1. You take 81 cards and write 1 on nine of them, 2 on nine of them,3 on nine of them,..., 9 on nine of them. Then you lay out the cards of the initial board configuration face up (so the number is in evidence).Then you think. Then you find your solution and put the other cards face down.
Now when the Verifier comes, you offer any row, any column, or any 3x3 Sudoku box. The verifier points to one or the other. You take those 9 cards make them all face down and shuffle them. After you have shuffled them sufficiently, you demonstrate to the Verifier that you have all numbers between 1 and 9.
2. You send me out of the room and you take a few grains, say some number between 1 and 30, after covering the bucket with a cape. You call me back in. If I know how many you've removed, then I'm either very lucky or I may have the ability I claim. If you try this test 100 times, then you might be convinced.
The problem of the sand counter can be solved by most kids 10 or older. If you wish, try it out on a kid you know. The Sudoku zero knowledge proof was developed by Professor Michael Rabin.
Reading: The origin of zero-knowledge proofs comes from a now-classic paper by Shafi Goldwasser, Silvio Micali, and Charles Racoff. The paper was rejected several times before it first appeared. What do program committees know?