Macmillan Academy The Scholar Lab

Maths · KS5 · problem solving

A Proof Puzzle

At university, the answer matters less than the argument. This is a taste of admissions-style maths — less calculation, more watertight reasoning.

The challenge

Consider the sum of the first n odd numbers: 1, 1+3, 1+3+5, 1+3+5+7, …

Part A. Make a conjecture for a simple formula for the sum of the first n odd numbers.

Part B. Prove it two different ways: once with a picture (a visual argument), and once by induction (algebra).

A proof must convince for every value of n — not just the ones you tried. Checking examples is how you find a pattern, not how you prove it.

How to get started

  1. Work out the first few sums by hand. What numbers appear? (1, 4, 9, 16, …) — what are these?
  2. State your conjecture as a clean formula in terms of n.
  3. For the picture: try arranging dots into a square and adding the odd numbers as L-shaped layers.
  4. For induction: show it's true for n = 1, then show that IF it's true for n = k it must be true for n = k+1.

Stuck? Open a hint

Try the challenge first — then reveal these one at a time.

Hint 1
The running totals are the square numbers: 1, 4, 9, 16, … so the conjecture is that the sum of the first n odd numbers is n².
Hint 2
Picture: an n×n square of dots can be built from L-shaped layers of size 1, 3, 5, 7, … — each new odd number completes the next bigger square.
Hint 3
Induction: assume 1+3+…+(2k−1) = k². Add the next odd number, 2k+1: k² + (2k+1) = (k+1)². Done.

Take it further

Try a classic proof by contradiction: show that √2 cannot be written as a fraction (is irrational). Assume it can, and find the contradiction.

Then explore real admissions material via the free STEP Support Programme from Cambridge.

Produce & share

Make something: Submit your two proofs that the first n odd numbers sum to n² — the visual ('picture') proof and the proof by induction — written out neatly. Add the √2 proof if you tried the extension.

When it's ready, email your work to Mr King. The best pieces may be featured in the student showcase.

What you'll practise

Rigorous proof, mathematical induction, proof by contradiction, and communicating an argument clearly — exactly what STEP and TMUA reward.