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

Why does Pascal's triangle make a fractal?

AA HLAA SL Solid Also in: Art & music, Fractals & chaos

Research question

Why do the odd entries of Pascal's triangle form the Sierpiński triangle, and how many odd entries are there in row n?

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Why it makes a good exploration

Colour the odd numbers in Pascal's triangle and a fractal appears. Explaining why — and finding that the number of odd entries in row n is 2 to the power of the number of 1s in n's binary expansion — is a beautiful, surprising result.

The mathematics you'll need

Binomial coefficients and Pascal's rule
Arithmetic modulo 2
Binary representation
Proof by induction
Self-similarity

Where the data comes from

Generate rows with a spreadsheet; check counting sequences on OEIS.

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

A possible outline

Generate and colour the triangle mod 2.
Observe self-similarity after 2ᵏ rows.
Conjecture and prove the doubling structure.
Count odd entries per row and link to binary.
Reflect on other moduli (mod 3, mod 5).

Pitfalls that cost marks

Pictures without proof.
Proof steps that skip the key idea (Pascal's rule mod 2).
Claiming Lucas' theorem without explanation.

Showing personal engagement

Make the triangle by hand and describe the moment the pattern appeared.
Explore mod 3 and compare.
Relate to a fractal you have seen elsewhere.

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	Binomial coefficients and Pascal's rule; Arithmetic modulo 2
AA HL	Good fit	Binomial coefficients and Pascal's rule; Arithmetic modulo 2
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: pictures without proof; and proof steps that skip the key idea (Pascal's rule mod 2). Say how each one affects your answer.

Use of mathematics (E)

SL: binomial coefficients and Pascal's rule; arithmetic modulo 2; binary representation; proof by induction; self-similarity, 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

Prove the odd-entry count formula, or compute the fractal dimension log 3 / log 2.

Extending it for HL

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

See a complete IA, marked

Is a hanging chain a parabola? Comparing catenary and quadratic models (AA 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 HL and AA SL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Binomial coefficients and Pascal's rule, arithmetic modulo 2, binary representation, proof by induction and self-similarity.

Where can I get data for this IA?

Generate rows with a spreadsheet; check counting sequences on OEIS. Checked sources and tools: OEIS (On-Line Encyclopedia of Integer Sequences).

Turn this idea into your IA

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