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.
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 4 - Other Surfaces and VariationsDan Freeman
This document summarizes Dan Freeman's presentation on chessboard puzzles involving surfaces other than the standard chessboard. It discusses knight's tours, domination numbers, and other concepts on toroidal, cylindrical, Klein bottle, and Mobius strip surfaces. Key results include every rectangular board having a closed knight's tour on a torus, formulas for domination numbers of various pieces on these surfaces, and examples of puzzles on different board geometries.
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.
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.
The document is a thesis that investigates rational points on elliptic curves and computing their rank. It begins with introductions to algebraic geometry concepts like affine and projective spaces. It defines elliptic curves and discusses their group structure. It explores points of finite order on elliptic curves and relates them to the curve's discriminant. Later sections analyze the group of rational points using Mordell's theorem and descent methods. It also examines specific problems like congruent numbers and constructing elliptic curves with high rank. The thesis provides mathematical foundations and examples to study rational points and ranks of elliptic curves.
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.
Worksheet - Refraction in difference mediumNeed Ntk
This document describes an experiment to investigate refraction through a semicircular block with a different refractive index than air. Students are instructed to take angle of incidence and refraction measurements for light passing from air into the block. They will use these measurements to generate graphs of the sine of the angles and determine the refractive index. Comparing this value to the known refractive index allows them to verify Snell's law. The document also prompts students to apply their understanding by relating the results back to the original problem of why pencils appear separated in water and discussing other examples of refraction in daily life.
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 4 - Other Surfaces and VariationsDan Freeman
This document summarizes Dan Freeman's presentation on chessboard puzzles involving surfaces other than the standard chessboard. It discusses knight's tours, domination numbers, and other concepts on toroidal, cylindrical, Klein bottle, and Mobius strip surfaces. Key results include every rectangular board having a closed knight's tour on a torus, formulas for domination numbers of various pieces on these surfaces, and examples of puzzles on different board geometries.
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.
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.
The document is a thesis that investigates rational points on elliptic curves and computing their rank. It begins with introductions to algebraic geometry concepts like affine and projective spaces. It defines elliptic curves and discusses their group structure. It explores points of finite order on elliptic curves and relates them to the curve's discriminant. Later sections analyze the group of rational points using Mordell's theorem and descent methods. It also examines specific problems like congruent numbers and constructing elliptic curves with high rank. The thesis provides mathematical foundations and examples to study rational points and ranks of elliptic curves.
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.
Worksheet - Refraction in difference mediumNeed Ntk
This document describes an experiment to investigate refraction through a semicircular block with a different refractive index than air. Students are instructed to take angle of incidence and refraction measurements for light passing from air into the block. They will use these measurements to generate graphs of the sine of the angles and determine the refractive index. Comparing this value to the known refractive index allows them to verify Snell's law. The document also prompts students to apply their understanding by relating the results back to the original problem of why pencils appear separated in water and discussing other examples of refraction in daily life.
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.
Haiku Deck is a presentation tool that allows users to create Haiku style slideshows. The tool encourages users to get started making their own Haiku Deck presentations which can be shared on SlideShare. In just a few sentences, it pitches the idea of using Haiku Deck to easily create visually engaging slideshows.
The document discusses finding the number of ways to place two queens on an nxn chessboard so that they do not attack each other. It first explores the average number of squares controlled by a queen based on its position. It then uses similar methods applied to the "stone problem" and "rooks problem" to derive a formula for the queen problem. The formula is n2(n2-Sq)/2, where Sq represents the average number of squares controlled by a queen. Testing various values of n verifies the formula works for any board size. A graph shown exponential growth in the number of possibilities as the board size n increases.
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.
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.
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.
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.
This document discusses network theory concepts like degree, type, and k-core as they relate to a social network graph of visitors to two cafes, Costa and Starbucks, on a Saturday afternoon. It presents the network, examines degree and type of connections, looks at network clusters, discusses a take away subgraph, and shows a k-core value of 3 for the overall network graph.
Mathematicians often solve practical problems using graph theory and problem solving steps. Graph theory represents problems as graphs with vertices and edges. A minimum spanning tree is a graph with no cycles and the minimum total edge weight connecting all vertices. Mathematicians analyze problems, represent them as graphs, develop and test solution strategies, and verify solutions work for all cases to solve problems like optimizing routes for aerial repair vans or hospital department layouts.
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 permutation puzzles and their connection to group theory. It specifically examines three puzzles: the Rubik's Cube, Pyraminx, and Megaminx. For the Rubik's Cube, it provides the history, establishes notation for the sides, cubies (pieces), and basic moves, and discusses how the cube's moves form a non-abelian group with specific structure and properties. The Pyraminx and Megaminx are similarly introduced, with notation and an overview of how their moves relate to group theory.
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.
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.
Haiku Deck is a presentation tool that allows users to create Haiku style slideshows. The tool encourages users to get started making their own Haiku Deck presentations which can be shared on SlideShare. In just a few sentences, it pitches the idea of using Haiku Deck to easily create visually engaging slideshows.
The document discusses finding the number of ways to place two queens on an nxn chessboard so that they do not attack each other. It first explores the average number of squares controlled by a queen based on its position. It then uses similar methods applied to the "stone problem" and "rooks problem" to derive a formula for the queen problem. The formula is n2(n2-Sq)/2, where Sq represents the average number of squares controlled by a queen. Testing various values of n verifies the formula works for any board size. A graph shown exponential growth in the number of possibilities as the board size n increases.
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.
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.
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.
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.
This document discusses network theory concepts like degree, type, and k-core as they relate to a social network graph of visitors to two cafes, Costa and Starbucks, on a Saturday afternoon. It presents the network, examines degree and type of connections, looks at network clusters, discusses a take away subgraph, and shows a k-core value of 3 for the overall network graph.
Mathematicians often solve practical problems using graph theory and problem solving steps. Graph theory represents problems as graphs with vertices and edges. A minimum spanning tree is a graph with no cycles and the minimum total edge weight connecting all vertices. Mathematicians analyze problems, represent them as graphs, develop and test solution strategies, and verify solutions work for all cases to solve problems like optimizing routes for aerial repair vans or hospital department layouts.
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 permutation puzzles and their connection to group theory. It specifically examines three puzzles: the Rubik's Cube, Pyraminx, and Megaminx. For the Rubik's Cube, it provides the history, establishes notation for the sides, cubies (pieces), and basic moves, and discusses how the cube's moves form a non-abelian group with specific structure and properties. The Pyraminx and Megaminx are similarly introduced, with notation and an overview of how their moves relate to group theory.
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.
This document provides an overview and table of contents for the book "1-2-3 Draw Knights, Castles, and Dragons" by Freddie Levin. The book contains step-by-step instructions for drawing knights, castles, dragons, and other elements of medieval times. It begins with an introduction to basic shapes and how to draw a basic human figure. Subsequent chapters provide guidance for drawing more complex subjects like royalty, castles, heraldry, weapons, and fantasy creatures. The document also includes an order form and list of other titles in the "1-2-3 Draw" series.
This book examines leadership lessons that can be learned from Alexander the Great by analyzing stories and events from his life. The book is organized around four leadership processes: reframing problems, building alliances, establishing identity, and directing symbols. Through short chapters that describe pivotal moments, the book aims to distill enduring lessons about leadership that remain relevant for modern business executives and leaders.
The cluster package provides methods for cluster analysis in R. It extends the original cluster analysis methods from Kaufman and Rousseeuw (1990). The package contains functions for hierarchical clustering (agnes, diana), partitioning clustering (pam, clara), and other clustering visualization and validation methods. It allows clustering on both data matrices and dissimilarity matrices. The package is widely used for cluster analysis in R.
[New Chapter - The GLGL Wheel.] The Sefer Yetzirah ('Book of Formation') is the ancient source text from which the contemporary design of the kabbalistic Tree of Life diagram has evolved. The original 'Tree of Yetzirah' design however, is different in certain important respects to our modern day versions of the Tree diagram. Sefer Yetzirah Magic includes an exposition of the original design of the Tree and develops a system of magic (and 'initiation') that's based on the Sefer Yetzirah's original metaphysical blueprint.
This document is a dissertation presented for a Doctor of Philosophy degree. It explores methods for introducing continuous parameters into circle packings to overcome limitations in discrete analytical function theory due to the discreteness of circle packings. Topics covered include packings with overlapping circles, fractional branching which assigns fractional edge lengths, and shift-points which allow edges to be shifted along circles. Computational examples demonstrate some previously non-realizable discrete functions can be realized using these techniques. Necessary theoretical developments are also presented, such as generalizing definitions for overlapping packings and expressing singularities associated with faces.
This document describes a final project for a bachelor's degree in computer engineering on developing an intelligent chess game. It provides an overview of the necessary components to computerize chess, including representing the chess board in memory, generating legal moves according to the rules of chess, using techniques to choose moves and evaluate positions, and a user interface. The goal is to develop a fully computerized version of chess that can play intelligently against a human player by making its own decisions.
The document provides instructions for using foldable study guides in social studies classes. It explains that foldables allow students to organize information in an interactive way that helps them retain concepts. Various folding techniques are demonstrated that can be used to create study guides, projects, charts, and diagrams for different topics in world history. Teachers are encouraged to make this book available as a resource for students to learn new ways to present information.
This thesis analyzes bargaining behavior at the Grand Bazaar in Istanbul using game theory. It conducts a field study surveying buyers and sellers at the bazaar to understand how buyers' characteristics influence prices. The study aims to determine if real-life bargaining can be modeled with games and provide tourists insights into negotiating better prices.
This booklet provides information and instructions for Scouts working on the Pioneering Merit Badge, including requirements, knot tying instructions, lashing techniques, projects like building bridges and towers, and safety information. Various rope knots, hitches, bends, loop knots and lashings are defined and illustrated. Requirements cover first aid, knot tying, lashings, rope handling, splices, rope making, projects involving signal towers and bridges.
antimicrobial and antioxidant activity of selected turkish spices english ver...Gioacchino dell'Aquila
This document provides an overview of several spices and their properties. It discusses the taxonomy, distribution, ethnobotany, history, and bioactive constituents of five specific spices: Thymbra spicata, Rhus coriaria, Ocimum basilicum, Mentha spicata, and Origanum vulgare. It also reviews the history and modern methods of biopreservation using natural compounds from spices and herbs. The objectives are to evaluate the antimicrobial and antioxidant activities of these five spices against common foodborne pathogens and their potential application as natural preservatives.
This thesis examines how greenway routes that provide experiences of nature can help commuters recover from mental fatigue during their daily commutes. It reviews literature on Attention Restoration Theory and how being in natural settings can aid recovery. Case studies of existing walking/cycling routes near Metro stations in Northern Virginia are analyzed using ART principles. Based on this, Huntington station is selected as the site for a proposed greenway design. The design aims to incorporate natural elements that maximize the route's potential for mental restoration according to ART. The framework developed can guide greenway planning to improve quality of life and address issues like traffic and public health.
The pst-euclide package allows drawing of geometric figures in LaTeX using macros that specify mathematical constraints. It defines points that can be used to construct figures through common transformations and intersections. Basic objects like points, lines, circles, arcs, and curves can be drawn. The package also includes macros for geometric transformations, intersections of objects, and other special objects like midpoints, bisectors, and centers of triangles.
This document provides a table of contents for a book on practical astronomy. The table of contents lists 24 chapters that cover various topics related to tools and techniques in astronomy, the interstellar medium, stars, galaxies, and more. Some of the chapter topics include stellar classification, the Hertzsprung–Russell diagram, nebulae, star formation, the life cycles of stars of different masses, pulsating stars, planetary nebulae, white dwarfs, supernovae, galaxies, and galaxy classification. The document provides an overview of the breadth of content included in the book.
This package provides functions for multivariate statistical analysis and is an R companion to the book "Introduction to Multivariate Statistical Analysis in Chemometrics". It contains functions for multivariate methods, diagnostics, calibration, cross-validation, clustering, and more. The package can be used to analyze example datasets from the book or other user data.
This document provides an introduction to mathematical and physics exploration studies for students at St. George's College in 2015. It lists tutors available and encourages students to attempt problems to develop problem solving skills. Key people and theorems in mathematics and physics are outlined that students should familiarize themselves with, covering topics from Euclid to Einstein. The document then continues with chapters on various mathematical and physics topics.
This document lists LATEX commands for 5913 symbols across various packages and fonts. It includes commands for symbols that can be used in both text and math mode, as well as commands for mathematical symbols like operators, relations, and variables. The document is intended to provide a comprehensive reference for obtaining symbols in LATEX.
The document is a curriculum for jazz theory to accompany instrumental jazz courses from levels 10 to 30. It covers topics such as scales, intervals, chord progressions, and analysis. The introduction explains that jazz theory was developed by analyzing the music played by great jazz artists, rather than being a set of rules they followed. It emphasizes applying theory to playing over just studying it abstractly.
Chessboard Puzzles Part 4 - Other Surfaces and Variations
1. Chessboard Puzzles: Other Surfaces and Variations
Part 4 of a 4-part Series of Papers on the Mathematics of the Chessboard
by Dan Freeman
May 19, 2014
2. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
Table of Contents
Table of Figures .............................................................................................................................. 3
Introduction ..................................................................................................................................... 4
Knight’s Tour Revisited .................................................................................................................. 4
2
Knight’s Tour on a Torus ............................................................................................................ 5
Knight’s Tour on a C ylinder ....................................................................................................... 6
Knight’s Tour on a K lein Bottle .................................................................................................. 7
Knight’s Tour on a Möbius Strip ................................................................................................ 8
Rooks and Bishops Domination on a Torus ................................................................................... 9
Kings Domination and Independence on a Torus ......................................................................... 11
Knights Domination on a Torus.................................................................................................... 13
Queens Domination on a Torus .................................................................................................... 13
The 8-queens Problem on a Cylinder............................................................................................ 14
Domination and Independence on the Klein Bottle ...................................................................... 16
Independent Domination Number................................................................................................. 19
Upper Domination Number .......................................................................................................... 20
Irredundance Number ................................................................................................................... 22
Upper Irredundance Number ........................................................................................................ 23
Total Domination Number ............................................................................................................ 26
Conclusion .................................................................................................................................... 28
Sources Cited ................................................................................................................................ 32
3. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
Table of Figures
Image 1: Knight Movement ............................................................................................................ 4
Image 2: Chessboard on a Torus..................................................................................................... 5
Image 3: C losed Knight’s Tour on 2x7 Torus ................................................................................ 6
Image 4: Closed Knight's Tour on 3x8 Torus ................................................................................. 6
Image 5: Closed Knight's Tour on 4x9 Torus ................................................................................. 6
Image 6: Klein Bottle ...................................................................................................................... 7
Image 7: Closed Knight's Tour on 6x2 Klein Bottle....................................................................... 8
Image 8: Closed Knight's Tour on 6x4 Klein Bottle....................................................................... 8
Image 9: Möbius Strip .................................................................................................................... 9
Image 10: Rook Movement .......................................................................................................... 10
Image 11: Bishop Movement ........................................................................................................ 10
Image 12: There Are n Distinct Diagonals (Shown in Red) on an nxn Torus .............................. 11
Image 13: King Movement ........................................................................................................... 11
Image 14: Nine Kings Dominating a Regular 7x7 Board ............................................................ 12
Image 15: Seven Kings Dominating a 7x7 Torus ......................................................................... 12
Image 16: Queen Movement......................................................................................................... 14
Image 17: Eight Queens Fail to Be Independent on 8x8 Cylinder ............................................... 15
Image 18: Row, Column and Negative Diagonal Labeling of 8x8 Chessboard .......................... 16
Image 19: Fifteen Kings Dominating a 7x14 Klein Bottle ........................................................... 18
Image 20: Thirteen Kings Dominating a 14x7 Klein Bottle ......................................................... 18
Image 21: Queens Domination Number, Independent Domination Number
and Independence Number on 4x4 Chessboard ........................................................................... 20
Image 22: Queens Domination Number, Independent Domination Number,
Independence Number and Upper Domination Number on 6x6 Chessboard.............................. 21
Image 23: Maximal Irredundant Sets of 8 Kings and 9 Kings on 7x7 Chessboard ...................... 23
Image 24: Maximum Irredundant Set of 16 Kings on 7x7 Chessboard ....................................... 24
Image 25: Maximum Irredundant Set of 12 Rooks on 8x8 Chessboard ....................................... 24
Image 26: Maximum Irredundant Set of 18 Bishops on 8x8 Chessboard .................................... 25
Image 27: Maximum Irredundant Set of 7 Bishops on 6x6 Chessboard ...................................... 25
Image 28: Totally Dominating Sets of 5 Queens and 8 Rooks on 8x8 Chessboard ..................... 26
Image 29: Totally Dominating Sets of 10 Bishops and 14 Knights on 8x8 Chessboard .............. 27
Table 1: Knights Domination N umbers on Regular C hessboard and Torus for 1 ≤ n ≤ 8 ........... 13
Table 2: Queens Domination N umbers on Regular Chessboard and Torus for 1 ≤ n ≤ 10 .......... 14
Table 3: K ings Total Domination N umbers for 1 ≤ n ≤ 12........................................................... 27
Table 4: Upper Bounds for K ings Total Domination N umbers for 13 ≤ n ≤ 25........................... 28
Table 5: Domination Number Formulas on the Torus by Piece ................................................... 29
Table 6: Chessboard Number Formulas by Piece ......................................................................... 31
3
4. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
Introduction
4
In this fourth and final paper in my series on mathematical chessboard puzzles, I will
build on the three major topics in domination, independence and the knight’s tour that I
examined in my first three papers by looking at them in the context of irregular surfaces such as
the torus, cylinder, Klein bottle and Möbius strip. I will then explore some other concepts
related to domination and independence such as the independent domination number, upper
domination number, irredundance number, upper irredundance number and total domination
number. Since bringing such irregular surfaces and other variations to the table results in a vast
amount of combinations that one could potentially explore, this paper is by no means exhaustive.
Rather, it should serve as an overview of the mathematical properties and formulas of these
variations to the concepts of domination, independence and the knight’s tour and how these
variations compare and contrast with the original concepts. My hope is that this paper will
provide the reader with a sense of the vast diversity among mathematical chessboard puzzles.
Knight’s Tour Revisited
Recall that knights move two squares in one direction (either horizontally or vertically)
and one square in the other direction, thus making the move resemble an L shape. Knights are
the only pieces that are allowed to jump over other pieces. In Image 1, the white and black
knights can move to squares with circles of the corresponding color [4].
Image 1: Knight Movement
5. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
5
Also, recall from my preceding paper that a knight’s tour is a series of moves made by a
knight that visits every square on an mxn1 chessboard once and only once. A knight’s tour may
fall into one of two categories, closed or open, defined as follows:
A closed knight’s tour is one in which the knight’s last move in the tour places it a single
move away from where it started.
An open knight’s tour is one in which the knight’s last move in the tour places it on a
square that is not a single move away from where it started.
Knight’s Tour on a Torus
Like any of the surfaces we will be studying in this paper, a torus is a topological surface
with a very specific definition. However, for our purposes, we can simply view a torus as a
donut-shaped surface in which both the rows and columns wrap around on their edges. That is,
when moving to the right beyond the right edge of the board, one returns to the left edge and
when moving up beyond the top edge of the board, one returns to the bottom edge [1, p. 65]. A
toroidal chessboard is illustrated in Image 2 [5].
Image 2: Chessboard on a Torus
In 1997, John Watkins and his student, Becky Hoenigman, proved the remarkable result
that every mxn rectangular chessboard has a closed knight’s tour on a torus! Images 2, 3 and 4
show examples of closed knight’s tours on a 2x7, 3x8 and 4x9 tori, respectively [1, p. 67]. The
1 Throughout this paper, m and n refer to arbitrary positive integers denoting the number of rows and columns of a
chessboard, respectively.
6. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
up arrows on either side of the board and the right-pointing arrows above and below the board
are commonly used notation to indicate that the board is on a torus.
Image 3: Closed Knight’s Tour on 2x7 Torus
Image 4: Closed Knight's Tour on 3x8 Torus
Image 5: Closed Knight's Tour on 4x9 Torus
1
Knight’s Tour on a Cylinder
6
1
Unlike a torus, a cylinder only wraps in one dimension, not both. Since the knight’s tours
on the 3x8 and 4x9 tori in Images 4 and 5 above only make use of horizontal wrapping, these
tours would also work on a cylinder. In 2000, John Watkins proved that a knight’s tour exists on
an mxn cylindrical chessboard unless one of the following two conditions holds:
7. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
7
1) m = 1 and n > 1; or
2) m = 2 or 4 and n is even [1, p. 71].
It is easy to see why the above cases are excluded. If m = 1, a knight can’t move at all. If
m = 2 and n is even, then each move would take the knight left or right by two columns and so
then only at most half of the columns would be visited. Lastly, if m = 4 and n is even, then the
coloring argument by Louis Pósa from my previous paper on knight’s tours holds [1, p. 71].
Knight’s Tour on a Klein Bottle
The Klein bottle, due to German mathematician Felix Klein in 1882 [6], operates like a
torus, except when wrapping horizontally, the rows reverse order due to the half-twist in the
construction of the surface [1, p. 79]. A Klein bottle is represented in Image 6 [7].
Image 6: Klein Bottle
John Watkins proved that, just like with a torus, every rectangular chessboard has a
knight’s tour on a Klein bottle [1, p. 81]. Examples of closed knight’s tours on 6x2 and 6x4
Klein bottles are shown in Images 7 and 8. The up arrow on the left side of the board, the down
arrow on the right side of the board and the right-pointing arrows above and below the board are
commonly used notation to indicate that the board is on a Klein bottle.
8. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
Image 7: Closed Knight's
Tour on 6x2 Klein Bottle
Image 8: Closed Knight's
Tour on 6x4 Klein Bottle
Knight’s Tour on a Möbius Strip
8
A very famous one-sided surface known as the Möbius strip shares properties of both the
cylinder and the Klein bottle. It is like the cylinder in that it only wraps in one dimension but is
like the Klein bottle in that it makes a half-twist when wrapping, thereby reversing the order of
the rows [1, p. 82]. A Möbius strip is shown in Image 9 [8].
9. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
9
Image 9: Möbius Strip
John Watkins proved that a closed knight’s tour exists on a Möbius strip unless one or
more of the following three conditions hold:
1) m = 1 and n > 1; or n = 1 and m = 3, 4 or 5;
2) m = 2 or 4 and n is even; or
3) n = 4 and m = 3 [1, p. 82]
Rooks and Bishops Domination on a Torus
Now, let’s turn our attention to the concept of domination, which we explored in my first
paper in this series, applied to a torus. First, let’s look at the uninteresting case of rooks
domination on a torus. Recall that 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. In Image 10, 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 [4].
10. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
10
Image 10: Rook Movement
Since it doesn’t make any difference whether a rook is on a regular board or on a torus, it
follows that γtor(Rnxn) = γ(Rnxn) = n, where the subscript “tor” indicates domination on a torus [1,
p. 144].
Almost as boring as the rook’s case is the bishop’s case. Recall that bishops move
diagonally any number of squares 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 11, 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 [4].
Image 11: Bishop Movement
Since the number of distinct diagonals in either direction drops from 2n – 1 on a regular
square chessboard to n on a torus, it is easy to see that γtor(Bnxn) = γ(Bnxn) = n [1, p. 144]. Image
12 illustrates this fact.
11. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
Image 12: There Are n Distinct Diagonals
(Shown in Red) on an nxn Torus
Kings Domination and Independence on a Torus
11
Recall that 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 13, the king can move to any of the squares
with a white circle [4].
Image 13: King Movement
Kings domination on a torus is more interesting than that of rooks or bishops because the
kings domination number on a torus is quite distinct from the formula on a regular chessboard2.
2 Throughout this paper, the symbols ‘└’ and ‘┘’ will be used to indicate the greatest integer or floor function and the
symbols ‘┌’ and ‘┐’ will be used to indicate the least integer or ceiling function.
12. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
While γ(Knxn) = └(n + 2) / 3┘
12
2, γtor(Knxn) = ┌(n / 3)*┌ n / 3 ┐┐ [1, p. 147]. The former uses the
floor function while the latter uses the ceiling function. However, both share a factor of 1/3
because regardless of whether a king is on a torus or not, it can control squares on at most three
rows and at most three columns (in fact, on a torus, a king always covers squares on exactly three
rows and three columns, since a torus technically has no edges). The kings domination number
on a torus can be generalized to a rectangular board using the following formula: γtor(Kmxn) =
max{┌(m / 3)*┌ n / 3 ┐┐, ┌(n / 3)*┌m / 3 ┐┐} [1, p. 149].
It is worth pointing out that, in general, fewer kings are needed to cover a torus than a
regular chessboard of the same size. For example, 9 kings are required to cover a 7x7 regular
board while only 7 kings are needed to cover a 7x7 board on a torus. This fact is illustrated in
Images 14 and 15.
Image 15: Nine Kings Dominating
a Regular 7x7 Board
Image 14: Seven Kings Dominating
a 7x7 Torus
13. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
13
The formulas for the kings independence number on a torus take a similar form to those
for the kings domination number except that the floor function is used instead of the ceiling
function and a minimum value, not a maximum value, is sought in the rectangular formula. The
formulas are as follows: βtor(Knxn) = └(½*n)*└½*n┘┘ and βtor(Kmxn) = min{└(½*m)*└
½*n
┘┘,
└(½*n)*└
½*m
┘┘}. Note that the factor of ½ appears here in the toroidal formulas just as it does
in the formula on the regular chessboard (recall that β(Knxn) = └½*(n + 1)┘
2). This is due to the
fact that any 2x2 block of squares can contain at most one independent king, regardless of
surface [1, p. 194].
Knights Domination on a Torus
The knights domination numbers on both a regular board and a torus for 1 ≤ n ≤ 8 appear
in Table 1 [1, p. 140]. Note that, shockingly, the value for γtor is lower for n = 8 than it is for n =
7! Also, each value of γtor is unique up to n = 8. This may or may not be the case in general.
Table 1: Knights Domination Numbers on Regular
Chessboard and Torus for 1 ≤ n ≤ 8
n γ(Nnxn) γtor(Nnxn)
1 1 1
2 4 2
3 4 3
4 4 4
5 5 5
6 8 6
7 10 9
8 12 8
Queens Domination on a Torus
Recall that queens move horizontally, vertically and diagonally any number of squares 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 11, the queen can move to any of the squares with
a black circle [4].
14. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
14
Image 16: Queen Movement
The queens domination numbers on both a regular board and a torus for 1 ≤ n ≤ 10 appear
Table 2 [1, p. 140]. Note that the only case where the two numbers differ is n = 8. In this case,
one fewer queen is needed to dominate the board on a torus than is required to cover the regular
board.
Table 2: Queens Domination Numbers on Regular
Chessboard and Torus for 1 ≤ n ≤ 10
n γ(Qnxn) γtor(Qnxn)
1 1 1
2 1 1
3 1 1
4 2 2
5 3 3
6 3 3
7 4 4
8 5 4
9 5 5
10 5 5
The 8-queens Problem on a Cylinder
Unlike the queens independence number on a regular chessboard (recall that this is just n
on an nxn square board for all n other than 2 or 3), a formula for this number on the cylinder,
denote it βcyl(Qnxn), has not yet been found. While βcyl(Q5x5) = β(Q5x5) = 5 and βcyl(Q7x7) =
β(Q7x7) = 7, βcyl(Q8x8) = 6 ≠ β(Q8x8) = 8 [1, pp. 193-194]. Image 17 illustrates why one particular
arrangement of eight queens that would be independent on a regular 8x8 board fails to be
15. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
independent on an 8x8 cylinder [1, p. 192]. The queen in the first column from the left resides in
the same negative diagonal as the queen in the fourth column while the queen in the second
column lies on the same positive diagonal as the queen in the seventh column. The right-pointing
15
arrows above and below the board are commonly used notation to indicate that the
board is on a cylinder.
Image 17: Eight Queens Fail to Be
Independent on 8x8 Cylinder
As it turns out, none of the twelve fundamental solutions to the 8-queens problem on a
regular 8x8 chessboard are independent on an 8x8 cylinder. One can simply check this by brute
force. E. Gik provided an alternative proof by labeling each square of an 8x8 board with an
ordered triple (i, j, k), where i, j and k represent the following:
i – the row number starting with 1 from the bottom of the board,
j – the column number starting with 1 from the left side of the board and
k – the number of the negative diagonal in reverse order from 8 through 1starting on the
diagonal right below the main diagonal [1, p. 192].
This (i, j, k) labeling on an 8x8 board is shown in Image 18. By design, the sum of each of the
three coordinates i, j and k is divisible by 8 for each of the 64 squares on the board. Now
suppose that we have 8 independent queens on the board. Then all three of the coordinates must
be distinct for each square, otherwise at least two queens would share the same row, column or
16. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
negative diagonal, contradicting the fact that the set of queens is independent. Therefore, the i
coordinates for these 8 queens sum to 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, and the j and k
coordinates sum to 36 as well. But then the sum of all three coordinates for the 8 queens would
be 108, which is not divisible by 8, contradicting the fact that i + j + k for each square is divisible
by 8. Therefore, our assumption that the 8 queens are independent is false [1, p. 192].
16
Image 18: Row, Column and Negative
Diagonal Labeling of 8x8 Chessboard
One can then check by exhausting all possibilities that no arrangement of 7 independent
queens exists on an 8x8 cylinder. In the 1 5 8 6 3 7 2 4 solution to the 8-queens problem on the
regular 8x8 chessboard shown in Image 17, one can simply remove one of the two queens lying
on the same positive diagonal and one of the two queens lying on the same negative diagonal to
arrive at an independent set of 6 queens on an 8x8 cylinder. Therefore, βcyl(Q8x8) = 6 [1, p. 193].
Domination and Independence on the Klein Bottle
Kings domination on a Klein bottle is somewhat convoluted so I will merely state the
formula rather than prove it. The formula on a square nxn chessboard is as follows [1, p. 151]:
17. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
γKlein(K
γKlein(K
=
17
(1/9)*n2 if n = 3k
(1/3(n + 1))2 if n = 3k + 2
(1/3(n + 2))2 – (1/6)*(n – 1) if n = 6k + 1
(1/3(n + 2))2 – (1/6)*(n + 2) if n = 6k + 4
nxn
) =
The preceding formula can be generalized to a rectangular mxn chessboard as below.
While it looks nothing like the formula for the square board, this does in fact reduce to the square
formula when m = n [1, p. 152].
┌m/6┐*┌2n/3┐ – ┌(n – 1)/3┐ if m ≡ 1, 2 or 3 mod 6
┌m/6┐*┌2n/3┐ if m ≡ 4, 5 or 6 mod 6
m xn
)
I find it fascinating that, unlike with a regular chessboard and even a torus, it does
actually matter whether one chooses to orientate the board horizontally or vertically as far as the
domination number is concerned on an mxn rectangular Klein bottle. That is, γKlein(P
m xn
) need
not be the same as γKlein(P
nxm
) for some chess piece P. This is certainly the case with kings, as
γKlein(K
7x14
) = 15 while γKlein(K
14x7
) = 13. This phenomenon, exhibited in Images 19 and 20, is
due to the half-twist that the Klein bottle makes as it wraps horizontally. The 7x14 board has one
3x28 band (the 3x14 band in dark orange at the top of the board merged with the 3x14 band at
the bottom in Image 19) that requires 10 kings to dominate. An additional 5 kings are needed to
cover the single row in the middle of the board (highlighted in gray in Image 19) for a total of 15
kings. However, on the 14x7 board, five kings are needed to cover the two 3x14 bands (the 3x7
band in dark orange at the top of the board merged with the 3x7 band at the bottom and the two
3x7 bands in light orange just below and above the dark orange bands in Image 20) and only
three kings are needed to cover the two rows in the middle of the board (highlighted in gray in
Image 20) for a total of 13 kings [1, pp. 160-161].
18. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
18
Image 19: Fifteen Kings Dominating a 7x14 Klein Bottle
Image 20: Thirteen Kings Dominating a 14x7 Klein Bottle
19. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
βKlein(K
βKlein(K
19
A formula is known for the bishops domination number on an nxn Klein bottle. Since a
bishop is able to cover twice as much ground on a Klein bottle as it can on a regular chessboard
(due to wrapping), it should come as no surprise then that γKlein(Knxn) = ┌½*n┐. More
specifically, if n is even, then γKlein(Knxn) = ½*n and if n is odd, then γKlein(Knxn) = ½*(n + 1) [1,
pp. 153-155].
In 2002, John Watkins and B. McVeigh provided a formula for the kings independence
number on an mxn rectangular Klein bottle, as follows [1, p. 196]:
└½*m┘*└½*n┘ if n even
n*└(1/4)*m┘ – 1 if n odd, m ≡ 0 mod 4
n*└(1/4)*m┘ if n odd, m ≡ 1 mod 4
n*└(1/4)*m┘ + └½*n┘ if n odd, m ≡ 2 or 3 mod 4
) =
m xn
As you can see, the above formula considers of four different cases for m and n. The
formula simplifies a little bit (but maybe not as much as we would like) when considering only
square nxn chessboards. Since it is impossible to have n odd and n ≡ 0 mod 4, the second case in
the above formula disappears, reducing the formula for the kings independence number on an
nxn square Klein bottle to the following:
└½*n┘
2 if n even
n*└(1/4)*n┘ if n ≡ 1 mod 4
n*└(1/4)*n┘ + └½*n┘ if n ≡ 3 mod 4
nxn
) =
Independent Domination Number
The independent domination number for a given piece P and a given mxn chessboard is
the minimum size of an independent dominating set, denoted i(Pmxn). This quantity need not
equal γ(Pmxn) or β(Pmxn), as shown in the arrangements of queens placed on 4x4 chessboards in
Image 21 [1, p. 198].
20. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
20
Image 21: Queens Domination Number, Independent Domination
Number and Independence Number on 4x4 Chessboard
From Image 21, we see that γ(Q4x4) = 2, i(Q4x4) = 3 and β(Q4x4) = 4, and so in this case,
the domination number, independent domination number and independence number are all
distinct from one another. This strict ordering is somewhat rare. In fact, it has been conjectured
by G. H. Fricke and others that i(Qnxn) = γ(Qnxn) for sufficiently large n. However, the following
inequality does hold for all P, m and n: γ(Pmxn) ≤ i(Pmxn) ≤ β(Pmxn). The first comparison in the
inequality is clear because the minimum independent dominating set is always at least as large as
the minimum dominating set overall. Also, since we have already seen that a maximum
independent set is also a dominating set, it then immediately follows that the minimum size of an
independent dominating set is no larger than a maximum independent set, that is, i(Pmxn) ≤
β(Pmxn) [1, p. 198].
For rooks, bishops and kings, the independent domination number is always the same as
the independence number. The reasons for these equalities are straightforward. Rooks that are
placed along the main diagonal both are independent and dominate the board. Bishops that are
placed down a central column also are independent while covering the board. Lastly, in our
construction of minimum dominating sets of kings from my first paper in this series on
mathematical chessboard puzzles, no two kings were adjacent to each other, hence the kings
were independent [1, p. 199].
Upper Domination Number
A dominating set, call it D, of chess pieces of type P is said to be a minimal dominating
set if the removal of any of the pieces from the board makes it so that the remaining pieces fail to
21. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
dominate the board. This implies that no proper subset of D is a dominating set. The upper
domination number is defined to be the maximum size of a minimal dominating set of pieces P,
denoted as Γ(Pmxn). Since a maximum independent set is also a minimal dominating set, we can
expand our chain of inequalities to the following: γ(Pmxn) ≤ i(Pmxn) ≤ β(Pmxn) ≤ Γ(Pmxn) [1, pp.
201-202].
21
From Image 22, we see that γ(Q6x6) = 3, i(Q6x6) = 4, β(Q6x6) = 6 and Γ(Q6x6) = 7 [1, p.
201]. The minimal dominating set of 7 queens in Image 22 was found by Weakley, who showed
that for n ≥ 5, it follows that Γ(Pnxn) ≥ 2n – 5. The largest value of n for which Γ(Qnxn) is known
is 7 and Γ(Q7x7) = 9 [1, p. 202].
Image 22: Queens Domination Number, Independent Domination Number,
Independence Number and Upper Domination Number on 6x6 Chessboard
McRae produced a minimal dominating set of 37 kings on a 12x12 board; whether this is
maximum or not has yet to be verified. However, this shows that β(K12x12) = 36 ≤ Γ(K12x12) [1, p.
202].
22. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
22
For rooks, bishops and knights, it turns out that Γ = β. A minimal dominating set of
rooks will have exactly one rook in each row or exactly one rook in each column; hence, Γ(Rnxn)
= n [1, p. 202]. Thus, overall, we have that γ(Rnxn) = i(Rnxn) = β(Rnxn) = Γ(Rnxn) = n. A minimal
dominating set of bishops can have no more than one bishop on each of the 2n – 1 positive
diagonals and cannot have bishops occupying both the upper left-hand corner and the lower
right-hand corner. Therefore, Γ(Bnxn) ≤ 2n – 2. Since Γ(Bnxn) ≥ β(Bnxn) = 2n – 2, it follows that
Γ(Bnxn) = β(Bnxn) = 2n – 2. As for knights, the fact that Γ(Nnxn) = β(Nnxn) stems from a graph
theory argument using bipartite graphs due to Cockayne, Favaron Payan and Thomason [1, p.
203].
Irredundance Number
An irredundant set of chess pieces is one in which each piece in the set either occupies a
square that is not covered by another piece or else it covers a square that no other piece covers.
A maximal irredundant set is one that is not a proper subset of any irredundant set. The
irredundance number for a given piece P and a given mxn chessboard is the minimum size of a
maximal irredundant set, denoted by ir(Pmxn). Since a minimum dominating set doesn’t have any
pieces that are redundant (otherwise, it wouldn’t be minimum) and must be maximal (otherwise,
it wouldn’t be dominating), it follows that a minimum dominating set is a maximal irredundant
set. Therefore, we can expand our chain of inequalities to the following: ir(Pmxn) ≤ γ(Pmxn) ≤
i(Pmxn) ≤ β(Pmxn) ≤ Γ(Pmxn) [1, pp. 204-205].
In order to illustrate the concept of irredundance, Image 23 contains two maximal
irredundant sets of kings on 7x7 chessboards. Neither set is the proper subset of any irredundant
set. However, the set of eight kings on the left fails to cover the entire board while the set of
nine kings on the right is a minimum dominating set. Thus, ir(K7x7) ≤ 8 < γ(K7x7) = 9. This
example shows that ir(Pmxn) ≠ γ(Pmxn) in general [1, p. 204].
23. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
23
Image 23: Maximal Irredundant Sets of
8 Kings and 9 Kings on 7x7 Chessboard
As one might expect, the irredundance number for both rooks and bishops on a square
nxn chessboard is n. However, no formulas are available for kings, queens or knights [1, p. 205].
Upper Irredundance Number
The upper irredundance number of a chess piece P on an mxn chessboard is the
maximum size of an irredundant set of such pieces, denoted by IR(Pmxn). For example, the
irredundant set of 16 kings in Image 24 is a maximum irredundant set because there exists no
irredundant set of kings on a 7x7 board that is larger. Thus, IR(K7x7) = 16. Since a minimal
dominating set is irredundant by definition, it then follows that such a set will be no larger than a
maximum irredundant set. This gives us the inequality at the far right of our chain of
inequalities: ir(Pmxn) ≤ γ(Pmxn) ≤ i(Pmxn) ≤ β(Pmxn) ≤ Γ(Pmxn) ≤ IR(Pmxn) [1, p. 205].
24. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
24
Image 24: Maximum Irredundant Set
of 16 Kings on 7x7 Chessboard
Upper irredundance formulas are known for rooks, bishops and knights. For rooks,
unlike the other five chessboard numbers we’ve studied thus far, the upper irredundance number
is not simply n. Hedetniemi, Jacobson and Wallis have proved that for n ≥ 4, IR(Rnxn) = 2n – 4.
A maximum irredundant set of 12 rooks is shown in Image 25. For bishops, Fricke has shown
that for n ≥ 6, IR(Bnxn) = 4n – 14. A maximum irredundant set of 18 bishops is shown in Image
26. Lastly, the same theorem due to Cockayne, Favaron Payan and Thomason that states that
Γ(Nnxn) = β(Nnxn) also applies to the upper irredundance number for knights. Thus, IR(Nnxn) =
β(Nnxn) as well [1, p. 206].
Image 25: Maximum Irredundant Set
of 12 Rooks on 8x8 Chessboard
25. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
25
Image 26: Maximum Irredundant Set
of 18 Bishops on 8x8 Chessboard
Much less is known about the kings and queens upper irredundance numbers. For kings,
Fricke has supplied a pair of upper and lower bounds: for n ≥ 1, (1/3)*(n – 1)2 ≤ IR(Knxn) ≤ (1/3)*
n2. The irredundant set of 16 kings displayed in Image 24 along with the upper bound (1/3)*72 =
49/3 proves that IR(K7x7) = 16, as stated above. Even less is known about the upper irredundance
numbers for queens. Only a handful of values are known, including IR(Q5x5) = 5, IR(Q6x6) = 7,
IR(Q7x7) = 9 and IR(Q8x8) = 11. It is worth noting that in the latter three cases, the upper
irredundance number is larger than the corresponding independence number (recall that β(Qnxn) =
n for all n other than 2 or 3). A maximum irredundant set of 7 queens on a 6x6 chessboard is
shown in Image 27 [1, pp. 206-207].
Image 27: Maximum Irredundant Set
of 7 Bishops on 6x6 Chessboard
26. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
Total Domination Number
26
W.W. Rouse Ball introduced the concept of total domination in 1987. The total
domination number for a given chess piece P on a given mxn chessboard, denoted γt(Pmxn), is the
minimum number of such pieces that are required to attack every square on the board, including
occupied ones [1, p. 207]. Since occupied squares must also be under attack, total domination is
a more restrictive notion than domination. By virtue of this, we have that γ(Pmxn) ≤ γt(Pmxn).
Also, γt(Pmxn) ≤ 2γ(Pmxn) since a minimum dominating set of pieces of size k will require at most
an additional k pieces to cover the k squares that are occupied by the dominating set [3, p. 2].
Ball showed the total domination number on an 8x8 chessboard to be 5 for queens, 8 for
rooks, 10 for bishops and 14 for knights [1, p. 207]. Totally dominating arrangements of these
pieces on an 8x8 board are shown in Images 28 and 29 [1, p. 212].
Image 28: Totally Dominating Sets of 5 Queens and 8 Rooks on 8x8 Chessboard
27. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
27
Image 29: Totally Dominating Sets of 10 Bishops and 14 Knights on 8x8 Chessboard
In 1995, Garnick and Nieuwejaar gave the kings total domination number for the first 12
values of n, shown in Table 3 [3, p. 2]. It is worth noting that the kings total domination number
matches the kings domination number for n = 1, 4 and 7, while the upper bound of 2γ(Knxn) is
attained for n = 6.
Table 3: Kings Total Domination
Numbers for 1 ≤ n ≤ 12
n γt(Knxn)
1 1
2 2
3 2
4 4
5 5
6 8
7 9
8 12
9 15
10 18
11 21
12 24
While the kings total domination number is unknown beyond n = 12, Garnick and
Nieuwejaar have provided upper bounds for 13 ≤ n ≤ 25, as shown in Table 4 [3, p. 2].
28. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
28
Table 4: Upper Bounds for Kings Total
Domination Numbers for 13 ≤ n ≤ 25
n
Upper Bound
for γt(Knxn)
13 29
14 33
15 38
16 43
17 48
18 54
19 60
20 68
21 72
22 80
23 87
24 95
25 102
Conclusion
One can take away from this paper that the knight’s tour is a better understood concept on
irregular surfaces than domination and independence. After all, thanks primarily to John
Watkins, the knight’s tour existence problem has been solved on all the major surfaces such as
the torus, cylinder, Klein bottle and Möbius strip. Conversely, so much is still unknown
regarding domination and independence on these surfaces.
However, as I pointed out in my previous paper on the knight’s tour, much remains to be
desired regarding the counting of the number of different knight’s tours on different size
chessboards. It would be interesting to see how many more permutations emerge, say on the
torus, as compared to the regular chessboard. It would also be good to find out the number of
different knight’s tours on irregular surfaces for chessboards of sizes that don’t work on the
regular chessboard. For example, how many closed knight’s tours are there on a 5x5 torus
(recall that closed tour knight’s tours do not exist when both dimensions are odd)? Since so little
is known regarding knight’s tour combinatorics on the general chessboard (recall that the number
of permutations of closed and open tours is known only for chessboards up to size 8x8), it didn’t
29. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
surprise me that I was unable to find any material in the literature regarding knight’s tour
combinatorics on other surfaces. Furthermore, since so many questions remain regarding magic
and semi-magic knight’s tours on the regular chessboard, it naturally follows that these same
questions hold on the torus, cylinder, Klein bottle and Möbius strip.
29
Moreover, domination and independence on irregular surfaces remain an active area of
research. Table 4 provides a snapshot of what is known and unknown regarding the domination
number formulas for the various chess pieces on the torus. While formulas have been supplied
for the rook, bishop and king on both square and rectangular chessboards, much informat ion is
lacking regarding knights and queens domination on the torus. Likewise, formulas are known
for the toroidal rooks, bishops and kings independence number but not entirely for the knights
independence number and not at all for the queens independence number. I failed to mention
this earlier in this paper, but the formula for the knights independence number on the torus is the
same as that on the regular chessboard for even n ≥ 4 (that is, βtor(Nnxn) = ½*n2). However, due
to the wrapping nature of the torus, the alternating black and white color scheme fails when n is
odd, hence the odd formula on the regular chessboard does not work on the torus. In addition, as
we have observed that very little is known about the n-queens problem on the cylinder, it goes to
show that even less is known about the n-queens problem on the torus (due to the even greater
freedom of a queen’s ability to move on the torus, thereby making independence more difficult).
Table 5: Domination Number Formulas on the Torus by Piece
Piece (P) γtor(Pnxn) (Square) γtor(Pmxn) (Rectangular)
Rook n min(m, n) [2, p. 13]
Bishop n gcd(m, n) [2, p. 13]
max{┌(m / 3)*┌ n / 3 ┐┐,
King ┌(n / 3)*┌ n / 3 ┐┐
┌(n / 3)*┌m / 3 ┐┐}
Knight
Unknown, though values
up to n = 8 are known
Unknown
Queen
Unknown, though values
up to n = 10 are known
Unknown
Beyond the torus, an active research task remains to find a formula for the queens
independence number on the cylinder. Further insight into the n-queens problem on the cylinder
might aid with uncovering new information about the n-queens problem on the torus, Klein
30. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
bottle and Möbius strip. While ultimately arriving at a formula for the queens independence
number on any of these surfaces would appear to be a tall task, once a formula for one of the
surfaces is discovered, finding formulas for the other surfaces should be markedly easier. Also, I
have found that perhaps not enough attention has been paid to domination and independence in
general on the Möbius strip. This fascinating surface should definitely be a major focus of future
research in mathematical chessboard problems.
30
Other areas that would benefit from further research are the variants to domination and
independence that we studied in the latter part of this paper. Table 5 provides a summary of the
known and unknown formulas for the irredundance number, domination number, independent
domination number, independence number, upper domination number and upper irredundance
number for each of the five chess pieces that we’ve been studying. This table not only illustrates
that all six formulas are known for the rook and bishop, but also the relatively simplistic nature
of these two pieces, as many of the formulas are simply n (five for the rook and three for the
bishop). Things start to get a little murky when one considers the king, for which only three of
the six formulas are known. The same holds true for the knight, but one glaring hole here that
isn’t present with the king is that arguably the most fundamental of the formulas, the domination
number, remains unknown. Without question – and this comes as no surprise – the queen
remains the biggest mystery, as only the independence number has a known formula. Greater
computing power and further analysis of more general mxn rectangular boards will yield new
insights into the mathematics associated with the king, knight and queen.
31. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
MAT 9000 Graduate Math Seminar
31
Table 6: Chessboard Number Formulas by Piece
Piece (P) ir(Pnxn) γ(Pnxn) i(Pnxn) β(Pnxn) Γ(Pnxn) IR(Pnxn)
Rook n n n n n
2n – 4
for n ≥ 4
Bishop n n n 2n – 2 2n – 2
4n – 14
for n ≥ 6
King Unknown └(n + 2) / 3┘
2 └(n + 2) / 3┘
2
└½*(n + 1)┘
2
Unknown Unknown
Knight Unknown Unknown Unknown
4 if n = 2;
½*n2 if n ≥ 4,
n even;
½*(n2 + 1) if
n odd
4 if n = 2;
½*n2 if n ≥ 4,
n even;
½*(n2 + 1) if
n odd
4 if n = 2;
½*n2 if n ≥ 4,
n even;
½*(n2 + 1) if
n odd
Queen Unknown Unknown Unknown
1 if n = 2;
2 if n = 3;
n for all
other n
Unknown Unknown
All in all, I hope that this series of four papers on the mathematics of chessboard
problems has enabled the reader to appreciate just how fun and fascinating this topic really is.
These papers have covered a lot of ground in examining seven different chessboard numbers (the
six in Table 5 above plus the total domination number), four different surfaces (the torus,
cylinder, Klein bottle and Möbius strip) in addition to the regular chessboard, and the knight’s
tour problem and its relation with other mathematical concepts such as magic squares and Latin
squares. As these are only a small fraction of all the problems that have been studied and that
one could potentially study on the chessboard, needless to say, the possibilities for further
research and analysis in this area are practically endless. Fortunately, recreational
mathematicians won’t have to worry about running out of work to do for quite some time.
32. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations
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. DeMaio, W.P. Faust. Domination on the mxn Toroidal Chessboard by Rooks and Bishops.
Department of Mathematics and Statistics, Kennesaw State University.
[3] J. DeMaio, A. Lightcap. King's Total Domination Number on the Square of Side n.
Department of Mathematics and Statistics, Kennesaw State University.
[4] “Chess.” Wikipedia, Wikimedia Foundation. http://en.wikipedia.org/wiki/Chess
[5] “Joint International Meeting UMI – DMV, Perugia, 18-22 June 2007.” Dipartimento di
Matematica e Informatica. http://www.dmi.unipg.it/JointMeetingUMI-DMV/events-ughi.htm
[6] “Klein bottle.” Wikipedia, Wikimedia Foundation. http://en.wikipedia.org/wiki/Klein_bottle
[7] "Imaging Maths - Inside the Klein Bottle." Plus Magazine.
http://plus.maths.org/content/os/issue26/features/mathart/WhiteBlue
[8] “Möbius strip.” Wikipedia, Wikimedia Foundation.
http://en.wikipedia.org/wiki/M%C3%B6bius_strip
32