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Taking Sudoku Seriously
''Taking Sudoku Seriously: The math behind the world's most popular pencil puzzle'' is a book on the mathematics of Sudoku. It was written by Jason Rosenhouse and Laura Taalman, and published in 2011 by the Oxford University Press. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries. It was the 2012 winner of the PROSE Awards in the popular science and popular mathematics category. Topics The book is centered around Sudoku puzzles, using them as a jumping-off point "to discuss a broad spectrum of topics in mathematics". In many cases these topics are presented through simplified examples which can be understood by hand calculation before extending them to Sudoku itself using computers. The book also includes discussions on the nature of mathematics and the use of computers in mathematics. After an introductory chapter on Sudoku and its deductive puzzle-solving techniques (also touching ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Sudoku
The mathematics of Sudoku refers to the use of mathematics to study Sudoku puzzles to answer questions such as ''"How many filled Sudoku grids are there?"'', "''What is the minimal number of clues in a valid puzzle?''" and ''"In what ways can Sudoku grids be symmetric?"'' through the use of combinatorics and group theory. The analysis of Sudoku falls is generally divided between analyzing the properties of unsolved puzzles (such as the minimum possible number of given clues) and analyzing the properties of solved puzzles. Initial analysis was largely focused on enumerating solutions, with results first appearing in 2004. For classical Sudoku, the number of filled grids is 6,670,903,752,021,072,936,960 (), which reduces to 5,472,730,538 essentially different solutions under the validity preserving transformations. There are 26 types of symmetry, but they can only be found in about 0.005% of all filled grids. A puzzle with a unique solution must have at least 17 clues, and there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Enumeration
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets ''S''''i'' indexed by the natural numbers, enumerative combinatorics seeks to describe a ''counting function'' which counts the number of objects in ''S''''n'' for each ''n''. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. The twelvefold way provides a unified framework for counting permutations, combinations and partitions. The simplest such functions are ''closed formulas'', which can be expressed as a composition of elementary functions such as factorials, powers, and so on. For instance, as shown below, the number of different possible orderings of a deck of ''n'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Bevan (mathematician)
David Bevan is an English mathematician, computer scientist and software developer. He is known for Bevan's theorem, which gives the asymptotic enumeration of grid classes of permutations and for his work on enumerating the class of permutations avoiding the pattern 1324. He is also known for devising weighted reference counting, an approach to computer memory management that is suitable for use in distributed systems. Work and research Bevan is a lecturer in combinatorics in the department of Mathematics and Statistics at the University of Strathclyde. He has degrees in mathematics and computer science from the University of Oxford and a degree in theology from the London School of Theology. He received his PhD in mathematics from The Open University in 2015; his thesis, ''On the growth of permutation classes'', was supervised by Robert Brignall. In 1987, as a research scientist at GEC's Hirst Research Centre in Wembley, he developed an approach to computer memory m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Keith Devlin
Keith J. Devlin (born 16 March 1947) is a British mathematician and popular science writer. Since 1987 he has lived in the United States. He has dual British-American citizenship.Curriculum vitae Profkeithdevlin.com, accessed 3 February 2014. Biography He was born and grew up in England, in . There he attended a local primary school followed by Greatfield High School in Hull. In the last school year he was appointed head boy. Devlin earned a BSc (special) in mathem ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recreational Mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited to being an endeavor for amateurs, many topics in this field require no knowledge of advanced mathematics. Recreational mathematics involves mathematical puzzles and games, often appealing to children and untrained adults, inspiring their further study of the subject. The Mathematical Association of America (MAA) includes recreational mathematics as one of its seventeen Special Interest Groups, commenting: Mathematical competitions (such as those sponsored by mathematical associations) are also categorized under recreational mathematics. Topics Some of the more well-known topics in recreational mathematics are Rubik's Cubes, magic squares, fractals, logic puzzles and mathematical chess problems, but this area of mathematics incl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extremal Combinatorics
Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics. Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc.) can be, if it has to satisfy certain restrictions. Much of extremal combinatorics concerns classes of sets; this is called extremal set theory. For instance, in an ''n''-element set, what is the largest number of ''k''-element subsets that can pairwise intersect one another? What is the largest number of subsets of which none contains any other? The latter question is answered by Sperner's theorem, which gave rise to much of extremal set theory. Another kind of example: How many people can be invited to a party where among each three people there are two who know each other and two who don't know each other? Ramsey theory shows that at most five persons can attend such a party. Or, suppose we are given a finite set of nonzero integers, and are asked to mark as la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gröbner Basis
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps. Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems. Gröbner bases were introduced in 1965, together with an algorithm to compute them (Buchberger's algorithm), by Bruno Buchberger in his Ph.D. thesis. He named them after h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sudoku Graph
In the mathematics of Sudoku, the Sudoku graph is an undirected graph whose vertices represent the cells of a (blank) Sudoku puzzle and whose edges represent pairs of cells that belong to the same row, column, or block of the puzzle. The problem of solving a Sudoku puzzle can be represented as precoloring extension on this graph. It is an integral Cayley graph. Basic properties and examples On a Sudoku board of size n^2\times n^2, the Sudoku graph has n^4 vertices, each with exactly 3n^2-2n-1 neighbors. Therefore, it is a regular graph. The total number of edges is n^4(3n^2-2n-1)/2. For instance, the graph shown in the figure above, for a 4\times 4 board, has 16 vertices and 56 edges, and is 7-regular. For the most common form of Sudoku, on a 9\times 9 board, the Sudoku graph is a 20-regular graph with 81 vertices and 810 edges. The second figure shows how to count the neighbors of each cell in a 9\times 9 board. Puzzle solutions and graph coloring Each row, column, or block ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Precoloring Extension
In graph theory, precoloring extension is the problem of extending a graph coloring of a subset of the vertices of a graph, with a given set of colors, to a coloring of the whole graph that does not assign the same color to any two adjacent vertices. Complexity Precoloring extension has the usual graph coloring problem as a special case, in which the initially colored subset of vertices is empty; therefore, it is NP-complete. However, it is also NP-complete for some other classes of graphs on which the usual graph coloring problem is easier. For instance it is NP-complete on the rook's graphs, for which it corresponds to the problem of completing a partially filled-in Latin square. The problem may be solved in polynomial time for graphs of bounded treewidth, but the exponent of the polynomial depends on the treewidth. It may be solved in linear time for precoloring extension instances in which both the number of colors and the treewidth are bounded. Related problems Precoloring ext ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Search
{{no footnotes, date=January 2013 In computer science and artificial intelligence, combinatorial search studies search algorithms for solving instances of problems that are believed to be hard in general, by efficiently exploring the usually large solution space of these instances. Combinatorial search algorithms achieve this efficiency by reducing the effective size of the search space or employing heuristics. Some algorithms are guaranteed to find the optimal solution, while others may only return the best solution found in the part of the state space that was explored. Classic combinatorial search problems include solving the eight queens puzzle or evaluating moves in games with a large game tree, such as reversi or chess. A study of computational complexity theory helps to motivate combinatorial search. Combinatorial search algorithms are typically concerned with problems that are NP-hard. Such problems are not believed to be efficiently solvable in general. However, the vari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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