generated by the Collatz process. They are called hailstone numbers because the values often rise and fall dramatically—like hailstones in a thundercloud—before eventually falling to 1.

Enter any positive integer (e.g., 27) into the input box and click Calculate. The tool computes the complete hailstone sequence, displays key statistics (steps to reach 1, peak value, peak step), plots the sequence on a chart, and lists all sequence numbers. Use the Random button to explore a random starting number, and switch between linear and logarithmic scales for a different perspective.

When the rule 3n+1 is applied to an odd number, the result grows rapidly. For some starting numbers, the sequence can climb to unexpectedly high peaks before eventually decreasing. For example, starting with 27 the peak reaches 9,232, even though the number itself is modest.

No, the Collatz conjecture remains unproven as of today. It has been verified for all numbers up to at least 2⁶⁸ (approximately 2.95×10²⁰), but a general proof is still elusive. The problem is known for its deceptive simplicity and the difficulty of finding a universal proof.

The longest sequence found within the verified range has a stopping time of over 2,000 steps. The exact record evolves as larger numbers are tested, but interestingly the maximum number of steps often appears from relatively small starting values. For instance, the number 27 takes 111 steps to reach 1.

The tool uses JavaScript's native number type, which can precisely represent integers up to 2⁵³–1 (about 9 quadrillion). Very large numbers may lose precision or cause extremely long sequences that slow down the browser. We recommend starting with numbers below 10¹² for the best experience.