Hitori Number

Abstract

Hitori is a popular “pencil-and-paper” puzzle game. In n-hitori, we are given an n ×n rectangular grid of which each square is labeled with a positive integer, and the goal is to paint a subset of the squares so that the following three rules hold: Rule 1) No row or column has a repeated unpainted label; Rule 2) Painted squares are never (horizontally or vertically) adjacent; Rule 3) The unpainted squares are all connected (via horizontal and vertical connections). The grid is called an instance of n-hitori if it has a unique solution. In this paper, we introduce hitori number defined as follows: For every integer n ≥ 2, hitori number h(n) is the minimum number of different integers used in an instance where the minimum is taken over all the instances of n-hitori. We then prove that ⌈(2n − 1)/3 ⌉ ≤ h(n) ≤ 2 ⌈n/3 ⌉ + 1.