About

Proof you can hold in your hands

A combinatorial proof counts one collection two different ways. Because you counted the same thing, the two totals have to match — and that equality is a real, complete proof. CountCraft turns that idea into tiles: build a shape, count it one way, count it the other way, and the identity falls out. It’s the concrete, see-it-first rung of proof (before you ever write algebra).

What you’re really doing

• The staircase. Two copies of 1 + 2 + … + n tile an n × (n+1) rectangle, so the sum is exactly half of it: n(n+1)/2. (That’s also the number of handshakes among n+1 people.)
• The growing square. Adding the next odd number is adding an L-shaped layer to a square, so 1 + 3 + 5 + … + (2n−1) is always n².
• The cut square. A square of side (a+b) splits into four rectangles — a², two a·b’s, and b² — so (a+b)² = a² + 2ab + b².

Honest promise

CountCraft makes proof tactile — it helps you see why an identity is true before you write it symbolically. That’s a real, standards-adjacent skill (constructing viable arguments), but we don’t claim it raises a test score. Everything runs on your device: no account, no timer, no streak, nothing leaves your phone.

The picture-proof method is the art of combinatorial proof — the tradition of proving identities by counting a set two ways. All tilings, examples, and wording here are our own.