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Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Used for calculation, it is considered the most important subject. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. (Full article...)

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The knight's tour is a mathematical chess problem in which the piece called the knight is to visit each square on an otherwise empty chess board exactly once, using only legal moves. It is a special case of the more general Hamiltonian path problem in graph theory. (A closely related non-Hamiltonian problem is that of the longest uncrossed knight's path.) The tour is called closed if the knight ends on a square from which it may legally move to its starting square (thereby forming an endless cycle), and open if not. The tour shown in this animation is open (see also a static image of the completed tour). On a standard 8 × 8 board there are 26,534,728,821,064 possible closed tours and 39,183,656,341,959,808 open tours (counting separately any tours that are equivalent by rotation, reflection, or reversing the direction of travel). Although the earliest known solutions to the knight's tour problem date back to the 9th century CE, the first general procedure for completing the knight's tour was Warnsdorff's rule, first described in 1823. The knight's tour was one of many chess puzzles solved by The Turk, a fake chess-playing machine exhibited as an automaton from 1770 to 1854, and exposed in the early 1820s as an elaborate hoax. True chess-playing automatons (i.e., computer programs) appeared in the 1950s, and by 1988 had become sufficiently advanced to win a match against a grandmaster; in 1997, Deep Blue famously became the first computer system to defeat a reigning world champion (Garry Kasparov) in a match under standard tournament time controls. Despite these advances, there is still debate as to whether chess will ever be "solved" as a computer problem (meaning an algorithm will be developed that can never lose a chess match). According to Zermelo's theorem, such an algorithm does exist.

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Knot theory is the branch of topology that studies mathematical knots, which are defined as embeddings of a circle S1 in 3-dimensional Euclidean space, R3. This is basically equivalent to a conventional knotted string with the ends of the string joined together to prevent it from becoming undone. Two mathematical knots are considered equivalent if one can be transformed into the other via continuous deformations (known as ambient isotopies); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knots can be described in various ways, but the most common method is by planar diagrams (known as knot projections or knot diagrams). Given a method of description, a knot will have many descriptions, e.g., many diagrams, representing it. A fundamental problem in knot theory is determining when two descriptions represent the same knot. One way of distinguishing knots is by using a knot invariant, a "quantity" which remains the same even with different descriptions of a knot.

Research in knot theory began with the creation of knot tables and the systematic tabulation of knots. While tabulation remains an important task, today's researchers have a wide variety of backgrounds and goals. Classical knot theory, as initiated by Max Dehn, J. W. Alexander, and others, is primarily concerned with the knot group and invariants from homology theory such as the Alexander polynomial.

The discovery of the Jones polynomial by Vaughan Jones in 1984, and subsequent contributions from Edward Witten, Maxim Kontsevich, and others, revealed deep connections between knot theory and mathematical methods in statistical mechanics and quantum field theory. A plethora of knot invariants have been invented since then, utilizing sophisticated tools as quantum groups and Floer homology. (Full article...)

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