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Walk-through

How to Use This Calculator

3 steps

1

Enter a starting number

Type any positive whole number into the Starting number field — for example 27. The calculator immediately generates the full Collatz sequence, applying the 3n+1 rule at each step until the sequence reaches 1. No Calculate button — the result updates as you type.

2

Read the step count and peak value

The Steps tab shows the headline result: how many steps it takes to reach 1, plus the peak value the sequence climbs to along the way. The plain-language interpretation line below the result spells both numbers out.

3

Explore the full sequence and the peak

Switch to the Sequence tab to see every number in the chain and a line chart of the value at each step, or the Peak Value tab to see exactly which step produced the highest number. Use Share to copy a link that reopens your exact starting number, or CSV to download the full step-by-step sequence.

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Reference

Formula & Methodology

3 formulas

The 3n+1 rule

n -> n/2 if n is even; n -> 3n + 1 if n is odd; repeat until n = 1

Starting from any positive whole number n, apply the rule above repeatedly. If the current value is even, halve it; if it is odd, triple it and add 1. Every starting number tried so far eventually reaches 1 — but proving this happens for every possible starting number is the unsolved Collatz conjecture.

Stopping time (steps)

steps = number of iterations until n first equals 1

The stopping time (or total stopping time) counts how many applications of the rule it takes to go from the starting number down to 1. This calculator's Steps tab reports this count as the headline result.

Peak value

peak = max(sequence)

The peak is the largest number the sequence reaches before collapsing to 1. Many starting numbers climb far above their own value before coming back down — for example 27 climbs to 9,232 before it finally descends.

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Glossary

Key Terms Explained

6 terms

Collatz conjecture The unproven hypothesis, posed by Lothar Collatz in 1937, that repeatedly applying the 3n+1 rule to any positive whole number will always eventually reach 1. It has been verified computationally for every starting number up to enormous bounds, but no general proof exists.

3n+1 rule The rule at the heart of the conjecture: if the current number is even, divide it by 2; if it is odd, multiply it by 3 and add 1. Applying this repeatedly generates the Collatz sequence for a given starting number.

Hailstone numbers A nickname for the values in a Collatz sequence, because they rise and fall unpredictably — like a hailstone bouncing up and down in a storm cloud — before eventually crashing down to 1.

Stopping time The number of steps it takes a starting number to reach 1 under the 3n+1 rule. Also called the total stopping time. This calculator's Steps tab reports this value.

Sequence The full ordered list of numbers produced by repeatedly applying the 3n+1 rule to a starting number, from the starting value down to 1. This calculator's Sequence tab lists every term and charts them.

Iteration A single application of the 3n+1 rule to the current value — one even-halving or one odd-tripling-plus-one — that produces the next term in the sequence.

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Scenarios

Real-World Examples

3 worked examples

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Math student

What is the Collatz sequence for 27?

Starting number 27

Starting from 27, the sequence climbs surprisingly high — reaching a peak of 9,232 — before finally descending to 1 after 111 steps. It's one of the most famous examples of how unpredictable the 3n+1 rule can be for a relatively small starting number.

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Curious tinkerer

Which starting numbers under 100 produce the highest peak?

Starting number 97

97 reaches a peak of 9,232 as well, taking 118 steps to get there — even higher step counts than 27 despite a similar peak. Comparing different starting numbers on the Peak Value tab shows just how little the size of the starting number predicts the size of the climb.

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Pattern spotter

Do powers of 2 always resolve fast?

Starting number 16

16 -> 8 -> 4 -> 2 -> 1 in just 4 steps, with the peak staying at the starting value of 16 the whole way — because every step halves an even number, there's never a 3n+1 jump to push the sequence higher. Any power of 2 behaves this way: log2(n) steps, no peak above the start.

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Deep Dive

Understanding the Collatz Conjecture

3 chapters · 4 min read

The Collatz conjecture is deceptively simple to state — take any positive whole number, halve it if it's even, triple it and add one if it's odd, and repeat — yet it has resisted proof for nearly a century. This calculator generates the sequence for any starting number, reporting the number of steps to reach 1, the peak value along the way, and the full step-by-step chain.

How the sequence is generated

Starting from a chosen whole number, the calculator applies the same two rules over and over: an even number is divided by 2, and an odd number is tripled and has 1 added to it. The process repeats until the value hits 1, at which point the sequence stops. Every rule application is one iteration, and the total number of iterations needed is called the stopping time. This calculator's Steps tab reports that count directly, while the Sequence tab lists every intermediate value.

Why the sequence is unpredictable

Even though the rule is fixed, the resulting sequence can behave wildly — climbing far above the starting number before eventually collapsing. That's why the values are nicknamed hailstone numbers: like hail bouncing up and down inside a storm cloud before finally falling, a Collatz sequence can rise for many steps before it starts its final descent. The starting number 27, for example, climbs all the way to 9,232 — over 340 times its starting value — before it settles down to 1 after 111 steps. This calculator's Peak Value tab pinpoints exactly which step produced that maximum.

What's proven and what isn't

Every starting number ever tested — computationally verified into the trillions — eventually reaches 1. But the Collatz conjecture itself, first posed by German mathematician Lothar Collatz in 1937, claims this holds for every positive whole number, and nobody has proven it. It remains one of the most famous open problems in mathematics: easy enough to explain to a child, yet unsolved by professional mathematicians for over 85 years. Related tools like the Modulo Calculator and Sequences Calculator are useful for exploring other number patterns in the same spirit.

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Questions

Frequently Asked Questions

6 questions

What is the rule behind the Collatz conjecture?+

If the current number is even, divide it by 2. If it is odd, multiply it by 3 and add 1. Repeat this process on the result, and the conjecture claims you will always eventually reach 1, no matter which positive whole number you start from.

Does every starting number always reach 1?+

It's conjectured to, and every number tested so far — computationally verified well into the trillions — does reach 1. But no one has proven this holds for every possible positive whole number, which is exactly why it's still called a conjecture rather than a theorem.

What's the Collatz sequence for 27?+

Starting from 27, the sequence takes 111 steps to reach 1, climbing to a peak value of 9,232 along the way. It's one of the most commonly cited examples because a relatively small starting number produces such a long, high-climbing sequence.

What does the peak value mean?+

The peak is the highest number the sequence reaches before it eventually comes back down to 1. Some starting numbers barely rise above themselves — like powers of 2, which only ever go down — while others climb to many times their starting value before descending.

Has the Collatz conjecture been proven?+

No. It was first posed by Lothar Collatz in 1937 and remains an unsolved problem in mathematics. It has been verified by computer for an enormous range of starting numbers, but a general mathematical proof covering every positive whole number has not been found.

What units does the calculator use?+

The starting number and every value in the sequence are plain whole numbers (no units). The step count and peak value reported by the calculator are also unitless counts — steps taken and the largest number reached, respectively.

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Collatz Conjecture Calculator

n → n/2 (even) or 3n+1 (odd)

Result

How to Use This Calculator

Enter a starting number

Read the step count and peak value

Explore the full sequence and the peak

Formula & Methodology

Key Terms Explained

Real-World Examples

Math student

Curious tinkerer

Pattern spotter

Understanding the Collatz Conjecture

How the sequence is generated

Why the sequence is unpredictable

What's proven and what isn't

Frequently Asked Questions

Get a branded PDF of your results

Collatz Conjecture Calculator

n → n/2 (even) or 3n+1 (odd)

Result

How to Use This Calculator

Enter a starting number

Read the step count and peak value

Explore the full sequence and the peak

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the Collatz Conjecture

How the sequence is generated

Why the sequence is unpredictable

What's proven and what isn't

Frequently Asked Questions

Get a branded PDF of your results

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