Physical Zero-Knowledge Proofs for Five Cells

Five Cells is a logic puzzle consisting of a rectangular grid, with some cells containg a number. The player has to partition the grid into pentominoes such that the number in each cell must be equal to the number of edges of that cell that are borders of pentominoes. In this paper, we propose two physical zero-knowledge proof protocols for Five Cells using a deck of playing cards, which allow a prover to physically show that he/she knows a solution of the puzzle without revealing it. In the optimization of our first protocol, we also develop a technique to reduce the number of required cards from quadratic to linear in the number of cells, which can be used in other zero-knowledge proof protocols related to graph coloring as well.