IA idea · Pure maths, number & proof
Why does Pascal's triangle make a fractal?
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?
Adapt it: change the place, the data or the comparison until the question is yours.
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
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
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.
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
- 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.
Taking it further
Prove the odd-entry count formula, or compute the fractal dimension log 3 / log 2.