IdentityForge

See why the identities are true. Cut a square, watch the pieces, and the algebra becomes a picture.

(a + b)² = a² + 2ab + b²

Box It Build (a+b)², (a+b)(c+d) and numbers as area — and snip-test an identity. Concepts 16 focused rounds across the identities — squares, differences, cubes, Pascal & the seven remarkable identities.

More ways to play

Identity Run

Meet the cast — the characters who teach this

Each one stands for an idea you’ll practice. Tap a character to read their illustrated story and hear their audio drama.

Tila The whole square (hero) - Mayor Tila the tortoise, whose square field of side (a+b) is made of four lots: the a-square, the b-square, and the two matching a-by-b strips; count every lot and you have counted (a+b) squared
Miro & Moro The matching twins - two otters whose two a-by-b strips are congruent; turn one a quarter-turn and it lands exactly on the other, so the two strips together are 2ab, never a lonely single ab
Kerf The snip test - a stag beetle who cuts a square anywhere; a claim that still balances after cut after cut, at any value, is not a lucky answer, it is an identity
Boxer The box method - a badger who draws (a+b)(c+d) as a grid; each room is one partial product (ac, ad, bc, bd), every pair gets its own room, and the sum of the rooms is the product
Overlap The corner counted twice - an octopus who shrinks a square to side (a-b) by removing two strips; the little corner where they cross is removed twice, so add it back once: a-squared minus 2ab plus b-squared
Dreamer The protege - a daydreaming fox who keeps only the two corner squares and forgets the two middle strips (the Freshman's Dream); the friend the learner teaches to count all four lots

Explore all the stories →

Every piece and every check is computed by a deterministic engine — the picture is a proof, not a decoration. On-device · no accounts · nothing leaves the device.