The Fewest Clues Problem

When analyzing the computational complexity of well-known puzzles, most papers consider the algorithmic challenge of solving a given instance of (a generalized form of) the puzzle. We take a different approach by analyzing the computational complexity of designing a "good" puzzle. We assume a puzzle maker designs part of an instance, but before publishing it, wants to ensure that the puzzle has a unique solution. Given a puzzle, we introduce the FCP (fewest clues problem) version of the problem: Given an instance to a puzzle, what is the minimum number of clues we must add in order to make the instance uniquely solvable? We analyze this question for the Nikoli puzzles Sudoku, Shakashaka, and Akari. Solving these puzzles is NP-complete, and we show their FCP versions are Sigma_2^P-complete. Along the way, we show that the FCP versions of 3SAT, 1-in-3SAT, Triangle Partition, Planar 3SAT, and Latin Square are all Sigma_2^P-complete. We show that even problems in P have difficult FCP versions, sometimes even Sigma_2^P-complete, though "closed under cluing" problems are in the (presumably) smaller class NP; for example, FCP 2SAT is NP-complete

The Fewest Clues Problem of Picross 3D

Picross 3D is a popular single-player puzzle video game for the Nintendo DS. It is a 3D variant of Nonogram, which is a popular pencil-and-paper puzzle. While Nonogram provides a rectangular grid of squares that must be filled in to create a picture, Picross 3D presents a rectangular parallelepiped (i.e., rectangular box) made of unit cubes, some of which must be removed to construct an image in three dimensions. Each row or column has at most one integer on it, and the integer indicates how many cubes in the corresponding 1D slice remain when the image is complete. It is shown by Kusano et al. that Picross 3D is NP-complete. We in this paper show that the fewest clues problem of Picross 3D is Sigma_2^P-complete and that the counting version and the another solution problem of Picross 3D are #P-complete and NP-complete, respectively

Herugolf and Makaro are NP-complete

Herugolf and Makaro are Nikoli\u27s pencil puzzles. We study the computational complexity of Herugolf and Makaro puzzles. It is shown that deciding whether a given instance of each puzzle has a solution is NP-complete

Zig-Zag Numberlink is NP-Complete

When can $t$ terminal pairs in an $m \times n$ grid be connected by $t$ vertex-disjoint paths that cover all vertices of the grid? We prove that this problem is NP-complete. Our hardness result can be compared to two previous NP-hardness proofs: Lynch's 1975 proof without the ``cover all vertices'' constraint, and Kotsuma and Takenaga's 2010 proof when the paths are restricted to have the fewest possible corners within their homotopy class. The latter restriction is a common form of the famous Nikoli puzzle \emph{Numberlink}; our problem is another common form of Numberlink, sometimes called \emph{Zig-Zag Numberlink} and popularized by the smartphone app \emph{Flow Free}

Solving Yin-Yang Puzzles Using Exhaustive Search and Prune-and-Search Algorithms

We investigate some algorithmic and mathematical aspects of Yin-Yang/Shiromaru-Kuromaru puzzles. Specifically, we discuss two algorithms for solving arbitrary Yin-Yang puzzles, namely the exhaustive search approach and the prune-and-search technique. We show that both algorithms have an identical asymptotic running time of O(max{mn, 2^(mn−h)}) for finding all solutions of a Yin-Yang instance with h hints of size m x n. Nevertheless, our experiments show that the practical running time of the prune-and-search technique outperforms the conventional exhaustive search approach

Tatamibari Is NP-Complete

In the Nikoli pencil-and-paper game Tatamibari, a puzzle consists of an m x n grid of cells, where each cell possibly contains a clue among ?, ?, ?. The goal is to partition the grid into disjoint rectangles, where every rectangle contains exactly one clue, rectangles containing ? are square, rectangles containing ? are strictly longer horizontally than vertically, rectangles containing ? are strictly longer vertically than horizontally, and no four rectangles share a corner. We prove this puzzle NP-complete, establishing a Nikoli gap of 16 years. Along the way, we introduce a gadget framework for proving hardness of similar puzzles involving area coverage, and show that it applies to an existing NP-hardness proof for Spiral Galaxies. We also present a mathematical puzzle font for Tatamibari