The proof shop

Count it two ways. Prove it forever.

Here is a maths superpower: if you can count the same pile of tiles two different ways, the two answers must be equal — and that is a proof. No algebra needed to see it is true. Build the picture, count both ways, and the shortcut appears.

Build it Count Two Ways Eight picture proofs you build with tiles: the staircase (1+2+…+n), the twin staircase, the growing square (odd numbers), the cut square ((a+b)²), the cube tower (Σk³), the Fibonacci spiral, the rings of eight (odd squares), and the three-colour triangle (the sum of squares, 1²+2²+…+n²).

The idea Double counting Count one collection two ways → the counts are equal → an identity is proven. The oldest trick in combinatorics, made from tiles.

More About CountCraft Why picture proofs work, what maths you’re really doing, and the honest promise.

See it, then say it

Every proof here is a picture first. Once you can see why 1 + 2 + … + n has a shortcut, the formula stops being something to memorise and becomes something you could re-draw from scratch. Want the counting facts behind it? Explore permutations and combinations at DiscreteQuest.

Honest promise: this makes proof tactile and gives you a picture you can trust — it isn’t a claim to raise a test score. On-device, no account, no timer, no streak, nothing leaves your phone. Inspired by the art of combinatorial proof (credit on the About page).