An illustrated introduction to Caïssan squares: the magic of chess

We study various properties of n × n Caïssan magic squares. Following the seminal 1881 article by "Ursus" [Henry James Kesson (b. c. 1844)] in The Queen, we define a magic square to be Caïssan whenever it is pandiagonal and knight-Nasik so that all paths of length n by a chess bishop are magic (pandiagonal, Nasik, CSP1-magic) and by a (regular) chess knight are magic (CSP2-magic). We also study Caïssan beauties, which are pandiagonal and both CSP2- and CSP3-magic; a CSP3-path is by a special knight that leaps over 3 instead of 2 squares. Our paper ends with a bibliography of over 100 items (many with hyperlinks) listed chronologically from the 14th century onwards. We give special attention to items by (or connected with) "Ursus": Henry James Kesson (b. c. 1844), Andrew Hollingworth Frost (1819–1907), Charles Planck (1856–1935), and Pavle Bidev (1912–1988). We have tried to illustrate our findings as much as possible, and whenever feasible, with images of postage stamps or other philatelic items

Symmetry in constraint programming

Constraint programming is an invaluable tool for solving many of the complex NP-complete problems that we need solutions to. These problems can be easily described as Constraint Satisfaction Problems (CSPs) and then passed to constraint solvers: complex pieces of software written to solve general CSPs efficiently. Many of the problems we need solutions to are real world problems: planning (e.g. vehicle routing), scheduling (e.g. job shop schedules) and timetabling problems (e.g. staff rotas) to name but a few. In the real world, we place structure on objects to make them easier to deal with. This manifests itself as symmetry. The symmetry in these real world problems make them easier to deal with for humans. However, they lead to a great deal of redundancy when using computational methods of problem solving. Thus, this thesis examines some of the many aspects of utilising the symmetry of CSPs to reduce the amount of computation needed by constraint solvers. In this thesis we look at the ease of use of previous symmetry breaking methods. We introduce a new and novel method of describing the symmetries of CSPs. We look at previous methods of symmetry breaking and show how we can drastically reduce their computation while still breaking all symmetry. We give the first detailed investigation into the behaviour of breaking only subsets of all symmetry. We look at how this affects the performance of constraint solvers before discovering the properties of a good symmetry. We then present an original method for choosing the best symmetries to use. Finally, we look at areas of redundant computation in constraint solvers that no other research has examined. New ways of dealing with this redundancy are proposed with results of an example implementation which improves efficiency by several orders of magnitude

Products and Services

Today’s global economy offers more opportunities, but is also more complex and competitive than ever before. This fact leads to a wide range of research activity in different fields of interest, especially in the so-called high-tech sectors. This book is a result of widespread research and development activity from many researchers worldwide, covering the aspects of development activities in general, as well as various aspects of the practical application of knowledge