NUMBER GAME

any of various puzzles and games that involve aspects of mathematics. The mathematical games that survived from Greco-Roman antiquity and aroused renewed interest in the Middle Ages were primarily of two kinds: those involving the manipulation of objects and those requiring computation. The first books devoted solely to recreational problems in mathematics appeared in the 17th century, and the volume and variety of mathematical games and puzzles greatly expanded in the following two centuries. During this period interest focused largely on such amusements as numerical curiosities; simple geometric puzzles; arithmetical story problems; magic squares; and traditional diversions concerning angle trisection, the duplication of the cube, and the squaring of the circle. By the middle of the 20th century, interest had begun to swing toward more mathematically sophisticated topics: cryptograms; recreations involving modular arithmetic, numeration bases, and number theory; graphs and networks; lattices and group theory; topological curiosities; flexagons; the manipulation of geometric shapes and forms; combinatorial problems; probability theory; inferential problems; logical paradoxes; fallacies of logic; and paradoxes of the infinite. Modern mathematical games and recreations can be grouped into several general classes: arithmetic and algebraic recreations (e.g., number patterns, cryptograms, perfect numbers, and magic squares); geometric and topological recreations (including geometric fallacies, tangrams and mazes, and map-colouring problems); manipulative recreations (puzzles involving configurations, chessboard problems, and polyominoes); and problems of logical inference (logical puzzles involving overlapping groups and paradoxes). any of various puzzles and games that involve aspects of mathematics. Mathematical recreations comprise puzzles and games that vary from naive amusements to sophisticated problems, some of which have never been solved. They may involve arithmetic, algebra, geometry, theory of numbers, graph theory, topology, matrices, group theory, combinatorics (dealing with problems of arrangements or designs), set theory, symbolic logic, or probability theory. Any attempt to classify this colourful assortment of material is at best arbitrary. Included in this article are the history and the main types of number games and mathematical recreations and the principles on which they are based. Details, including descriptions of puzzles, games, and recreations mentioned in the article, will be found in the references listed in the bibliography. At times it becomes difficult to tell where pastime ends and serious mathematics begins. An innocent puzzle requiring the traverse of a path may lead to technicalities of graph theory; a simple problem of counting parts of a geometric figure may involve combinatorial theory; dissecting a polygon may involve transformation geometry and group theory; logical inference problems may involve matrices. A problem regarded in medieval timesor before electronic computers became commonplaceas very difficult may prove to be quite simple when attacked by the mathematical methods of today. Mathematical recreations have a universal appeal. The urge to solve a puzzle is manifested alike by young and old, by the unsophisticated as well as the sophisticated. An outstanding English mathematician, G.H. Hardy, observed that professional puzzle makers, aware of this propensity, exploit it diligently, knowing full well that the general public gets an intellectual kick out of such activities. The relevant literature has become extensive, particularly since the beginning of the 20th century. Some of it is repetitious, but surprisingly enough, successive generations have found the older chestnuts to be quite delightful, whether dressed in new clothes or not. Much newly created material is continually being added. Additional reading General works W.W. Rouse Ball and H.S.M. Coxeter, Mathematical Recreations and Essays, 12th ed. (1974); John H. Conway, On Numbers and Games (1976); Henry E. Dudeney, 536 Puzzles and Curious Problems, ed. by Martin Gardner (1967); Kobon Fujimura, The Tokyo Puzzles, trans. from the Japanese, ed. by Martin Gardner (1978); Martin Gardner, more than a dozen collections of mathematical recreations, including Mathematical Circus (1979, reissued 1981) and Wheels, Life and Other Mathematical Amusements (1983); Douglas R. Hofstadter, Gdel, Escher, Bach: An Eternal Golden Braid (1979, reprinted 1980); J.A.H. Hunter, Challenging Mathematical Teasers (1980); J.A.H. Hunter and Joseph S. Madachy, Mathematical Diversions, rev. ed. (1975); David A. Klarner, ed., The Mathematical Gardner (1981), a tribute to Martin Gardner; Boris A. Kordemsky, The Moscow Puzzles, trans. from the Russian by Albert Parry, ed. by Martin Gardner (1972); Maurice Kraitchik, Mathematical Recreations, 2nd rev. ed. (1953); Joseph S. Madachy, Madachy's Mathematical Recreations (1979); T.H. O'Beirne, Puzzles and Paradoxes (1965, reprinted 1984); Hubert Phillips (Caliban), Question Time: An Omnibus of Problems for a Brainy Day (1938); Frederik Schuh, The Master Book of Mathematical Recreations, trans. from the Dutch, ed. by T.H. O'Beirne (1968); Hugo Steinhaus, Mathematical Snapshots, 3rd U.S. ed. (1983; originally published in Polish, 1954). Books on special topics (Cube puzzles): John Ewing and Czes Kosniowski, Puzzle It Out: Cubes, Groups, and Puzzles (1982); P.A. MacMahon, New Mathematical Pastimes (1921); James G. Nourse, The Simple Solution to Cubic Puzzles (1981); Don Taylor and Leanne Rylands, Cube Games (1981); Ferdinand Winter, Das Spiel der 30 Bunten Wrfel (1934). (Dissections): V.G. Boltyanskii, Equivalent and Equidecomposable Figures (1963; originally published in Russian, 1956); Harry Lindgren, Geometric Dissections (1964). (Fallacies): V.M. Bradis, V.L. Minkovskii, and A.K. Kharcheva, Lapses in Mathematical Reasoning (1963; originally published in Russian, 2nd ed., 1959); Edwin A. Maxwell, Fallacies in Mathematics (1959, reprinted 1969). (Fibonacci numbers): Verner E. Hoggatt, Jr., Fibonacci and Lucas Numbers (1969). (Fractals): Benoit B. Mandelbrot, The Fractal Geometry of Nature, rev. ed. (1983). (Graphs): Oystein Ore, Graphs and Their Uses (1963). (Logical inference): Maxey Brooke, 150 Puzzles in Crypt-Arithmetic, 2nd rev. ed. (1969); Hubert Phillips (Caliban), My Best Puzzles in Logic and Reasoning (1961); Raymond M. Smullyan, What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles (1978), and This Book Needs No Title (1980); George J. Summers, Test Your Logic: 50 Puzzles in Deductive Reasoning (1972); Clarence R. Wylie, 101 Puzzles in Thought and Logic (1957). (Manipulative puzzles and games): Maxey Brooke, Fun for the Money (1963); Solomon W. Golomb, Polyominoes (1965); Ronald C. Read, Tangrams: 330 Puzzles (1965); T. Sundara Row, Geometric Exercises in Paper Folding, 2nd ed. (1905, reprinted 1966); Sid Sackson, A Gamut of Games (1969, reissued 1982). (Mazes): Walter Shepherd, Mazes and Labyrinths: A Book of Puzzles, rev. ed. (1961). (Polytopes): H.S.M. Coxeter, Regular Polytopes, 3rd ed. (1973); H. Martyn Cundy and A.P. Rollett, Mathematical Models, 2nd ed. (1961, reprinted 1977); L. Fejes Tth, Regular Figures (1964). (Probability): Warren Weaver, Lady Luck: The Theory of Probability (1963, reprinted 1982). Periodicals Journal of Recreational Mathematics (quarterly; United States); Pythagoras (4 issues a year; The Netherlands, England, and the United States). William L. Schaaf