Physical Zero-Knowledge Proof for Ripple Effect

Ripple Effect is a logic puzzle with an objective to fill numbers into a rectangular grid divided into rooms. Each room must contain consecutive integers starting from 1 to its size. Also, if two cells in the same row or column have the same number $x$, the space separating the two cells must be at least $x$ cells. In this paper, we propose a physical protocol of zero-knowledge proof for Ripple Effect puzzle using a deck of cards, which allows a prover to physically show that he/she knows a solution without revealing it. In particular, we develop a physical protocol that, given a secret number $x$ and a list of numbers, verifies that $x$ does not appear among the first $x$ numbers in the list without revealing $x$ or any number in the list.