Hitori numbers

Hitori is a popular “pencil-and-paper” puzzle defined as follows. In n-hitori, we are given an n×n rectangular grid in 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 are satisfied: 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 and maximum hitori number which are defined as follows: For every integer n, 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. For every integer n, maximum hitori number h(n) is the maximum number of different integers used in an instance where the maximum is taken over all the instances of n-hitori. We then prove that ⌊(2n - 1)/3⌋ ≤ h(n) ≤ 2⌊n/3⌋ + 1 for n ≥ 2 and ⌊(4n² - 4n + 11)/5⌋ ≥ h(n) ≤ (4n² + 2n - 2)/5 for n ≤ 3.