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 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 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.
This document discusses linear independence, basis, and dimension in linear algebra. It defines linear independence as vectors being linearly independent if the only solution that produces the zero vector is the trivial solution with all coefficients equal to zero. A basis is defined as a set of linearly independent vectors that span the vector space. The dimension of a vector space is the number of vectors in any basis of that space. The dimensions of the four fundamental subspaces (row space, column space, nullspace, and left nullspace) of a matrix are defined in terms of the rank of the matrix.
How to Create a Dungeons and Dragons Characteramychamy
This document provides instructions for creating a Dungeons & Dragons character. It outlines the steps to choose a race, class, background, ability scores, proficiencies, equipment, and other character details. Players fill out character sheets by recording race and class abilities, skill proficiencies, saving throw bonuses, equipment, and other numerical stats like hit points and armor class. The document explains how ability scores, modifiers, attack bonuses, damage, and other numbers are determined based on class, race, skills and equipment.
This document discusses sequences and their properties. It defines a sequence as a list of numbers written in a definite order. The nth term of a sequence is denoted as an. It provides examples of describing sequences using notation, defining formulas, and listing terms. It defines convergent and divergent sequences and gives examples testing for convergence or divergence. It also discusses bounded sequences and decreasing sequences, giving examples and proofs.
The document describes m-way search trees, B-trees, heaps, and their related operations. An m-way search tree is a tree where each node has at most m child nodes and keys are arranged in ascending order. B-trees are similar but ensure the number of child nodes falls in a range and all leaf nodes are at the same depth. Common operations like searching, insertion, and deletion are explained for each with examples. Heaps store data in a complete binary tree structure where a node's value is greater than its children's values.
This document provides an overview of sequences and series in algebra 2. It defines a sequence as an ordered list of numbers that can be arithmetic, geometric, or neither. A series is the sum of the terms in a sequence. The document explains how to determine the nth term of arithmetic and geometric sequences using their respective formulas. It also discusses how to identify if a sequence is arithmetic, geometric, or neither based on having a common difference or ratio. Examples of sequences in real world contexts like loan interest and training distances are provided.
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 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.
This document discusses linear independence, basis, and dimension in linear algebra. It defines linear independence as vectors being linearly independent if the only solution that produces the zero vector is the trivial solution with all coefficients equal to zero. A basis is defined as a set of linearly independent vectors that span the vector space. The dimension of a vector space is the number of vectors in any basis of that space. The dimensions of the four fundamental subspaces (row space, column space, nullspace, and left nullspace) of a matrix are defined in terms of the rank of the matrix.
How to Create a Dungeons and Dragons Characteramychamy
This document provides instructions for creating a Dungeons & Dragons character. It outlines the steps to choose a race, class, background, ability scores, proficiencies, equipment, and other character details. Players fill out character sheets by recording race and class abilities, skill proficiencies, saving throw bonuses, equipment, and other numerical stats like hit points and armor class. The document explains how ability scores, modifiers, attack bonuses, damage, and other numbers are determined based on class, race, skills and equipment.
This document discusses sequences and their properties. It defines a sequence as a list of numbers written in a definite order. The nth term of a sequence is denoted as an. It provides examples of describing sequences using notation, defining formulas, and listing terms. It defines convergent and divergent sequences and gives examples testing for convergence or divergence. It also discusses bounded sequences and decreasing sequences, giving examples and proofs.
The document describes m-way search trees, B-trees, heaps, and their related operations. An m-way search tree is a tree where each node has at most m child nodes and keys are arranged in ascending order. B-trees are similar but ensure the number of child nodes falls in a range and all leaf nodes are at the same depth. Common operations like searching, insertion, and deletion are explained for each with examples. Heaps store data in a complete binary tree structure where a node's value is greater than its children's values.
This document provides an overview of sequences and series in algebra 2. It defines a sequence as an ordered list of numbers that can be arithmetic, geometric, or neither. A series is the sum of the terms in a sequence. The document explains how to determine the nth term of arithmetic and geometric sequences using their respective formulas. It also discusses how to identify if a sequence is arithmetic, geometric, or neither based on having a common difference or ratio. Examples of sequences in real world contexts like loan interest and training distances are provided.
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.
Gauss Jorden and Gauss Elimination method.pptxAHSANMEHBOOB12
Gauss elimination and Gauss-Jordan elimination are methods for solving systems of linear equations. Gauss elimination puts the matrix of coefficients into row-echelon form using elementary row operations, while Gauss-Jordan elimination further reduces the matrix to reduced row-echelon form. These methods can also be used to find the rank of a matrix, calculate the determinant, and compute the inverse of an invertible square matrix. Examples demonstrate applying the methods to solve systems of 3 equations with 3 unknowns.
The document provides examples of finding the general term of different sequences. For each sequence, it identifies the pattern between the terms and determines the formula for the nth term. The general terms provided are:
1) f(n) = 2n for the sequence 2, 4, 6, 8,...
2) f(n) = 2n - 1 for the sequence 1, 3, 5, 7,...
3) f(n) = 2n for the sequence 2, 4, 8, 16,...
4) f(n) = n^2 for the sequence 1, 4, 9, 16,...
It also provides guidelines for finding the general term such as looking for a common difference,
El documento proporciona varias sugerencias para proteger los recursos de un jugador de los granjeros automáticos y otros jugadores, incluyendo usar desventajas en el Salón del Trono, protección de ascensión, arcanos, comodines protegidos y puertos automáticos. Recomienda configurar el Salón del Trono y Watchtower para la defensa, y mover recursos a través de comodines o puertos automáticos cuando no se pueda jugar activamente.
Double integration in polar form with change in variable (harsh gupta)Harsh Gupta
This document outlines how to perform double integration in polar coordinates when changing variables. It introduces the concept of double integration over a region R in polar coordinates, defined by angles α and β and curves r=f1(θ) and r=f2(θ). It provides examples of transforming Cartesian to polar coordinates when integrating, as well as real-world applications like calculating mass flow into an F1 car airbox using velocity profiles and double integrals. Finally, it works through an example problem of finding the volume of a region under a sphere, above a plane, and inside a cylinder using double integration in polar coordinates.
The document discusses the bonuses provided by the Redoubt Tower and Arcane Temple buildings in the game. The Redoubt Tower has shorter range but higher attack, life, defense, and accuracy than the Defensive Tower. It is more resilient against archer units. The Arcane Temple and alliance bonuses can further increase the tower stats up to a 25% boost. Throne room items can also enhance tower stats, applying to both the tower and redoubt.
The document discusses solving systems of 3 linear equations with 3 variables. It provides steps to set up the equations in standard form, eliminate one variable using two equations, eliminate the same variable from another pair of equations to get a system of 2 equations with 2 variables, solve this system to find the values of two variables, substitute these values into the original third equation to solve for the remaining variable, and check the solution. An example problem demonstrates applying these steps to solve for x, y, and z. The summary notes that the solution may not be unique depending on whether eliminating variables results in a true or false statement.
The document discusses the Fundamental Theorem of Calculus (FTC), which links differentiation and integration. It defines the FTC in two parts: the first part states that if a function f is continuous on an interval [a,b] and F is defined by integrating f, then F is uniformly continuous and differentiable. The second part states that if f is differentiable on [a,b] and its derivative f' is integrable, then the integral of f' from a to b is equal to f(b)-f(a). Proofs of both parts of the FTC are provided.
The document discusses different types of algebraic structures including semigroups, monoids, groups, and abelian groups. It defines each structure based on what axioms they satisfy such as closure, associativity, identity element, and inverses. Examples are given of sets that satisfy each structure under different binary operations like addition, multiplication, subtraction and division. The properties of algebraic structures like commutativity, associativity, identity, inverses and cancellation laws are also explained.
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.
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 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.
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.
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.
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.
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.
Gauss Jorden and Gauss Elimination method.pptxAHSANMEHBOOB12
Gauss elimination and Gauss-Jordan elimination are methods for solving systems of linear equations. Gauss elimination puts the matrix of coefficients into row-echelon form using elementary row operations, while Gauss-Jordan elimination further reduces the matrix to reduced row-echelon form. These methods can also be used to find the rank of a matrix, calculate the determinant, and compute the inverse of an invertible square matrix. Examples demonstrate applying the methods to solve systems of 3 equations with 3 unknowns.
The document provides examples of finding the general term of different sequences. For each sequence, it identifies the pattern between the terms and determines the formula for the nth term. The general terms provided are:
1) f(n) = 2n for the sequence 2, 4, 6, 8,...
2) f(n) = 2n - 1 for the sequence 1, 3, 5, 7,...
3) f(n) = 2n for the sequence 2, 4, 8, 16,...
4) f(n) = n^2 for the sequence 1, 4, 9, 16,...
It also provides guidelines for finding the general term such as looking for a common difference,
El documento proporciona varias sugerencias para proteger los recursos de un jugador de los granjeros automáticos y otros jugadores, incluyendo usar desventajas en el Salón del Trono, protección de ascensión, arcanos, comodines protegidos y puertos automáticos. Recomienda configurar el Salón del Trono y Watchtower para la defensa, y mover recursos a través de comodines o puertos automáticos cuando no se pueda jugar activamente.
Double integration in polar form with change in variable (harsh gupta)Harsh Gupta
This document outlines how to perform double integration in polar coordinates when changing variables. It introduces the concept of double integration over a region R in polar coordinates, defined by angles α and β and curves r=f1(θ) and r=f2(θ). It provides examples of transforming Cartesian to polar coordinates when integrating, as well as real-world applications like calculating mass flow into an F1 car airbox using velocity profiles and double integrals. Finally, it works through an example problem of finding the volume of a region under a sphere, above a plane, and inside a cylinder using double integration in polar coordinates.
The document discusses the bonuses provided by the Redoubt Tower and Arcane Temple buildings in the game. The Redoubt Tower has shorter range but higher attack, life, defense, and accuracy than the Defensive Tower. It is more resilient against archer units. The Arcane Temple and alliance bonuses can further increase the tower stats up to a 25% boost. Throne room items can also enhance tower stats, applying to both the tower and redoubt.
The document discusses solving systems of 3 linear equations with 3 variables. It provides steps to set up the equations in standard form, eliminate one variable using two equations, eliminate the same variable from another pair of equations to get a system of 2 equations with 2 variables, solve this system to find the values of two variables, substitute these values into the original third equation to solve for the remaining variable, and check the solution. An example problem demonstrates applying these steps to solve for x, y, and z. The summary notes that the solution may not be unique depending on whether eliminating variables results in a true or false statement.
The document discusses the Fundamental Theorem of Calculus (FTC), which links differentiation and integration. It defines the FTC in two parts: the first part states that if a function f is continuous on an interval [a,b] and F is defined by integrating f, then F is uniformly continuous and differentiable. The second part states that if f is differentiable on [a,b] and its derivative f' is integrable, then the integral of f' from a to b is equal to f(b)-f(a). Proofs of both parts of the FTC are provided.
The document discusses different types of algebraic structures including semigroups, monoids, groups, and abelian groups. It defines each structure based on what axioms they satisfy such as closure, associativity, identity element, and inverses. Examples are given of sets that satisfy each structure under different binary operations like addition, multiplication, subtraction and division. The properties of algebraic structures like commutativity, associativity, identity, inverses and cancellation laws are also explained.
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.
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 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.
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.
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.
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.
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.
This document describes a chess variant called Hex Chess that is played on a board with an hexagonal shape containing 96 equilateral triangles. It summarizes the movements of the different pieces, which generally follow standard chess movements but adapted to the triangular geometry. It also discusses the development of distance formulas specific to this variant to calculate distances traveled by kings, queens, and potentially other pieces. Finally, it outlines some open problems for further developing and analyzing the geometry and rules of Hex Chess.
This document presents an overview of the N-Queen problem and its solution using backtracking. It discusses how the N-Queen problem was originally proposed as a chess puzzle in 1848 and involved placing N queens on an N×N chessboard so that no two queens attack each other. It then explains how backtracking can be used to systematically place queens on the board one by one and remove placements that result in conflicts until all queens are placed or no more placements are possible. Examples are given showing the backtracking process and solution trees for 4x4 boards. The time complexity of this backtracking solution is analyzed to be O(N!).
Analysis & Design of Algorithms
Backtracking
N-Queens Problem
Hamiltonian circuit
Graph coloring
A presentation on unit Backtracking from the ADA subject of Engineering.
A presentation about Infinite Chess and the difference between man and machines. From works by C.D.A. Evans and J.D. Hamkins. Presented during the International Interdisciplinary Seminar of London, January 2018.
The document discusses the 8 queens problem and how backtracking can be used to solve it. The 8 queens problem aims to place 8 queens on a chessboard so that no two queens attack each other. Backtracking is an algorithm that builds candidate solutions incrementally and abandons partial solutions ("backtracks") that cannot be completed. It explains that backtracking works by placing queens in columns, removing placements that lead to conflicts, and backtracking to try other placements. The document also provides the number of solutions for placing different numbers of queens on boards of corresponding sizes.
Backtracking is an algorithmic technique for solving problems recursively by trying to build candidates for the solutions incrementally, and abandoning each partial candidate ("backtracking") as soon as it is determined that the candidate cannot possibly be completed to a valid solution. It is useful for constraint satisfaction problems and involves systematically trying choices until finding one that "works". The eight queens problem, which involves placing eight queens on a chessboard so that none attack each other, is a classic example solved using backtracking.
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.
The document discusses the history of mathematics and various patterns in numbers such as magic squares, magic stars, triangular numbers, Fibonacci numbers, and Pascal's triangle. It provides examples and properties of each type of pattern. For instance, it explains that a magic square of order 8 forms another magic square if columns are rearranged and that triangular numbers represent the number of dots that can form an equilateral triangle.
The document describes the board and pieces used in chess. It explains that the board is made up of 64 squares arranged in 8 rows and 8 files. Each square has a unique name combining its file letter and rank number. It then details the standard starting position of all pieces, allowed piece movements, and the goal of checkmating the opponent's king to win the game.
The document provides an overview of the basic rules of chess, including:
- How to set up the board with the white square in the lower right corner and ranks and files labeled from a-h and 1-8.
- How the different pieces (pawns, knights, bishops, rooks, queen, king) move and their relative values. Pawns can move 1 or 2 squares forward on their first move.
- How pieces are captured by moving to their space, with pawns capturing diagonally. The goal is checkmate, where the opponent's king cannot escape being captured.
- Special moves include castling to move the king and rook, and capturing en passant to immediately capture a
This presentation summarizes Karnaugh maps, which are a graphical technique for simplifying Boolean expressions. Karnaugh maps arrange the terms of a truth table in a two-dimensional grid, making common factors between terms visible. They can be used for functions with up to five variables. Examples show how to identify groupings of terms and simplify expressions using Karnaugh maps for two, three, and four variables. The presentation concludes with an example of a five variable Karnaugh map.
This document provides information about the rules and movement of each chess piece: pawns, rooks, queens, knights, and kings. It explains how each piece can move and includes practice activities to help learn the movement. The summary also notes the relative value of each piece from most to least valuable: queen, rook, bishop, knight, then pawn.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
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- 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.
Similar to Chessboard Puzzles Part 1 - Domination (18)
1. Chessboard Puzzles
Part 1: Domination
Dan Freeman
February 20, 2014
Villanova University
MAT 9000 Graduate Math Seminar
2. Brief Overview of Chess
• Chess is a classic board game that has
been played for at least 1,200 years
• Chess is a 2-player turn-based game
played on an 8x8 board
• There are six different types of pieces:
pawn, knight, bishop, rook, queen and king
• The objective of the game is to put the
opponent’s king in a position in which it
cannot escape attack; this position is
known as checkmate
4. Domination Defined
• A dominating set of chess pieces is one
such that every square on the 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” or “cover” the board
• Domination numbers are denoted by
γ(Pmxn) where P represents the type of
chess piece (see legend to the right) and m
and n are the number of rows and columns
of the board, respectively
Domination
Number
Notation
King – K
Queen – Q
Rook – R
Bishop – B
Knight – N
5. Rook Movement
• Rooks move horizontally and vertically
• Rooks 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 the example below, 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
6. Rooks Domination
• Domination among rooks is the simplest of
all chess pieces
• For a square nxn chessboard, the rooks
domination number is simply n
• For a general rectangular mxn chessboard,
γ(Rmxn) = min(m, n)
7. Proof that γ(Rnxn) = n
• Two Russian brothers Akiva and Isaak
Yaglom proved this:
– 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.
Thus, γ(Rnxn) ≥ n.
– Second, if n rooks are placed along a single row
or down a single column, the entire board is
clearly dominated. That is, γ(Rnxn) ≤ n.
– Lastly, since γ(Rnxn) ≥ n and γ(Rnxn) ≤ n, we
conclude that γ(Rnxn) = n.
• The fact that γ(Rmxn) = min(m, n) follows
immediately from the above
8. Bishop Movement
• Bishops move diagonally
• Bishops 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 the example below, 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
9. Bishops Domination
• As with rooks, γ(Bnxn) = n (though in
general, γ(Bmxn) ≠ min(m, n))
• The proof that γ(Bnxn) = n for bishops is
more involved than that for rooks
• The proof starts with rotating the
chessboard 45 degrees, as shown below
45°
5x4 Black Square
10. Proof that γ(Bnxn) = n
• Yaglom and Yaglom proved this:
– Suppose n = 8 (the following argument works for
all even n).
– Clearly, from rotating the board 45 degrees, we
see a 5x4 construction of dark (black) squares in
the middle of the board. Therefore, at least 4
bishops are needed to cover all of the black
squares. By symmetry, at least 4 bishops are
needed to cover all of the light (white) squares.
Therefore, γ(B8x8) ≥ 4 + 4 = 8.
– On the other hand, if we place 8 bishops in the
fourth column of a chessboard, we find that the
entire board is covered. Thus, γ(B8x8) ≤ 8.
– Since γ(B8x8) ≥ 8 and γ(B8x8) ≤ 8, it follows that
γ(B8x8) = 8 and for general even n, γ(Bnxn) = n.
Bishops Domination
on 8x8 Board
11. Proof that γ(Bnxn) = n
– Now suppose n is odd and let n = 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; 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 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, the entire
board is covered.
– In conclusion, γ(Bnxn) = for all n.
12. King Movement
• 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 the example below, the king can move to
any of the squares with a white circle
13. Kings Domination
• Shown are examples of
9 kings dominating 7x7,
8x8 and 9x9 boards
• For 7 ≤ n ≤ 9,
γ(Knxn) = 9
14. Kings Domination
• No matter where one
places a king on any of
the 7x7, 8x8 or 9x9
boards, only one of the
nine dark orange
squares will be covered
15. Kings Domination
• Thus, 32 = 9 is the domination number for
square chessboards where n = 7, 8 or 9
• This triplet pattern continues for larger
boards:
– For n = 10, 11 and 12, γ(Knxn) = 42 = 16.
– For n = 13, 14 and 15, γ(Knxn) = 52 = 25.
• Thus, the general formula for the kings
domination number can be written making
use of the greatest integer or floor function:
– γ(K) = └(n + 2) / 3┘
2.
nxn• Generalizing even further, the formula for
rectangular boards is:
– γ(Kmxn) = └(m + 2) / 3┘* └(n + 2) / 3┘.
16. Knight Movement
• Knights move two squares in one direction
(either horizontally or vertically) and one
square in the other direction as long as they
do not take the place of a friendly piece
• Knights’ moves resemble an L shape
• Knights are the only pieces that are allowed
to jump over other pieces
• In the example below, the white and black
knights can move to squares with circles of
the corresponding color
17. Knights Domination
• 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 the table to the
right.
• As can be seen from the table,
as n increases, γ(Nnxn) increases
in no discernible pattern
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
19. Queen Movement
• Queens move horizontally, vertically and
diagonally
• Queens 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 the example below, the queen can move
to any of the squares with a black circle
20. Queens Domination
• Domination among queens is the most
complicated and interesting of all chess
pieces, as well as the least understood
• No formula is known for the queens
domination number, but lower and upper
bounds have been established
Arrangement of
5 Queens
Dominating
8x8 Board
21. Queens Diagonal Domination
• However, a formula for the queens diagonal
domination number is known
• The queens diagonal domination number,
denoted diag(Qnxn), is the minimum number
of queens all placed along the main
diagonal required to cover the board
• For all n, diag(Qnxn) = n – max(|mid-point
free, all even or all odd, subset of
{1, 2, 3, …, n}|)
• A mid-point free set is a set in which for any
given pair of elements in the set, the
midpoint of those two numbers is not in the
set
22. Upper and Lower Bounds
• Upper bound for queens domination
number: Welch proved that for n = 3m + r,
0 ≤ r < 3, γ(Qnxn) ≤ 2m + r
• Lower bound for queens domination
number: Spencer proved that for any n,
γ(Qnxn) ≥ ½*(n – 1)
• Weakley improved on Spencer’s lower
bound by showing that if the lower bound is
attained, that is, if γ(Qnxn) = ½*(n – 1), then
n ≡ 3 mod 4
• Corollaries of Spencer’s improved lower
bound include:
– γ(Q7x7) = 4.
– For n = 4k + 1, γ(Qnxn) ≥ ½*(n + 1) = 2k + 1.
23. Queens Domination Numbers
• The lower bounds on the previous slide
allow us to narrow the number of
possibilities considerably for the queens
domination number for larger chess
boards, as shown in the table below
n γ(Qnxn)
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
24. Sources Cited
• J.J. Watkins. Across the Board: The
Mathematics of Chessboard Problems.
Princeton, New Jersey: Princeton
University Press, 2004.
• J. Nunn. Learn Chess. London, England:
Gambit Publications, 2000.
• “Chess.” Wikipedia, Wikimedia Foundation.
http://en.wikipedia.org/wiki/Chess
• “A006075 – OEIS.” http://oeis.org/A006075