What is the Collatz conjecture?

The Collatz conjecture (also called the 3n+1 problem) states that for any positive integer, if you repeatedly apply the rule — divide by 2 if even, multiply by 3 and add 1 if odd — the sequence will always eventually reach 1. Despite its simplicity, this conjecture remains unproven and is one of the most famous unsolved problems in mathematics.

Why is it called the hailstone sequence?

The sequence is nicknamed the hailstone sequence because the values rise and fall unpredictably, similar to how hailstones are carried up and down by air currents inside a cloud before eventually falling to the ground (reaching 1).

Has the Collatz conjecture been proven?

No, the Collatz conjecture has not been proven or disproven despite decades of research. It has been verified computationally for all numbers up to approximately 2^68, but a general proof remains elusive. Mathematician Paul Erdős famously said that mathematics is not yet ready for such problems.

What starting number under 10,000 has the longest sequence?

Among numbers under 10,000, the number 6,171 has one of the longest Collatz sequences, taking 261 steps to reach 1. The number 9,663 takes 184 steps. You can use this tool to explore and compare different starting numbers.

Why does the sequence always seem to reach 1?

While no proof exists, the intuition is that the ÷2 operation reduces values more frequently and more powerfully than the ×3+1 operation increases them. On average, about two-thirds of the values in a sequence are even, so the halving steps dominate, gradually driving the sequence down to 1.

What is the maximum value a Collatz sequence can reach?

There is no known upper bound on peak values. For example, starting at 27, the sequence peaks at 9,232 — over 340 times the starting value. Starting at 871, the peak exceeds 190,000. Larger starting numbers can produce dramatically higher peaks before converging to 1.

Can I use this tool for programming assignments?

Yes, this tool is perfect for verifying your own Collatz sequence implementations. You can compare your program's output against the step-by-step results shown here, including total steps, peak values, and the full sequence to ensure correctness.

Is the sequence generation done locally?

Yes, all Collatz sequence computation happens entirely in your browser using JavaScript. No data is sent to any server, ensuring complete privacy. The algorithm runs instantly for all starting numbers up to 10,000.

Collatz Sequence Generator

Generate Collatz sequences (3n+1 problem) for any starting number with step tracking, max value analysis, and sequence visualization. Explore the famous unsolved conjecture with detailed hailstone sequence generation and statistical breakdowns.

collatz

3n+1

hailstone

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Notable starting numbers for the Collatz conjecture

Collatz Sequence Options

Enter a starting number to generate the 3n+1 sequence

Starting Number

Enter a positive integer between 1 and 10,000

How it works: The Collatz conjecture (3n+1 problem) applies a simple rule repeatedly: if n is even, divide by 2; if n is odd, multiply by 3 and add 1. The conjecture states that every positive integer eventually reaches 1, though this remains unproven.

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