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?

Adapt it: change the place, the data or the comparison until the question is yours.

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

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to 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.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

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.

Taking it further

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

Turn this idea into your IA

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