Five Cells is a pencil 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 Shikaku 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 be used in other protocols related to graph coloring.

翻译：《Five Cells》是一种铅笔谜题，由矩形网格构成，其中部分单元格包含数字。玩家需将该网格划分为由五个单元格组成的块，使得每个单元格中的数字等于该单元格作为块边界的边数。本文提出了一种基于扑克牌的《五格》物理零知识证明协议，使得证明者能够在不泄露谜题解的情况下，物理性地展示自己知晓该解。更重要的是，在优化过程中，我们开发了一种技术，用于验证图着色中任意两个相邻顶点颜色不同，且不泄露任何着色信息。该技术将协议所需卡牌数量从单元格数的二次量级降至线性量级，并可用于其他与图着色相关的协议。