I’ve just uploaded to youtube a video I made with Katie Steckles to demonstrate why zero-knowledge protocols exist and how one works.
Katie is a habitual liar, so we followed the zero-knowledge protocol described in the paper, “Cryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles” which you can download from http://www.mit.edu/~rothblum/papers/sudoku.pdf
By following this protocol, Katie can prove that she isn’t lying to me about being able to solve the puzzle, without revealing anything about how she solved it.
The paper I mentioned, “How to explain zero-knowledge protocols to your children” is an excellent explanation of the ideas behind zero-knowledge proof. You can get it from http://pages.cs.wisc.edu/~mkowalcz/628.pdf