Nurikabe is a logic-based puzzle that originated in Japan. The name "Nurikabe" translates to "painted wall" in English. It involves filling a grid with "walls" represented by cells, following specific rules to create an interesting path or area. ### Rules of Nurikabe: 1. **Grid Structure**: The puzzle is played on a rectangular grid. Some cells contain numbers, which indicate how many contiguous cells are part of a "region" or island.
Not-All-Equal 3-Satisfiability (NAE-3-SAT) is a variation of the standard 3-Satisfiability (3-SAT) problem in the field of computational complexity and logic. In general, the 3-SAT problem involves determining whether there is an assignment of truth values to variables in a boolean formula written in conjunctive normal form (CNF), such that the entire formula evaluates to true.
NP-completeness is a concept from computational complexity theory that classifies certain decision problems based on their solvability and the difficulty of solving them. Let's break down the relevant terms: 1. **P (Polynomial Time)**: This class consists of decision problems (problems with a yes/no answer) that can be solved by a deterministic Turing machine in polynomial time. In simpler terms, these are problems for which an efficient (i.e., fast) algorithm exists.
Minesweeper is a classic single-player puzzle video game that involves logic and strategy. It gained widespread popularity with its inclusion in Microsoft Windows in the early 1990s, although it has origins dating back to earlier games. ### Gameplay Mechanics: - **Objective**: The primary goal of Minesweeper is to clear a rectangular board containing hidden "mines" or bombs without detonating any of them. Players aim to uncover all safe squares while avoiding the mines.
Masyu is a type of logic puzzle that typically appears on puzzle websites and in puzzle books. The objective of Masyu is to draw a single loop through a grid, following certain rules indicated by special circles within the grid. Here are the basic rules of Masyu: 1. **Loop Structure**: The loop must be a continuous, non-intersecting path that forms a closed loop.
Mastermind is a classic board game that involves strategy and logic. Originally created in the 1970s, it is typically played by two players. The game consists of a code maker and a code breaker. ### Game Components: - **Board**: A long, horizontal board with rows of pegs. - **Pegs**: Available in several colors. - **Key pegs**: Used to give feedback to the code breaker.
NP-complete problems are a class of problems in computational complexity theory that are particularly important because they are both in NP (nondeterministic polynomial time) and as hard as any problem in NP. If any NP-complete problem can be solved in polynomial time, then every problem in NP can also be solved in polynomial time.
Light Up is a type of logic puzzle that involves placing "lights" in a grid to illuminate the entire area according to specific rules. The goal is to position lights (often represented as circles or bulbs) in such a way that all cells in the grid are lit up, while adhering to certain constraints. Here are some key features of Light Up puzzles: 1. **Grid and cells**: The puzzle is typically played on a rectangular grid made up of individual cells.
Richard Karp’s seminal 1972 paper, "Reducibility Among Combinatorial Problems," identified 21 specific problems that are NP-complete, which means they are among the most challenging problems in computational complexity theory. Below is a list of these 21 problems: 1. **Clique Problem**: Given a graph \( G \) and an integer \( k \), is there a complete subgraph (clique) of size \( k \)?
Kakuro is a number puzzle that is sometimes referred to as a "cross-sum" puzzle. It is similar to a crossword puzzle but uses numbers instead of words. The objective of Kakuro is to fill in the blank cells in a grid with digits from 1 to 9, such that the numbers in each group of contiguous cells sum up to the specified clues provided in the grid.
Hitori is a logic-based puzzle game that originated in Japan. The objective of Hitori is to fill in cells in a grid based on certain rules, which typically involve the numbers provided in the cells. The game is played on a rectangular grid filled with numbers. Here are the basic rules: 1. **Grid Structure**: The grid consists of a set of numbers, where each number represents how many times that number appears in its respective row and column.
Hashiwokakero is a logic-based puzzle game that originated in Japan. The name translates to "Build Bridges" in English, which describes the game's objective. The game is played on a grid that contains various islands, represented by circles. The goal is to connect these islands with bridges, following specific rules: 1. Each island has a number that indicates how many bridges must connect to it. 2. Bridges can only be drawn horizontally or vertically between islands.
The Generalized Assignment Problem (GAP) is an optimization problem that involves assigning a set of tasks to a set of agents (or resources) in a way that minimizes costs or maximizes efficiency while respecting certain constraints. Unlike the classical assignment problem, where each task is assigned to exactly one agent, the Generalized Assignment Problem allows each agent to handle multiple tasks, but with additional limitations.
Battleship is a logic-based puzzle game that is reminiscent of the classic naval combat board game. In the puzzle format, the player is presented with a grid where they must place a set number of ships of varying lengths without overlapping them, while also ensuring that they adhere to specific clues provided in the puzzle. The grid typically has numbers along the top and left sides, which indicate how many segments of ships are located in each row and column.
Weakly NP-complete problems are a subset of decision problems that are considered to be NP-complete, but with a specific characteristic: they can be solved in polynomial time given a solution in the form of a polynomial-size numerical representation. This means that the problem can be solved in polynomial time when the numerical parameters or inputs are bounded by a polynomial size.
Strongly NP-complete problems are a subset of NP-complete problems that remain NP-complete even when the numerical values in the input are bounded by a polynomial in the size of the input. This contrasts with "weakly NP-complete" problems, which can be solved in polynomial time when the numbers involved are small (i.e., their magnitude is polynomially bounded) but may be hard in the general case where numerical values can be arbitrary.
SMT solvers, or Satisfiability Modulo Theories solvers, are tools designed to determine the satisfiability of logical formulas with respect to specific theories. They extend the capabilities of traditional SAT solvers, which deal only with propositional logic, by incorporating more complex theories such as: 1. **Arithmetic**: Integers, reals, etc. 2. **Bit-vectors**: Operations on sequences of bits.
SAT solvers are algorithms or software tools designed to solve the Boolean satisfiability problem (SAT). The SAT problem involves determining whether there exists an assignment of truth values (true or false) to a set of variables such that a given Boolean formula evaluates to true. The problem is typically represented in conjunctive normal form (CNF), which is a conjunction (AND) of clauses, where each clause is a disjunction (OR) of literals (variables or their negations).
Number partitioning is a problem in combinatorial mathematics and computer science that involves dividing a set of integers into two or more subsets such that certain conditions are met. The most common form of the problem involves partitioning a set of integers into two subsets that have equal sums, known as the "Partition Problem." The Partition Problem can be formally defined as follows: Given a set of integers, can it be divided into two subsets such that the sum of the elements in each subset is equal?
Nonograms, also known as "Picross," "Griddlers," or "Paint by Numbers," are logic puzzle games involving a grid. The objective is to fill the grid with black squares according to a set of numerical clues provided for each row and column. These clues indicate consecutive blocks of filled squares, and players use them to deduce which squares should be filled in and which should remain empty.