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 blocks, each consisting of five cells, such that the number in each cell must be equal to the number of edges of that cell that are borders of blocks. In this paper, we propose a physical zero-knowledge proof protocol for Five Cells using a deck of playing cards, which allows a prover to physically show that he/she knows a solution of the puzzle without revealing it. More importantly, in the optimization we develop a technique to verify a graph coloring that no two adjacent vertices have the same color without revealing any information about the coloring. This technique reduces the number of required cards in our protocol from quadratic to linear in the number of cells, and can also be used in other protocols related to graph coloring.

翻译："五格拼图"是一种逻辑谜题，包含一个矩形网格，部分单元格中含有数字。玩家需要将网格划分为若干区域，每个区域由五个单元格组成，且每个单元格中的数字必须等于该单元格作为区域边界的边数。本文提出了一种基于扑克牌的物理零知识证明协议，使得证明者能够在不泄露谜题解法的前提下，物理性地展示其知晓该解法的能力。更重要的是，在优化过程中，我们开发了一种可验证图着色（相邻顶点颜色互异）且不泄露着色信息的技术。该技术将协议所需卡牌数量从单元格数的二次方降至线性，并可用于其他与图着色相关的协议。