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

How many times can you fold a sheet of paper?

AA SLAA HL Solid Also in: Pure maths

Research question

Does Britney Gallivan's folding-limit formula predict how many times I can fold strips of different paper, and how does the required length grow with the number of folds?

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Why it makes a good exploration

In 2002 a high-school student, Britney Gallivan, derived a formula for the paper needed to fold n times in one direction and then folded paper 12 times. Testing her formula with your own paper is a real piece of mathematics with a great story.

The mathematics you'll need

Exponential growth and 2ⁿ
Sequences and series (the length lost at each fold)
Deriving a formula from a geometric model
Logarithms to solve for n

Modelling the data, step by step

This idea usually needs an exponential model. See it worked step by step, with a criterion tip at every step: Exponential y = a e^(kx) + c and y = a·bˣ.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

Where the data comes from

Measure paper thickness (stack of 100 sheets), fold strips of known length in one direction until you cannot, and record the result; use toilet paper or tracing paper for long strips.

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 folding model: each fold loses a semicircle of paper.
Derive the length formula L = (πt/6)(2ⁿ + 4)(2ⁿ − 1) step by step.
Test predictions for several papers and strip lengths.
Solve for the maximum n given a strip length.
Reflect on why real folds lose more than the model predicts.

Pitfalls that cost marks

Quoting the formula without deriving it.
Not measuring thickness accurately (it dominates the result).
Confusing one-direction and alternate-direction folding.

Showing personal engagement

Predict your maximum folds before trying.
Do the experiment with friends in a corridor and document it.
Derive the formula for alternate-direction folding yourself.

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	Exponential growth and 2ⁿ; Sequences and series (the length lost at each fold)
AA HL	Good fit	Exponential growth and 2ⁿ; Sequences and series (the length lost at each fold)
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

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: quoting the formula without deriving it; and not measuring thickness accurately (it dominates the result). Say how each one affects your answer.

Use of mathematics (E)

SL: exponential growth and 2ⁿ; sequences and series (the length lost at each fold); deriving a formula from a geometric model; logarithms to solve for n, each explained in your own words and carried out correctly.

How modelling explorations reach the top bands in general: Modelling with functions IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

HL: prove the length-lost series by induction and explore how sensitive n is to measurement error in thickness.

Extending it for HL

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

See a complete IA, marked

When can we launch? Modelling the tide at our sailing club's slipway (AA SL) is a complete modelling exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

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

What mathematics does this IA use?

Exponential growth and 2ⁿ, sequences and series (the length lost at each fold), deriving a formula from a geometric model and logarithms to solve for n.

Where can I get data for this IA?

Measure paper thickness (stack of 100 sheets), fold strips of known length in one direction until you cannot, and record the result; use toilet paper or tracing paper for long strips. Checked sources and tools: OEIS (On-Line Encyclopedia of Integer Sequences).

Turn this idea into your IA

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