# Hitori numbers

Akira Suzuki, Masashi Kiyomi, Yota Otachi, Kei Uchizawa, Takeaki Uno

Research output: Contribution to journalArticlepeer-review

## Abstract

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.

Original language English 695-707 13 Journal of Information Processing 25 https://doi.org/10.2197/ipsjjip.25.695 Published - 2017 Aug

## Keywords

• Hitori
• Pencil-and-paper puzzle
• Unique solution

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