The IdentityForge cast
Each friend shows you one idea by how they act. Meet them all and see how the ideas fit together.
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Tila The whole square - 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