A zero-knowledge proof convinces a verifier that you know a secret without revealing it. Here: graph 3-coloring. Peggy knows a valid 3-coloring of a graph. Each round she commits to a random permutation of colors (hidden), Victor picks an edge, Peggy reveals its endpoints. Completeness: honest Peggy always passes. Soundness: a cheater passes each round with probability (E−1)/E.