Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

The Collatz conjecture: what can we say about stopping times?

AA SLAA HLAI SLAI HL Solid Also in: Statistics, Probability, Cryptography

Research question

How are the stopping times of the Collatz sequence distributed for starting values up to 10⁶, and does a simple probabilistic model explain why sequences tend to fall?

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Why it makes a good exploration

The Collatz conjecture is unsolved, so you cannot prove it — but you can investigate it. A heuristic argument that on average each step multiplies by about 3/4 explains why sequences usually come down, and is a lovely use of expected value and logarithms.

The mathematics you'll need

Iteration and sequences
Statistics of stopping times
Expected value and geometric means
Logarithms to model average decrease
Honest discussion of heuristic vs proof

Where the data comes from

Generate data with a spreadsheet or short program; compare with OEIS records.

OEIS (On-Line Encyclopedia of Integer Sequences) — Check a sequence you have found and read its known formulas and references.

A possible outline

Explain the rule and the conjecture.
Generate stopping times and analyse their distribution.
Build the probabilistic model.
Test the model's prediction of average stopping time against log n.
Reflect on why heuristics are not proofs.

Pitfalls that cost marks

Claiming to prove the conjecture.
Huge tables with no analysis.
Not explaining the program used.

Showing personal engagement

Find a starting number with a long sequence yourself.
Explore a variant (5n + 1) and see it fail.
Discuss why famous mathematicians consider it hard.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Iteration and sequences; Statistics of stopping times
AA HL	Good fit	Iteration and sequences; Statistics of stopping times
AI SL	Good fit	Iteration and sequences; Statistics of stopping times
AI HL	Good fit	Iteration and sequences; Statistics of stopping times

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: claiming to prove the conjecture; and huge tables with no analysis. Say how each one affects your answer.

Use of mathematics (E)

SL: iteration and sequences; statistics of stopping times; expected value and geometric means; logarithms to model average decrease; honest discussion of heuristic vs proof, each explained in your own words and carried out correctly.

How pure maths explorations reach the top bands in general: Pure maths, number & proof IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Fit stopping time against ln n and compare the gradient with your model's prediction.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL pure maths exploration adds.

See a complete IA, marked

Should I lease or buy my first car? (AI SL) is a complete pure maths exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Iteration and sequences, statistics of stopping times, expected value and geometric means, logarithms to model average decrease and honest discussion of heuristic vs proof.

Where can I get data for this IA?

Generate data with a spreadsheet or short program; compare with OEIS records. Checked sources and tools: OEIS (On-Line Encyclopedia of Integer Sequences).

Turn this idea into your IA

The planner opens with “The Collatz conjecture: what can we say about stopping times?” filled in: the question, the maths and the data sources.

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