Collatz Sequence Calculator

Frequently Asked Questions

What is the Collatz conjecture?

It states that if you repeatedly halve even numbers and triple-plus-one odd numbers, any positive starting integer eventually reaches one. It has been verified for enormous ranges of numbers but remains mathematically unproven.

Why does the number 27 take so long?

Starting from 27, the sequence climbs to a peak above nine thousand and takes over a hundred steps before settling to one. Small starting numbers can produce surprisingly long and high sequences, which is part of the puzzle's appeal.

What are the rules of the Collatz sequence?

Start with any positive whole number. If the current value is even, halve it; if it is odd, triple it and add one. Repeat the rule until you reach one. The calculator reports the number of steps taken and the highest value the sequence reached along the way.

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How This Calculator Works

The Collatz sequence is a simple iterative process applied to a positive whole number. If the current value is even, you halve it; if it is odd, you triple it and add one. You repeat this rule over and over. The famous Collatz conjecture states that no matter which positive integer you start with, the sequence always eventually reaches one, though no one has proven it.

How To Use It

Enter any starting integer of one or more. The calculator runs the rule until the value reaches one and reports the number of steps it took and the highest value the sequence climbed to along the way. For safety the iteration stops after ten thousand steps, which is far more than any small starting number needs.

A Worked Example

Start at 7, an odd number. Tripling and adding one gives 22. That is even, so halve it to 11, then 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, and finally 1. The path took 16 steps and peaked at 52 even though we began with a single-digit number. This zig-zag of climbs and drops is typical: every odd value pushes the sequence up, and each run of halving pulls it back down toward one.

Hailstone Numbers and Terminology

Because the values rise and fall like hailstones tossed inside a storm cloud before dropping to the ground, the numbers a Collatz sequence visits are nicknamed hailstone numbers. Mathematicians distinguish two related counts: the total stopping time is the number of steps to reach one, while the stopping time is the shorter count of steps needed to first drop below the starting value. The conjecture has been verified by computer for every starting value up to roughly 2 to the 68th power, yet a general proof has resisted every attempt since Lothar Collatz proposed it in 1937.

What The Numbers Mean

The step count is the total number of operations needed to reach one, also called the stopping time. The peak value shows how high the sequence soared before collapsing back down. Some innocent-looking starting numbers produce surprisingly long sequences and very large peaks before settling.

Tips

Even numbers always shrink; odd numbers grow first.

The number 27 famously takes over a hundred steps.

The conjecture remains unproven for all integers.

Collatz Sequence Calculator

Frequently Asked Questions

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