This document is the first part of a 4-part series on the mathematics of chessboard domination. It introduces the concept of domination and covers domination numbers for various chess pieces including rooks, bishops, kings, knights, and queens on square and rectangular chessboards. For rooks and bishops, the domination number on an nxn board is simply n. For kings, the number increases as the square of the number of rows/columns needed to cover the board. Knights and queens domination numbers are more complex with no known closed-form formulas.
Chessboard Puzzles Part 2 - IndependenceDan Freeman
This document is part 2 of a 4-part series on the mathematics of chessboard puzzles. It examines the concept of independence among different chess pieces on a chessboard, which is defined as the maximum number of pieces that can be placed without any attacking each other. Formulas for the independence number are derived for rooks, bishops, kings, and knights on both square and rectangular boards. The 8-queens problem and its generalization to n-queens are also discussed. Proofs are provided for many of the independence numbers and permutation counts.
This document discusses domination numbers in chessboard puzzles. It defines domination as a set of pieces covering every square or attacking every square. Formulas are given for the domination numbers of rooks, bishops, and kings on an nxn board. For queens and knights, only specific values and bounds have been determined due to greater complexity. The movement rules for each piece are also outlined.
Knights tour on chessboard using backtrackingAbhishek Singh
The knight is placed o any block of an empty chess board and moving according to the rules of chess, it must visit each square exacty once. Here in the ppt the algorithm along with some visualisation and explanation is given and problem is solved using backtracking approach.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise boosts blood flow, releases endorphins, and promotes changes in the brain which help enhance one's emotional well-being and mental clarity.
This document discusses planar graphs, including:
- Definitions of planar graphs and examples.
- Theorems like Kuratowski's and Euler's theorems about planar graphs.
- Algorithms for planarity testing and drawing planar graphs.
- Applications of planar graphs in VLSI circuits, computer vision, and electrical circuit design.
Chessboard Puzzles Part 3 - Knight's TourDan Freeman
This document summarizes research on knight's tours on chessboards. It defines closed and open knight's tours, provides examples of tours on boards of different sizes, and discusses some related topics like magic squares, Latin squares, and the mathematical proofs investigating the existence of tours on certain board sizes. It also cites several sources for further information on knight's tour puzzles and combinatorics.
4.2 standard form of a quadratic function (Part 1)leblance
Standard form of a quadratic function is f(x) = ax^2 + bx + c. The graph is a parabola that opens up if a > 0 and opens down if a < 0. The axis of symmetry is the line x = -b/2a and the vertex is (-b/2a, f(-b/2a)). To graph in standard form, identify a, b, c, find the axis of symmetry and vertex, then plot the y-intercept and use reflection to sketch the parabola. The document provides an example of using standard form to identify the vertex, axis of symmetry, minimum/maximum value, and range of a parabola.
Chessboard Puzzles Part 2 - IndependenceDan Freeman
This document is part 2 of a 4-part series on the mathematics of chessboard puzzles. It examines the concept of independence among different chess pieces on a chessboard, which is defined as the maximum number of pieces that can be placed without any attacking each other. Formulas for the independence number are derived for rooks, bishops, kings, and knights on both square and rectangular boards. The 8-queens problem and its generalization to n-queens are also discussed. Proofs are provided for many of the independence numbers and permutation counts.
This document discusses domination numbers in chessboard puzzles. It defines domination as a set of pieces covering every square or attacking every square. Formulas are given for the domination numbers of rooks, bishops, and kings on an nxn board. For queens and knights, only specific values and bounds have been determined due to greater complexity. The movement rules for each piece are also outlined.
Knights tour on chessboard using backtrackingAbhishek Singh
The knight is placed o any block of an empty chess board and moving according to the rules of chess, it must visit each square exacty once. Here in the ppt the algorithm along with some visualisation and explanation is given and problem is solved using backtracking approach.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise boosts blood flow, releases endorphins, and promotes changes in the brain which help enhance one's emotional well-being and mental clarity.
This document discusses planar graphs, including:
- Definitions of planar graphs and examples.
- Theorems like Kuratowski's and Euler's theorems about planar graphs.
- Algorithms for planarity testing and drawing planar graphs.
- Applications of planar graphs in VLSI circuits, computer vision, and electrical circuit design.
Chessboard Puzzles Part 3 - Knight's TourDan Freeman
This document summarizes research on knight's tours on chessboards. It defines closed and open knight's tours, provides examples of tours on boards of different sizes, and discusses some related topics like magic squares, Latin squares, and the mathematical proofs investigating the existence of tours on certain board sizes. It also cites several sources for further information on knight's tour puzzles and combinatorics.
4.2 standard form of a quadratic function (Part 1)leblance
Standard form of a quadratic function is f(x) = ax^2 + bx + c. The graph is a parabola that opens up if a > 0 and opens down if a < 0. The axis of symmetry is the line x = -b/2a and the vertex is (-b/2a, f(-b/2a)). To graph in standard form, identify a, b, c, find the axis of symmetry and vertex, then plot the y-intercept and use reflection to sketch the parabola. The document provides an example of using standard form to identify the vertex, axis of symmetry, minimum/maximum value, and range of a parabola.
2D transformations can be represented by matrices and include translations, rotations, scalings, and reflections. Translations move objects by adding a translation vector. Rotations rotate objects around the origin by pre-multiplying the point coordinates with a rotation matrix. Scaling enlarges or shrinks objects by multiplying the point coordinates with scaling factors. Composite transformations represent multiple transformations applied in sequence, with the overall transformation represented as the matrix product of the individual transformations. The order of transformations matters as matrix multiplication is not commutative.
This document discusses logic and truth tables which are used in mathematics and computer science. It defines primitive statements, logical connectives like conjunction, disjunction, negation, implication and biconditional. Truth tables are used to determine the truth values of compound statements formed using these connectives. Examples are given to show how compound statements can be written symbolically and their truth values determined from truth tables. Decision structures like if-then and if-then-else used in programming languages are also discussed.
This document presents a summary of a project on solving the knight's tour problem. It discusses the knight's tour problem, concepts related to open and closed tours, and approaches to solving it through backtracking for small boards and divide-and-conquer for larger boards. It provides sample outputs for boards of sizes 5x5, 6x6, and 10x10, and discusses challenges faced and future work in solving the problem for general m x n boards.
Integer Representations & Algorithms
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
The document describes and analyzes two algorithms for finding the convex hull of a set of points: a brute force algorithm and a divide and conquer algorithm.
The brute force algorithm iterates through all points three times, checking all possible line combinations, resulting in O(n3) time complexity.
The divide and conquer algorithm recursively divides the point set into halves at each step by finding the furthest point from the current left-right boundary line. It has O(n log n) time complexity.
An experiment comparing runtimes on sample point sets showed the divide and conquer approach was significantly faster than the brute force approach.
The document discusses the rules for matrix multiplication. It states that two matrices can only be multiplied if the number of columns of the first matrix is equal to the number of rows of the second matrix. It provides examples of multiplying different matrices and explains how to calculate each element of the resulting matrix by taking the dot product of the corresponding row and column. It also gives an example of using matrix multiplication to calculate total sales and revenue from sales data organized in matrices.
Using matrices to transform geometric figures, including translations, dilations, reflections, and rotations. Translations use a matrix with the distances of movement in each row. Dilations multiply coordinates by a scalar factor. Reflections across an axis involve changing the sign of coordinates on one side of the axis. Rotation matrices involve trigonometric functions to rotate the figure a specified number of degrees clockwise or counterclockwise. Examples show setting up and performing each type of transformation on sample polygons.
This document discusses trigonometric functions. It begins by defining trigonometric functions as generalizations of trigonometric ratios to any angle measure, in terms of radian measure. It defines the six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - in terms of the x-coordinate and y-coordinate of a point on a unit circle. Key properties discussed include the periodic nature of the functions and their values for quadrantal and other common angles.
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
Benginning Calculus Lecture notes 2 - limits and continuitybasyirstar
This document discusses limits and continuity in calculus. It begins by defining limits and providing examples of computing limits of functions. It then covers one-sided limits, properties of limits, and using direct substitution to evaluate limits. The document also discusses limits of trigonometric functions and infinite limits. The overall goal is to determine the existence of limits, compute limits, understand continuity of functions, and connect the ideas of limits and continuity.
The document discusses 2D geometric transformations including translation, rotation, scaling, and matrix representations. It explains that transformations can be combined through matrix multiplication and represented by 3x3 matrices in homogeneous coordinates. Common transformations like translation, rotation, scaling and reflections are demonstrated.
Lecture 2 predicates quantifiers and rules of inferenceasimnawaz54
1) Predicates become propositions when variables are quantified by assigning values or using quantifiers. Quantifiers like ∀ and ∃ are used to make statements true or false for all or some values.
2) ∀ (universal quantifier) means "for all" and makes a statement true for all values of a variable. ∃ (existential quantifier) means "there exists" and makes a statement true if it is true for at least one value.
3) Predicates with unbound variables are neither true nor false. Binding variables by assigning values or using quantifiers turns predicates into propositions that can be evaluated as true or false.
Regula Falsi Method, For Numerical analysis. working matlab code. numeric analysis Regula Falsi method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Regula Falsi Method
Introduction to graph theory (All chapter)sobia1122
1) Graph theory can be used to model and solve problems in many fields like physics, chemistry, computer science, and more. Certain problems can be formulated as problems in graph theory.
2) Graph theory has developed from puzzles and practical problems, like the Königsberg bridge problem inspiring Eulerian graph theory and the "Around the World" game inspiring Hamiltonian graph theory.
3) Connectivity in graphs measures how connected a graph is, and how the removal of vertices or edges affects connectivity. Connectivity is important for applications like communication networks.
The document discusses minimum spanning trees (MST) and two algorithms for finding them: Prim's algorithm and Kruskal's algorithm. Prim's algorithm operates by building the MST one vertex at a time, starting from an arbitrary root vertex and at each step adding the cheapest connection to another vertex not yet included. Kruskal's algorithm finds the MST by sorting the edges by weight and sequentially adding edges that connect different components without creating cycles.
An earlier version 1.0 can be found here: https://www.slideshare.net/xqin74/how-to-write-papers-part-1-principles/edit?src=slideview
5 Simple Steps to Write a Good Research Paper Title
1. Ask yourself these questions and make note of the answers What is my paper about? What techniques/ designs were used? Who/what is studied? What were the results?
2. Use your answers to list key words.
3. Create a sentence that includes the key words you listed.
4. Delete all unnecessary/repetitive words and link the remaining.
5. Delete non-essential information and reword the title.
This document provides an overview of Lex and Yacc. It describes Lex as a tool that generates scanners to tokenize input streams based on regular expressions. Yacc is described as a tool that generates parsers to analyze tokens based on grammar rules. The document outlines the compilation process for Lex and Yacc, describes components of a Lex source file including regular expressions and transition rules, and provides examples of Lex and Yacc usage.
Increasing and decreasing functions ap calc sec 3.3Ron Eick
The document discusses increasing and decreasing functions and the first derivative test. It defines that a function is increasing if the derivative is positive, decreasing if the derivative is negative, and constant if the derivative is zero. It provides examples of finding the intervals where a function is increasing or decreasing by identifying critical numbers and testing points in each interval. The document also summarizes the first derivative test, stating that a critical point is an extremum if the derivative changes sign there, and whether it is a maximum or minimum depends on if the derivative changes from negative to positive or positive to negative.
This document discusses basic matrix operations including:
- Defining a matrix as a rectangular arrangement of numbers in rows and columns with an order specified by the number of rows and columns.
- Adding and subtracting matrices requires they have the same order and involves adding or subtracting corresponding entries.
- Multiplying a matrix by a scalar involves multiplying each entry in the matrix by the scalar value.
- Matrix multiplication is not commutative and can only be done if the number of columns in the first matrix equals the number of rows in the second matrix. It involves multiplying entries and summing the products based on their positions.
Chessboard Puzzles Part 3 - Knight's TourDan Freeman
This document discusses knight's tours on chessboards. It defines closed and open knight's tours, and provides examples of each on various board sizes. Closed tours end where they start, while open tours can end anywhere. The document also discusses theorems about knight's tours and how they can be used to construct magic squares and Latin squares, showing relationships between knight's tours and other mathematical structures.
Chessboard Puzzles Part 2 - IndependenceDan Freeman
This document summarizes key concepts related to independence in chessboard puzzles. It defines independence as pieces not attacking each other and introduces independence numbers to represent the maximum independent pieces on a board. Formulas are provided for the independence numbers of different chess pieces, including rooks (n), bishops (2n-2), kings (└1⁄2(n+1)┘), and queens (n). The document also discusses permutations of independent pieces and solutions to the n-queens problem.
2D transformations can be represented by matrices and include translations, rotations, scalings, and reflections. Translations move objects by adding a translation vector. Rotations rotate objects around the origin by pre-multiplying the point coordinates with a rotation matrix. Scaling enlarges or shrinks objects by multiplying the point coordinates with scaling factors. Composite transformations represent multiple transformations applied in sequence, with the overall transformation represented as the matrix product of the individual transformations. The order of transformations matters as matrix multiplication is not commutative.
This document discusses logic and truth tables which are used in mathematics and computer science. It defines primitive statements, logical connectives like conjunction, disjunction, negation, implication and biconditional. Truth tables are used to determine the truth values of compound statements formed using these connectives. Examples are given to show how compound statements can be written symbolically and their truth values determined from truth tables. Decision structures like if-then and if-then-else used in programming languages are also discussed.
This document presents a summary of a project on solving the knight's tour problem. It discusses the knight's tour problem, concepts related to open and closed tours, and approaches to solving it through backtracking for small boards and divide-and-conquer for larger boards. It provides sample outputs for boards of sizes 5x5, 6x6, and 10x10, and discusses challenges faced and future work in solving the problem for general m x n boards.
Integer Representations & Algorithms
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
The document describes and analyzes two algorithms for finding the convex hull of a set of points: a brute force algorithm and a divide and conquer algorithm.
The brute force algorithm iterates through all points three times, checking all possible line combinations, resulting in O(n3) time complexity.
The divide and conquer algorithm recursively divides the point set into halves at each step by finding the furthest point from the current left-right boundary line. It has O(n log n) time complexity.
An experiment comparing runtimes on sample point sets showed the divide and conquer approach was significantly faster than the brute force approach.
The document discusses the rules for matrix multiplication. It states that two matrices can only be multiplied if the number of columns of the first matrix is equal to the number of rows of the second matrix. It provides examples of multiplying different matrices and explains how to calculate each element of the resulting matrix by taking the dot product of the corresponding row and column. It also gives an example of using matrix multiplication to calculate total sales and revenue from sales data organized in matrices.
Using matrices to transform geometric figures, including translations, dilations, reflections, and rotations. Translations use a matrix with the distances of movement in each row. Dilations multiply coordinates by a scalar factor. Reflections across an axis involve changing the sign of coordinates on one side of the axis. Rotation matrices involve trigonometric functions to rotate the figure a specified number of degrees clockwise or counterclockwise. Examples show setting up and performing each type of transformation on sample polygons.
This document discusses trigonometric functions. It begins by defining trigonometric functions as generalizations of trigonometric ratios to any angle measure, in terms of radian measure. It defines the six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - in terms of the x-coordinate and y-coordinate of a point on a unit circle. Key properties discussed include the periodic nature of the functions and their values for quadrantal and other common angles.
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
Benginning Calculus Lecture notes 2 - limits and continuitybasyirstar
This document discusses limits and continuity in calculus. It begins by defining limits and providing examples of computing limits of functions. It then covers one-sided limits, properties of limits, and using direct substitution to evaluate limits. The document also discusses limits of trigonometric functions and infinite limits. The overall goal is to determine the existence of limits, compute limits, understand continuity of functions, and connect the ideas of limits and continuity.
The document discusses 2D geometric transformations including translation, rotation, scaling, and matrix representations. It explains that transformations can be combined through matrix multiplication and represented by 3x3 matrices in homogeneous coordinates. Common transformations like translation, rotation, scaling and reflections are demonstrated.
Lecture 2 predicates quantifiers and rules of inferenceasimnawaz54
1) Predicates become propositions when variables are quantified by assigning values or using quantifiers. Quantifiers like ∀ and ∃ are used to make statements true or false for all or some values.
2) ∀ (universal quantifier) means "for all" and makes a statement true for all values of a variable. ∃ (existential quantifier) means "there exists" and makes a statement true if it is true for at least one value.
3) Predicates with unbound variables are neither true nor false. Binding variables by assigning values or using quantifiers turns predicates into propositions that can be evaluated as true or false.
Regula Falsi Method, For Numerical analysis. working matlab code. numeric analysis Regula Falsi method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Regula Falsi Method
Introduction to graph theory (All chapter)sobia1122
1) Graph theory can be used to model and solve problems in many fields like physics, chemistry, computer science, and more. Certain problems can be formulated as problems in graph theory.
2) Graph theory has developed from puzzles and practical problems, like the Königsberg bridge problem inspiring Eulerian graph theory and the "Around the World" game inspiring Hamiltonian graph theory.
3) Connectivity in graphs measures how connected a graph is, and how the removal of vertices or edges affects connectivity. Connectivity is important for applications like communication networks.
The document discusses minimum spanning trees (MST) and two algorithms for finding them: Prim's algorithm and Kruskal's algorithm. Prim's algorithm operates by building the MST one vertex at a time, starting from an arbitrary root vertex and at each step adding the cheapest connection to another vertex not yet included. Kruskal's algorithm finds the MST by sorting the edges by weight and sequentially adding edges that connect different components without creating cycles.
An earlier version 1.0 can be found here: https://www.slideshare.net/xqin74/how-to-write-papers-part-1-principles/edit?src=slideview
5 Simple Steps to Write a Good Research Paper Title
1. Ask yourself these questions and make note of the answers What is my paper about? What techniques/ designs were used? Who/what is studied? What were the results?
2. Use your answers to list key words.
3. Create a sentence that includes the key words you listed.
4. Delete all unnecessary/repetitive words and link the remaining.
5. Delete non-essential information and reword the title.
This document provides an overview of Lex and Yacc. It describes Lex as a tool that generates scanners to tokenize input streams based on regular expressions. Yacc is described as a tool that generates parsers to analyze tokens based on grammar rules. The document outlines the compilation process for Lex and Yacc, describes components of a Lex source file including regular expressions and transition rules, and provides examples of Lex and Yacc usage.
Increasing and decreasing functions ap calc sec 3.3Ron Eick
The document discusses increasing and decreasing functions and the first derivative test. It defines that a function is increasing if the derivative is positive, decreasing if the derivative is negative, and constant if the derivative is zero. It provides examples of finding the intervals where a function is increasing or decreasing by identifying critical numbers and testing points in each interval. The document also summarizes the first derivative test, stating that a critical point is an extremum if the derivative changes sign there, and whether it is a maximum or minimum depends on if the derivative changes from negative to positive or positive to negative.
This document discusses basic matrix operations including:
- Defining a matrix as a rectangular arrangement of numbers in rows and columns with an order specified by the number of rows and columns.
- Adding and subtracting matrices requires they have the same order and involves adding or subtracting corresponding entries.
- Multiplying a matrix by a scalar involves multiplying each entry in the matrix by the scalar value.
- Matrix multiplication is not commutative and can only be done if the number of columns in the first matrix equals the number of rows in the second matrix. It involves multiplying entries and summing the products based on their positions.
Chessboard Puzzles Part 3 - Knight's TourDan Freeman
This document discusses knight's tours on chessboards. It defines closed and open knight's tours, and provides examples of each on various board sizes. Closed tours end where they start, while open tours can end anywhere. The document also discusses theorems about knight's tours and how they can be used to construct magic squares and Latin squares, showing relationships between knight's tours and other mathematical structures.
Chessboard Puzzles Part 2 - IndependenceDan Freeman
This document summarizes key concepts related to independence in chessboard puzzles. It defines independence as pieces not attacking each other and introduces independence numbers to represent the maximum independent pieces on a board. Formulas are provided for the independence numbers of different chess pieces, including rooks (n), bishops (2n-2), kings (└1⁄2(n+1)┘), and queens (n). The document also discusses permutations of independent pieces and solutions to the n-queens problem.
Chessboard Puzzles Part 4 - Other Surfaces and VariationsDan Freeman
This document discusses various mathematical puzzles and problems related to chessboard configurations on different surfaces beyond the standard rectangular chessboard. It examines knight's tours, domination by different chess pieces, and other concepts on surfaces like the torus, cylinder, Klein bottle, and Möbius strip. The document provides examples of solutions to these puzzles and problems on irregular surfaces, compares them to the standard cases, and outlines some key formulas related to piece domination numbers on different board geometries.
The Knight's Tour is a mathematical problem involving a knight on a chessboard. The knight is placed on the empty board and, moving according to the rules of chess, must visit each square exactly once. A tour is called closed, if ending square is same as the starting. Otherwise it is called an open tour. The exact number of open tours is still unknown. Variations of the knight's tour problem involve chessboards of different sizes than the usual 8 × 8, as well as irregular (non-rectangular) boards.
Chess camp 3. checkmates with many piecesNelson ruiz
This document is a chess instruction book titled "Volume 3: Checkmates with Many Pieces" by Igor Sukhin. It contains copyright information and a table of contents listing various chapters on delivering checkmate in one move in different phases of the game using multiple pieces. The introduction notes the goals of helping students see patterns in complex positions, learn important tactical tools like pins and discovered checks, and gain experience with typical mating scenarios that arise in common openings.
The document discusses the knight's tour problem in chess, where the goal is to move a knight to every square of the chessboard only once. It describes the different types of knight's tours and algorithms that can solve them, including brute force search, divide and conquer, and Warnsdorff's rule. Warnsdorff's rule is an efficient algorithm that solves the problem in linear time by moving to the square with the fewest available moves at each step. The document also presents example C code to solve knight's tours and discusses applications in cryptography.
This document discusses the Knight's Tour problem in chess and two algorithms for solving it: a neural network approach and Warnsdorff's algorithm. It explains that the Knight's Tour problem involves finding a path for a knight to visit every square on a chessboard exactly once. It then summarizes that Warnsdorff's algorithm, which selects the next square with the fewest available future moves, is simpler and faster than the neural network approach and always produces a closed tour, making it a better solution to the Knight's Tour problem.
This document provides information on different types of tremors, including their causes, characteristics, and pathophysiology. It discusses rest tremor seen in Parkinson's disease and other conditions. It also covers postural tremor, physiological tremor, essential tremor, kinetic tremor, and cerebellar intention tremor. For each type of tremor, the summary highlights key details like involved areas of the brain, typical frequencies, symptoms, and potential treatments.
This document summarizes the n-queen problem, which involves placing N queens on an N x N chessboard so that no queen can attack any other. It describes the problem's inputs and tasks, provides examples of solutions for different board sizes, and outlines the backtracking algorithm commonly used to solve this problem. The backtracking approach guarantees a solution but can be slow, with complexity rising exponentially with problem size. It is a good benchmark for testing parallel computing systems due to its iterative nature.
This document discusses data structures and algorithms related to queues. It defines queues as first-in first-out (FIFO) linear lists and describes common queue operations like offer(), poll(), peek(), and isEmpty(). Implementations of queues using linked lists and circular arrays are presented. Applications of queues include accessing shared resources and serving as components of other data structures. The document concludes by explaining the eight queens puzzle and presenting an algorithm to solve it using backtracking.
Los Angeles Lakers tops, Miami Heat and Chicago Bulls follow as the most soci...Simplify360
Key Findings
Simplify360 has come up with March rankings of NBA Clubs, rating them as per their social score.
Here's a snapshot of the findings:-
1. The clubs are using social media to share updates of individual players and the club news in general. They also post updates of latest offers and ticket availability of match days.
2. Miami Heat is the most mentioned club on Twitter as well as on Facebook
3. Houston Rockets have the most engaged fan base on Facebook
4. LA Lakers are the most social club with an SSI score of 94
5. The big three clubs viz. LA Lakers, Miami Heat and Chicago Bulls have more than 47% of total fan following of the 30 NBA clubs on Facebook
6. Utah Jazz has the least SSI score. Only 16 of the 30 clubs have an SSI score of over 50 indicating the need for the clubs to concentrate on their social media strategy.
Drop in your comments please.
This document discusses different types of tremors, including their classification, causes, symptoms, and treatment. It covers rest tremors like Parkinsonian tremor and midbrain tremors. It also discusses postural tremors such as essential tremor and enhanced physiologic tremor. Treatment options covered include medications, botulinum toxin injections, deep brain stimulation, and thalamotomy.
The document describes the backtracking method for solving problems that require finding optimal solutions. Backtracking involves building a solution one component at a time and using bounding functions to prune partial solutions that cannot lead to an optimal solution. It then provides examples of applying backtracking to solve the 8 queens problem by placing queens on a chessboard with no attacks. The general backtracking method and a recursive backtracking algorithm are also outlined.
This document provides an overview of various movement disorders including chorea, athetosis, ballismus, myoclonus, Wilson's disease, tardive dyskinesia, essential tremor, and Tourette's syndrome. It describes the clinical manifestations and pathophysiology of each disorder and discusses treatment options. The disorders represent a spectrum of involuntary movements that can overlap and are often difficult to classify precisely. Accurate diagnosis relies on identifying structural lesions or genetic/environmental causes in the basal ganglia-thalamic motor circuits.
The document discusses solving the 8 queens problem using backtracking. It begins by explaining backtracking as an algorithm that builds partial candidates for solutions incrementally and abandons any partial candidate that cannot be completed to a valid solution. It then provides more details on the 8 queens problem itself - the goal is to place 8 queens on a chessboard so that no two queens attack each other. Backtracking is well-suited for solving this problem by attempting to place queens one by one and backtracking when an invalid placement is found.
This document discusses various problems that can be solved using backtracking, including graph coloring, the Hamiltonian cycle problem, the subset sum problem, the n-queen problem, and map coloring. It provides examples of how backtracking works by constructing partial solutions and evaluating them to find valid solutions or determine dead ends. Key terms like state-space trees and promising vs non-promising states are introduced. Specific examples are given for problems like placing 4 queens on a chessboard and coloring a map of Australia.
A Guide to SlideShare Analytics - Excerpts from Hubspot's Step by Step Guide ...SlideShare
This document provides a summary of the analytics available through SlideShare for monitoring the performance of presentations. It outlines the key metrics that can be viewed such as total views, actions, and traffic sources over different time periods. The analytics help users identify topics and presentation styles that resonate best with audiences based on view and engagement numbers. They also allow users to calculate important metrics like view-to-contact conversion rates. Regular review of the analytics insights helps users improve future presentations and marketing strategies.
How to Make Awesome SlideShares: Tips & TricksSlideShare
Turbocharge your online presence with SlideShare. We provide the best tips and tricks for succeeding on SlideShare. Get ideas for what to upload, tips for designing your deck and more.
SlideShare is a global platform for sharing presentations, infographics, videos and documents. It has over 18 million pieces of professional content uploaded by experts like Eric Schmidt and Guy Kawasaki. The document provides tips for setting up an account on SlideShare, uploading content, optimizing it for searchability, and sharing it on social media to build an audience and reputation as a subject matter expert.
Chess is a strategy game played between two opponents on a checkered board with 64 squares arranged in an 8x8 grid. The goal is to checkmate the opponent's king by placing it under an inescapable threat of capture. The document provides a summary of the history of chess, the basic equipment used including chess pieces and boards, an overview of the basic rules and strategies of chess such as castling and checkmate, and safety tips for playing chess.
The document provides information about the board game "Lord Of The Ring: The Sword Of Drastan" created by a group of students for their Creative Thinking Skills module. It includes the story line, objectives, materials used to create the gameboard and components, as well as processes for making the gameboard, cards, tokens, and packaging.
This document provides the rules for playing the classic board game RISK. It includes instructions for setup, placing initial armies, gameplay turns involving getting new armies, attacking, fortifying positions, and winning conditions. Variations are also described for 2 player games and shorter Capital RISK games.
This document provides a summary of the table of contents for a book about chess opening traps. It lists over 700 chess openings in alphabetical order, each with a brief description of traps or short games that can occur in that opening. The purpose is to help amateur chess players learn about potential traps in openings to either spring them on opponents or avoid them. It aims to show how poor opening strategies can be exploited to result in short games even at higher levels of play.
The document provides an overview of the game of chess, including:
- The origin and basic setup of the game, with two players using white and black pieces on a chessboard.
- The objective is to checkmate the opponent's king by placing it in a position where it cannot escape capture.
- Descriptions of the movement and capturing abilities of each type of chess piece - pawns, rooks, bishops, knights, queen, and king.
- Additional chess concepts covered include notation, promotion of pawns, relative piece values, exchanges, checks and checkmates, stalemate, repetition of moves, and castling.
This document provides the rules for Crusader Kings - The Board Game, based on the grand strategy computer game. Players lead historical dynasties in medieval Europe over 3 eras, each with 3 rounds of 2 turns. The goal is to spread influence, develop your dynasty and dominions, and gain the most victory points through controlling territories, achievements, and development cards by the game's end.
- Chess is a strategic board game played between two players on a checkered board consisting of 64 squares.
- Each player controls 16 pieces including a king, queen, two rooks, two bishops, two knights and eight pawns. The objective is to checkmate the opponent's king.
- Players take turns moving one piece per turn according to the rules of movement for each piece. A player can capture an opponent's piece by moving their piece to the same square. The captured piece is removed from play.
This document is a chapter from Fred Reinfeld's book "How to Win at Checkers" which provides instruction on checkers fundamentals, tactics, openings, endgames, and how to draw "lost" positions. The chapter summarized here discusses checkers fundamentals, explaining that most people play checkers badly despite it being easy, but that the game can offer as much challenge as chess if played properly. It aims to help readers increase their playing strength by developing finer points and winning methods previously unknown to amateurs.
Nakisha wheatley a beginner´s guide to become a better chess player+Izzquierdo
This document provides an overview of the chess game, including its setup, rules, notation, strategies, and terminology. It discusses the objective to checkmate the opponent's king and describes the movement and capturing rules for each piece. It also summarizes key concepts like checks, castling, en passant captures, promotions, recording moves in algebraic notation, and different stages of the game including openings, middlegames, and endgames. Evaluation of chess positions considers factors like material and pawn structure.
FICHEALL Lámhleabhair Handbook English Una O Boyle 2Una O Boyle
This document provides information about Úna O Boyle, including her contact information and credentials. It also contains a table of contents for a book or document about chess, listing sections on the game of chess, basic terms, the aim of the game, the pieces, points of the pieces, special moves, drawn games, notation, chess terms, and internet sites related to chess. There is also information provided about playing chess online and advice to have a parent present for online play.
FICHEALL Lámhleabhair Handbook English Una O Boyle 2Una O Boyle
This document provides information about Úna O Boyle, including her contact information and credentials. It also contains a table of contents for a book or document about chess, listing sections on the game of chess, basic terms, the aim of the game, the pieces, points of the pieces, special moves, drawn games, notation, chess terms, and internet sites related to chess. There is also information provided about playing chess online and advice to have a parent present for online play.
Similar to Chessboard Puzzles Part 1 - Domination (12)
FICHEALL Lámhleabhair Handbook English Una O Boyle 2
Chessboard Puzzles Part 1 - Domination
1. Chessboard Puzzles: Domination
Part 1 of a 4-part Series of Papers on the Mathematics of the Chessboard
by Dan Freeman
March 24, 2014
2. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
Table of Contents
Table of Figures .............................................................................................................................. 3
Motivation ....................................................................................................................................... 4
Overview of Chess .......................................................................................................................... 4
Definition of Domination................................................................................................................ 6
Rooks Domination .......................................................................................................................... 6
Bishops Domination........................................................................................................................ 8
Kings Domination ......................................................................................................................... 11
Knights Domination ...................................................................................................................... 14
Queens Domination....................................................................................................................... 18
2
The Spencer-Cockayne Construction ........................................................................................ 20
Upper and Lower Bounds for γ(Qnxn)........................................................................................ 24
Queens Diagonal Domination ................................................................................................... 27
Conclusion .................................................................................................................................... 29
Sources Cited ................................................................................................................................ 31
3. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
Table of Figures
Image 1: Chess Piece Symbols ....................................................................................................... 5
Image 2: Starting Chessboard Arrangement ................................................................................... 5
Image 3: Rook Movement............................................................................................................... 7
Image 4: Uncovered Square on 8x8 Board with 7 Rooks............................................................... 7
Image 5: Bishop Movement ............................................................................................................ 8
Image 6: Chessboard Rotated 45° ................................................................................................... 9
Image 7: Bishops Domination on 8x8 Board................................................................................ 10
Image 8: 5x5 White Square Inside 9x9 Board .............................................................................. 10
Image 9: 4x4 Black Square Inside 9x9 Board .............................................................................. 11
Image 10: King Movement ........................................................................................................... 11
Image 11: Kings Domination on 7x7, 8x8 and 9x9 Boards.......................................................... 12
Image 12: Each King Can Over Only One of the Orange Squares............................................... 13
Image 13: Knight Movement ........................................................................................................ 15
Image 14: Knights Domination on 4x4, 5x5, 6x6, 7x7 and 8x8 Boards....................................... 17
Image 15: Knights Domination on 11x11 Board .......................................................................... 18
Image 16: Queen Movement......................................................................................................... 19
Image 17: Five Queens Dominating an 8x8 Board...................................................................... 19
Image 18: Queen in Center of 5x5 Board ..................................................................................... 20
Image 19: Five Queens Inside a 5x5 Square Dominating a 9x9 Board ........................................ 21
Image 20: Five Queens on 11x11 Board....................................................................................... 22
Image 21: Nine Queens Inside an 11x11 Square Dominating a 15x15 Board ............................. 23
Image 22: Upper Bound of 8 Queens Covering 11x11 Board ...................................................... 25
Image 23: In Diagonal Domination, a Queen Lies Halfway Between Two Empty Columns ...... 29
Table 1: Domination Number Notation .......................................................................................... 6
Table 2: Knights Domination N umbers for 1 ≤ n ≤ 20 ................................................................. 15
Table 3: Queens Domination N umbers for 1 ≤ n ≤ 25.................................................................. 27
Table 4: Domination Number Formulas by Piece ........................................................................ 30
3
4. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
Motivation
4
In the past few years, I have become quite interested in the game of chess and have begun
to play it fairly regularly. Though I am by no means an expert in chess nor can I even be
considered a good player, I have noticed the undeniable relationship between the game and
several branches of mathematics, most notably number theory, one of my favorite areas of the
discipline. As a lifelong student of mathematics, in this and my subsequent three papers in this
series, I wish to survey most of the well-known problems and concepts associated with the
mathematics of the chessboard. The fact that chess is not only a fun game to play but also a
game with a long and rich history makes it that much more enjoyable to study the math behind it.
Overview of Chess
Chess is a classic board game that has been played for at least 1,200 years. Historical
evidence indicates that chess was being played back in A.D. 800, though a few earlier references
suggest that the game existed in India circa A.D. 600. Chess may have been played earlier than
that, but this is unclear because the ubiquitous 8x8, 64-square board on which it is played is used
for numerous other games as well [2, p. 6].
Chess is a 2-player turn-based game played on the aforementioned 8x8 board. The game
includes six different types of pieces: pawn, knight, bishop, rook, queen and king (see Image 1
for symbols representing each piece). To distinguish the pieces of the two players, one player’s
pieces are lighter in color than the other player’s; the former player is called “white” while the
latter player is called “black.” A game of chess always begins with the white player moving
first. Each player begins with eight pawns, two knights, two bishops, two rooks, one queen and
one king in the arrangement depicted on the board in Image 2 [5].
5. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
5
Image 1: Chess Piece Symbols
King
Queen
Rook
Bishop
Knight
Pawn
Image 2: Starting Chessboard Arrangement
While the objective of the game won’t directly tie into this paper, for the reader who may
be less familiar with chess, it is worth pointing out how a game of chess is won and lost. A
player wins by putting his or her opponent’s king in a position such that it cannot escape attack
from the winning player’s pieces. This position is known as checkmate. A game does not have
to end this way; it can also end in a draw or a stalemate, the details of which are outside the
scope of this paper.
6. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
Definition of Domination
6
A dominating set of chess pieces is one such that every square on an mxn1 chessboard is
either occupied by a piece in the set or under attack by a piece in the set. The domination
number for a certain piece and certain size chessboard is the minimum number of such pieces
required to “dominate” the board. The term “cover” is frequently used as a synonym for
“dominate” in the study of chessboard domination [1, pp. 95-97]. Domination numbers are
denoted by γ(Pmxn) where P represents the type of chess piece, as denoted in Table 1.
Table 1: Domination Number Notation
Piece Abbreviation
Knight N
Bishop B
Rook R
Queen Q
King K
Rooks Domination
Before exploring domination among rooks, we first need to establish how rooks move on
the chessboard. Rooks are permitted to move any number of squares either horizontally or
vertically, as long as they do not take the place of a friendly piece or pass through any piece
(own or opponent’s) currently on the board. As with any piece, rooks are allowed to move to a
square occupied by an enemy piece, thereby removing the enemy piece from the board (such a
move is known as a capture). In Image 3, the white rook can move to any of the squares with a
white circle and the black rook can move to any of the squares with a black circle [5].
1 Throughout this paper, m and n refer to arbitrary positive integers denoting the number of rows and columns of a
chessboard, respectively.
7. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
7
Image 3: Rook Movement
Domination among rooks is the simplest of all chess pieces. For a square nxn
chessboard, the rooks domination number is simply n [1, p. 99]. Moreover, for a general
rectangular mxn chessboard, γ(Rmxn) = min(m, n).
In 1964, two Russian brothers Akiva and Isaak Yaglom proved that γ(Rnxn) = n, as
follows. First, suppose there are fewer than n rooks placed on an nxn board. Then there must be
at least one row and at least one column that contain no rooks. Hence, the square where this
empty row and column intersect is uncovered, that is, it is not under attack by any of the rooks
(see Image 4). Thus, γ(Rnxn) ≥ n. Second, if n rooks are placed along a single row or down a
single column, the entire board is clearly covered. That is, γ(Rnxn) ≤ n. In conclusion, since
γ(Rnxn) ≥ n and γ(Rnxn) ≤ n, it follows that γ(Rnxn) = n [1, p. 99].
Image 4: Uncovered Square
on 8x8 Board with 7 Rooks
8. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
8
The fact that γ(Rmxn) = min(m, n) is immediately apparent from the Yaglom brothers’
proof above. Clearly, if m < n, then one need only place m rooks down a single column on the
board to cover all of the squares in each row. Likewise, if n < m, then one need only place n
rooks along a single row to cover all of the squares in each column. In either case, the rooks
domination number is the minimum of the number of rows m and the number of columns n.
Bishops Domination
Unlike rooks, bishops move diagonally, not horizontally and vertically. Bishops are
allowed to move any number of squares in one diagonal direction as long as they do not take the
place of a friendly piece or pass through any piece (own or opponent’s) currently on the board.
In Image 5, the white bishop can move to any of the squares with a white circle and the black
bishop can move to any of the squares with a black circle [5].
Image 5: Bishop Movement
As is the case with rooks, the domination number for bishops on a square nxn chessboard
is n (though in general, γ(Bmxn) ≠ min(m, n); in fact, no formula is known for γ(Bmxn) [4, p.
13]). However, the proof that this is the case requires a little more creativity than the proof for
rooks. As with the proof for the rooks domination number, the one for bishops was published by
the Yaglom brothers in 1964. The proof starts with rotating an 8x8 chessboard 45 degrees, as
shown in Image 6 [1, p. 100].
9. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
9
Image 6: Chessboard Rotated 45°
5x4 Black Square
We are fixing n to be 8, but the following argument works for all even positive integers.
From the rotated chessboard at the right of Image 5, we see a 5x4 square formed by the dark
orange squares inside the black-bordered rectangle (for general even n, an (½*n)x(½*n + 1)
square will emerge) . Therefore, at least 4 bishops (in general, ½*n) are needed to cover all of
the dark squares (from here on, the dark orange squares will be referred to as black and the light
orange squares will be referred to as white). By symmetry, at least 4 bishops (in general, ½*n)
are needed to cover all of the white squares as well. Thus, γ(B8x8) ≥ 4 + 4 = 8 (in general, γ(Bnxn)
≥ ½*n + ½*n = n). On the other hand, if we place 8 bishops on the fourth column of an 8x8
board as in Image 7, the entire board is covered [5]. Likewise, in general, if we place n bishops
on the (½*n)th column of an nxn board, then the board is dominated. Since γ(B8x8) ≥ 8 and
γ(B8x8) ≤ 8, it follows that γ(B8x8) = 8, and, similarly, for general even n, γ(Bnxn) = n [1, pp. 100-
101].
10. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
10
Image 7: Bishops Domination on 8x8 Board
Now suppose that n is odd, that is, n is of the form 2k + 1. The board corresponding to
squares of one color (without loss of generality, suppose this color is white) will contain a (k +
1)x(k + 1) group of squares (see Image 8 for white 5x5 square inside 9x9 board); hence, at least
k + 1 bishops are needed to cover the white squares. Likewise, the board corresponding to black
squares will contain a kxk group of squares (see Image 9 for a black 4x4 square inside a 9x9
board) and hence at least k bishops are needed to cover the black squares. Thus, at least (k + 1) +
k = 2k + 1 = n bishops are needed to dominate the entire nxn board. To see that γ(Bnxn) ≤ n,
observe that if n bishops are placed down the center column (more precisely, the (k + 1)st
column), the entire board is covered. Therefore, γ(Bnxn) = n for all odd n, and since we already
showed it to be true for even n, the result is proven for all n [1, p. 101].
Image 8 : 5x5 White Square Inside 9x9 Board
11. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
Kings Domination
11
Image 9: 4x4 Black Square Inside 9x9 Board
Kings are allowed to move exactly one square in any direction as long as they do not take
the place of a friendly piece. In Image 10, the king can move to any of the squares with a white
circle [5].
Image 10: King Movement
Domination among kings is a little bit more complicated than that of rooks and bishops,
yet is still completely determined formulaically. To arrive at a formula for the kings domination
number, it is first helpful to look at a set of kings dominating 7x7, 8x8 and 9x9 boards. In Image
11, nine kings each are covering a 7x7, 8x8 and 9x9 board. Therefore, for 7 ≤ n ≤ 9,
12. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
γ(Knxn) ≤ 9 [1, p. 102].
12
Image 11: Kings Domination on 7x7, 8x8 and 9x9 Boards
Furthermore, no matter where one places a king on any of the 7x7, 8x8 or 9x9 boards,
only one of the nine dark orange squares on each board in Image 12 will be covered. Therefore,
γ(Knxn) ≥ 9. So in fact γ(Knxn) = 9 for 7 ≤ n ≤ 9 [1, p. 102].
13. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
13
Image 12: Each King Can Over Only One of the Orange Squares
Note that 9 is the square of 3, the number of rows and columns of kings needed to cover a
7x7, 8x8 and 9x9 chessboard. For n = 10, 11 and 12, an additional row and column of kings is
needed to cover the board, so γ(Knxn) = 42 = 16 [1, p. 103]. Observe that this makes kings
domination very inefficient, as an additional 7 kings are required to cover the board when n
increases by just one from 9 to 10. In fact, it becomes increasingly inefficient as n increases,
since 9 more kings are needed to dominate a 13x13 board than what is required for a 12x12
board (25 total kings as compared to 16), 11 more kings are needed for n = 16 than for n = 15 (36
14. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
as compared to 25), and so on. Thus, the kings domination function, γ(Knxn), certainly behaves in
a non-linear fashion, unlike the rooks and bishops domination functions, which are simply equal
to n.
14
We can now see that γ(Knxn) is constant for each number n within a certain triplet of
successive of positive integers (1, 2 and 3; 4, 5 and 6; 7, 8 and 9, etc.). The kings domination
number only jumps every 3 values of n. Thus, the formula for γ(Knxn) can be expressed as three
separate equations as below (where k is a non-negative integer):
k2 = (n / 3)2 if n = 3k
γ(Knxn) = (k + 1)2 = ((n + 2) / 3)2 if n = 3k + 1
(k + 1)2 = ((n + 1) / 3)2 if n = 3k + 2
The above formula can be compressed into a single equation making use of the handy
greatest integer or floor function, as follows: γ(Knxn) = └(n + 2) / 3┘
2 [1, p. 103]. For rectangular
chessboards, this formula can be generalized to γ(Kmxn) = └(m + 2) / 3┘*└(n + 2) / 3┘, since the
kings domination number is directly related to the number of rows and columns of kings on the
board.
Knights Domination
Knights are allowed to move two squares in one direction (either horizontally or
vertically) and one square in the other direction as long as they don’t take the place of a friendly
piece. The full move resembles the letter L. Knights are unique in that they are the only pieces
allowed to jump over other pieces (both friendly and enemy). In Image 13, the white and black
knights can move to squares with circles of the corresponding color [5].
15. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
15
Image 13: Knight Movement
No explicit formula is known for the knights domination number. However, several
values of γ(Nnxn) have been verified; the first 20 knights domination numbers appear in Table 2
[6]. As can be seen from the table, as n increases, γ(Nnxn) increases in no discernible pattern.
Table 2: Knights Domination Numbers for 1 ≤ n ≤ 20
n γ(Nnxn)
1 1
2 4
3 4
4 4
5 5
6 8
7 10
8 12
9 14
10 16
11 21
12 24
13 28
14 32
15 36
16 40
17 46
18 52
19 57
20 62
16. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
16
The differences in domination numbers between successive values of n are not
monotonically increasing. For example, γ(N6x6) – γ(N5x5) = 8 – 5 = 3, while γ(N7x7) – γ(N6x6) = 10
– 8 = 2. In other words, the difference between the 6th and 5th knights domination numbers is 3
while the difference between the 7th and 6th knights domination numbers is only 2. In addition,
γ(N18x18) – γ(N17x17) = 52 – 46 = 6, while γ(N19x19) – γ(N18x18) = 57 – 52 = 5. This lack of
monotonicity makes it difficult to tell how quickly γ(Nnxn) grows as n becomes larger and larger.
In Image 14, a minimum number of dominating knights are placed on 4x4, 5x5, 6x6, 7x7
and 8x8 boards [1, p. 97]. Note that there appears to be much symmetry with respect to the
placement of these knights on each board. The four knights on the 4x4 board are placed in a
square in the center. The five knights on the 5x5 board are arranged in a plus sign sort of shape.
On the 6x6 board, four knights are arranged in a square in the center just like with the 4x4 board,
with an additional four knights occupying the corners. On the 7x7 board, two groups of five
knights are placed in horizontal lines on the rows just above and below the middle row. Lastly,
on the 8x8 board, four groups of three knights are arranged in right-angle patterns at symmetric
locations on the board.
17. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
17
Image 14: Knights Domination on 4x4, 5x5, 6x6, 7x7 and 8x8 Boards
However, Image 15 shows that the symmetry displayed among the knights on the boards
in Image 14 fails to hold in the 11x11 case. The 21 knights that dominate this board exhibit no
observable symmetry or pattern whatsoever. This breakdown in symmetry for larger values of n
gives a visual explanation of why domination among knights is not all that well understood. In
1971, Bernard Lemaire devised the arrangement of 21 knights in Image 15, and Alice McRae
showed that 21 was the minimum number of knights needed to cover the 11x11 board, that is,
γ(N11x11) = 21. Further developments were made in 1987 when Eleanor Hare and Stephen
18. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
Hedetniemi developed a linear-time algorithm for computing knights domination numbers on
rectangular mxn chessboards [1, p. 98].
Image 15: Knights Domination on 11x11 Board
Queens Domination
18
Queens are the most powerful chess piece and move horizontally, vertically and
diagonally. Similar to rooks and bishops, they are allowed to move any number of squares in
one direction as long as they do not take the place of a friendly piece or pass through any piece
(own or opponent’s) currently on the board. In Image 16, the queen can move to any of the
squares with a black circle [5].
19. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
19
Image 16: Queen Movement
Domination among queens is the most complicated and interesting of all chess pieces, as
well as the least understood. As with knights, no formula is known for the queens domination
number. Simply analyzing the standard 8x8 chessboard, one can already start to see the
complexities associated with domination among queens. Yaglom and Yaglom proved that are a
whopping 4,860 different ways to cover an 8x8 board with five queens [1, p. 113]. One such
arrangement is shown in Image 17.
Image 17: Five Queens Dominating
an 8x8 Board
20. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
The Spencer-Cockayne Construction
20
Additional evidence of the convoluted nature of queens domination is due to Spencer-
Cockayne in 1990 [1, pp. 116-117]. Starting with a 5x5 board and placing a queen in the center
square, we see that this queen clearly covers the 3x3 square in the center of the board, but leaves
open eight squares symmetrically placed along the four edges of the 5x5 board (for what it’s
worth, these eight squares all happen to be a knight’s move away from the queen). This 5x5
board is displayed in Image 18 with the eight uncovered squares colored in orange.
Image 18: Queen in
Center of 5x5 Board
Now if we place four queens symmetrically spaced apart on previously uncovered
squares on the 5x5 board, the five queens in total dominate a 9x9 board! See Image 19 for this
construction.
21. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
21
Image 19: Five Queens Inside a 5x5 Square
Dominating a 9x9 Board
Now consider an 11x11 board that surrounds this 9x9 board. The same five queens as
before now control all of the squares on the 11x11 board, except for eight symmetrically located
squares, as was the case with the lone queen on the 5x5 board [1, pp. 117-118]. These eight
uncovered squares are highlighted in orange in Image 20.
22. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
22
Image 20: Five Queens on 11x11 Board
By placing four additional queens in a symmetric fashion on squares that were previously
uncovered on the 11x11 board, the nine queens in total on the board now completely control a
15x15 board (see Image 21)!
23. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
23
Image 21: Nine Queens Inside an 11x11 Square Dominating a 15x15 Board
A natural question to ask at this point is whether the pattern associated with this
construction continues ad infinitum. That is, will 13 queens cover a 21x21 board, 17 queens
cover a 27x27 board, 21 queens cover a 33x33 board, etc.? Unfortunately, the answer is no.
While 13 queens do control a 21x21 board, 17 queens only dominate a 25x25 board, not a 27x27
board [1, p. 136]. This is an instance of why queens domination is so difficult. In addition, as of
today, we still do not know whether 9 is the minimum number of queens needed to cover a
15x15 board (see Table 3 for possible domination numbers) [1, p. 119]. Furthermore, only 11
queens are required to cover a 21x21 board, not 13 [1, p. 132]. Thus, in essence, the Spencer-
24. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
Cockayne construction provides us little information about what values γ(Qnxn) might be for
arbitrary values of n.
Upper and Lower Bounds for γ(Qnxn)
24
While there is still much to be discovered about queens domination, both upper and lower
bounds have been established for γ(Qnxn). L. Welch showed that for n = 3m + r, 0 ≤ r ≤ 3,
γ(Qnxn) ≤ 2m + r [3, p. 3]. He went about showing this by dividing a 3nx3n chessboard into nine
nxn blocks. He then placed n queens in the upper right-hand block and n queens in the lower
right-hand block such that the entire 3nx3n board was covered. Thus, it takes at most 2n queens
to cover a 3nx3n board. If n is not divisible by 3, then one can simply perform the same block
construction on only k rows and columns where k is the greatest multiple of 3 less than or equal
to n. Then one could take care of the remaining one or two rows and columns by placing one or
two queens, respectively, such that the remaining squares are covered. Therefore, we have just
proved exactly what Welch’s result suggests, that is, one needs at most (2/3)*k + n mod 3 queens
to cover a 3nx3n board. For example, for n = 11, the greatest multiple of 3 less than or equal to n
is 9, so k = 9. Also, 11 mod 3 ≡ 2. So γ(Qnxn) ≤ (2/3)*k + n mod 3 = (2/3)*9 + 11 mod 3 = 6 + 2
= 8 [1, p. 119]. This concept is illustrated in Image 22.
25. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
25
Image 22: Upper Bound of 8 Queens
Covering 11x11 Board
Spencer proved the following remarkably simple lower bound: γ(Qnxn) ≥ ½*(n – 1) [1, p.
121]. Weakley expanded on this lower bound by showing that if γ(Qnxn) = ½*(n – 1), then n ≡ 3
mod 4 [1, p. 124]. Both proofs are fairly involved so I will omit them. A couple of corollaries
that emerge from Spencer’s and Weakley’s theorems are as follows:
1. γ(Q7x7) = 4 [1, p. 128]
2. For n = 4k + 1, γ(Qnxn) ≥ ½*(n + 1) = 2k + 1 [1, p. 129]
I will omit the proofs of these corollaries but show how Spencer’s lower bound and the lower
bound from Corollary 2 can be used to narrow down the possibilities for γ(Qnxn), if not outright
determine γ(Qnxn).
26. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
26
For n = 9, by Corollary 2, γ(Q9x9) = ½*(9 + 1) = 5. Since there exists an arrangement of
five queens that dominate a 9x9 chessboard2, we conclude that γ(Q9x9) = 5. For n = 10, Corollary
2 can’t be used so we are forced to use Spencer’s lower bound. So γ(Q10x10) ≥ ½*(10 – 1) = 4.5.
Since 4.5 is not an integer, we can simply take the least integer greater than or equal to 4.5 (the
ceiling), which is 5, as the lower bound. Five queens can be arranged so as to dominate a 10x10
board. Therefore, γ(Q10x10) = 5. For n = 11, Spencer’s lower bound is also 5 and there exists an
arrangement of 5 queens that dominate an 11x11 board. Therefore, γ(Q11x11) = 5 as well. For n
= 12, Spencer’s lower bound gives ½*(12 – 1) = 5.5, which rounds up to 6. One can arrange six
queens so as to cover a 12x12 board, so γ(Q12x12) = 6. For n = 13, we can use Corollary 2 since
13 ≡ 1 mod 4. Therefore, γ(Q13x13) ≥ ½*(13 + 1) = 7. In 1994, Burger, Mynhardt and Cockayne
produced a covering of a 13x13 board with 7 queens. Thus, γ(Q13x13) = 7 [1, p. 129-130].
The first value of n for which γ(Qnxn) is not known is 14. Spencer’s lower bound tells us
that γ(Q14x14) ≥ 7. However, no one has been able to devise a placement of seven queens that
dominate a 14x14 board; the best known arrangements of seven queens leave just two squares
uncovered. An eighth queen can be placed so as to cover those two squares. Therefore, γ(Q14x14)
is either 7 or 8 [1, p. 130].
The same sort of reasoning shown above for n = 9 through 14 can be used to deduce
possible values of γ(Qnxn) for larger values of n. Possible γ values for 1 ≤ n ≤ 25 are shown in
Table 3 [1, pp. 124, 128-132].
2 In this and the examples that follow, none of the arrangements of queens dominating a chessboard will be shown.
The fact that such dominating arrangements exist is to be taken as given.
27. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
27
Table 3: Queens Domination Numbers for 1 ≤ n ≤ 25
n γ(Qnxn)
1 1
2 1
3 1
4 2
5 3
6 3
7 4
8 5
9 5
10 5
11 5
12 6
13 7
14 7 or 8
15 7, 8 or 9
16 8 or 9
17 9
18 9
19 9 or 10
20 10 or 11
21 11
22 11 or 12
23 11, 12 or 13
24 12 or 13
25 13
Queens Diagonal Domination
Before concluding my paper, I would like to touch on one last idea related to queens
domination. While queens domination itself has several complications, queens diagonal
domination is much easier to solve. The queens diagonal domination number, denoted
diag(Qnxn), is defined to be the minimum number of queens all placed along the main diagonal
such that the nxn board is dominated. Obviously, diag(Qnxn) ≥ γ(Qnxn) for all n because of the
limitation that queens must be placed on the main diagonal with diagonal domination as opposed
to just anywhere any the board with regular domination [1, pp. 114-115].
Unlike with γ(Qnxn), a formula for diag(Qnxn) is known. The formula is diag(Qnxn) = n –
max(|mid-point free, all even or all odd, subset of {1, 2, 3, …, n}|). Before we can proceed with
28. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
proving this formula, a couple of things need to be defined. First, the vertical bars | in the
definition denote the number of elements in the set in question. Second, a mid-point free set is a
set in which for any given pair of elements in the set, the midpoint or average of those two
numbers is not in the set [1, pp. 115-116].
28
To begin the proof of the formula for the queens diagonal domination number, suppose
that there are a minimum number of queens, all placed along the main diagonal that dominate an
nxn board. Without loss of generality, suppose the squares along the diagonal are white in color.
Now let C be the set of all columns that do not contain any queens. For any two columns i and j
in C, the corresponding square in the ith column and in the jth row (call it the (i, j) square for
short) is not under attack by any queen vertically or horizontally (since there are no queens in
columns i and j). Therefore, (i, j) must be under attack by a queen diagonally and hence the
square must be white. Thus, i + j is even, which implies that both i and j are odd or they are both
even. Since i and j are arbitrary, all numbers in C must be even or all of them must be odd.
Additionally, since the queen attacking the (i, j) square is along the diagonal, it must be on some
square of the form (k, k). Also, i + j = k + k, which means that k = ½*(i + j). In other words, the
queen in column k is exactly halfway between the unoccupied columns i and j. Therefore, given
any two unoccupied columns, the column midway between the two must be occupied, which
implies that C – the set of all unoccupied columns – is a mid-point free set. Consequently, in
order to minimize the number of queens placed along the diagonal needed to dominate a
chessboard, one must maximize the set of empty columns such that the columns are all of the
same parity and the set is mid-point free. Hence, the size of this maximum mid-point free set is
subtracted from n, which gives us the desired formula: diag(Qnxn) = n – max(|mid-point free, all
even or all odd, subset of {1, 2, 3, …, n}|) [1, pp. 115-116]. Image 23 illustrates the argument
laid out in this proof for an 11x11 chessboard.
29. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
29
Image 23: In Diagonal Domination, a Queen Lies
Halfway Between Two Empty Columns
If we let n = 11, we see that diag(Qnxn) can be different from γ(Qnxn). Let C = {2, 4, 8,
10}, which one can check to see that it is mid-point free. Therefore, if we place queens in the
columns not contained in C, that is, columns 1, 3, 5, 6, 7, 9 and 11, we have an arrangement of 7
queens on the diagonal that cover the board. Therefore, diag(Qnxn) = 7. However, as we already
observed earlier, γ(Q11x11) = 5 [1, p. 116].
Conclusion
Chessboard domination remains an unsolved problem in recreational mathematics today.
While domination among rooks, bishops and kings on square nxn chessboards has more or less
been completely characterized, knights and queens domination is still largely an enigma. These
30. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
facts are summarized in Table 4, which gives a compact view of what is known and unknown
about the domination numbers for the five chess pieces analyzed in this paper.
30
Table 4: Domination Number Formulas by Piece
Piece (P) γ(Pnxn) (Square) γ(Pmxn) (Rectangular)
Rook n min(n, m)
Bishop n Unknown
King └(n + 2) / 3┘
2 └(m + 2) / 3┘*└(n + 2) / 3┘
Knight Unknown Unknown
Queen
Unknown, though upper
and lower bounds exist
Unknown
It is no doubt that the irregular L-shaped movement of knights and the versatility of
queens with their vertical, horizontal and diagonal movement has caused domination among
these pieces to be difficult to analyze. I am not confident that a formula will be discovered in the
near future for the domination numbers for either of these two pieces. However, I believe that
the mathematical community is closer to solving the queens domination problem than the knights
domination problem by virtue of the fact that several upper and lower bounds have already been
established for the former. Computer analysis of large chessboards will certainly be key to
uncovering new information and patterns. In addition, I believe that further analysis of
rectangular boards may prove helpful in understanding how the domination functions γ(Pmxn)
behave in a broader sense (note that γ(Bmxn) is unknown so there is still considerable work to do
here).
In my next paper in this series, I will examine the notion of chessboard independence.
Exploring this idea and making the link between it and domination will give us a greater
understanding and appreciation of the mathematical dynamics at play with chess pieces and the
chessboard.
31. Dan Freeman Chessboard Puzzles: Domination
MAT 9000 Graduate Math Seminar
Sources Cited
[1] J.J. Watkins. Across the Board: The Mathematics of Chessboard Problems. Princeton, New
Jersey: Princeton University Press, 2004.
[2] J. Nunn. Learn Chess. London, England: Gambit Publications, 2000.
[3] E.J. Cockayne. Chessboard Domination Problems. Discrete Math, Volume 86, 1990.
[4] J. DeMaio, W.P. Faust. Domination on the mxn Toroidal Chessboard by Rooks and Bishops.
Department of Mathematics and Statistics, Kennesaw State University.
[5] “Chess.” Wikipedia, Wikimedia Foundation. http://en.wikipedia.org/wiki/Chess
[6] “A006075 – OEIS.” http://oeis.org/A006075
31