Mathematical Recreations and Essays

by Ball, W. W. Rouse; Coxeter, H. S. M.

• ISBN13: 9780486253572
• ISBN10: 0486253570
• Edition: 13th
• Format: Paperback
• Copyright: 2010-05-06
• Publisher: Dover Publications

Summary

This classic work offers scores of stimulating, mind-expanding games and puzzles: arithmetical and geometrical problems, chessboard recreations, magic squares, map-coloring problems, cryptography and cryptanalysis, much more. "A must to add to your mathematics library" The Mathematics Teacher.Index. References for Further Study. Includes 150 black-and-white line illustrations.

Author Biography

H. S. M. Coxeter: Through the Looking Glass
Harold Scott MacDonald Coxeter (1907–2003) is one of the greatest geometers of the last century, or of any century, for that matter. Coxeter was associated with the University of Toronto for sixty years, the author of twelve books regarded as classics in their field, a student of Hermann Weyl in the 1930s, and a colleague of the intriguing Dutch artist and printmaker Maurits Escher in the 1950s.

In the Author's Own Words:
"I'm a Platonist — a follower of Plato — who believes that one didn't invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."

"In our times, geometers are still exploring those new Wonderlands, partly for the sake of their applications to cosmology and other branches of science, but much more for the sheer joy of passing through the looking glass into a land where the familiar lines, planes, triangles, circles, and spheres are seen to behave in strange but precisely determined ways."

"Geometry is perhaps the most elementary of the sciences that enable man, by purely intellectual processes, to make predictions (based on observation) about the physical world. The power of geometry, in the sense of accuracy and utility of these deductions, is impressive, and has been a powerful motivation for the study of logic in geometry."

"Let us revisit Euclid. Let us discover for ourselves a few of the newer results. Perhaps we may be able to recapture some of the wonder and awe that our first contact with geometry aroused." — H. S. M. Coxeter

Table of Contents

I ARITHEMETICAL RECREATIONS
    To find a number selected by someone
    Prediction of the result of certain operations
    Problems involving two numbers
    Problems depending on the scale of notation
    Other problems with numbers in the denary scale
    Four fours problems
    Problems with a series of numbered things
    Arithmetical restorations
    Calendar problems
    Medieval problems in arithmetic
      The Josephus problem. Decimation
    Nim and similar games
      Moore's game
      Kayles
      Wythoff's game
    Addendum on solutions
II ARITHEMETICAL RECREATIONS (continued)
    Arithmetical fallacies
    Paradoxical problems
    Probability problems
    Permutation problems
    Bachet's weights problem
    The decimal expression for 1/n
    Decimals and continued fractions
    Rational right-angled triangles
    Triangular and pyramidal numbers
    Divisibility
    The prime number theorem
    Mersenne numbers
    Perfect numbers
    Fermat numbers
    Fermat's Last Theorem
    Galois fields
III GEOMETRICAL RECREATIONS
    Geometrical fallacies
    Geometrical paradoxes
    Continued fractions and lattice points
    Geometrical dissections
    Cyclotomy
    Compass problems
    The five-disc problem
    Lebesgue's minimal problem
    Kakeya's minimal problem
    Addendum on a solution
IV GEOMETRICAL RECREATIONS (continued)
    Statical games of position
      Three-in-a-row. Extension to p-in-a-row
      Tessellation
      Anallagmatic pavements
      Polyominoes
      Colour-cube problem
      Squaring the square
    Dynamical games of position
      Shunting problems
      Ferry-boat problems
      Geodesic problems
      Problems with counters or pawns
    Paradromic rings
    Addendum on solutions
V POLYHEDRA
    Symmetry and symmetries
    The five Platonic solids
      Kepler's mysticism
      "Pappus, on the distribution of vertices"
      Compounds
    The Archimedean solids
      Mrs. Stott's construction
    Equilateral zonohedra
    The Kepler-Poinsot polyhedra
    The 59 icosahedra
    Solid tessellations
    Ball-piling or close-packing
      The sand by the sea-shore
    Regular sponges
    Rotating rings of tetrahedra
    The kaleidoscope
VI CHESS-BOARD RECREATIONS
    Relative value of pieces
    The eight queens problem
    Maximum pieces problem
    Minimum pieces problem
    Re-entrant paths on a chess-board
      Knight's re-entrant path
      King's re-entrant path
      Rook's re-entrant path
      Bishop's re-entrant path
      Route's on a chess-board
      Guarini's problem
    Latin squares
      Eulerian squares
      Euler's officers problem
      Eulerian cubes
VII MAGIC SQUARE
    Magic squares of an odd order
    Magic squares of a singly-even order
    Magic squares of a doubly-even order
    Bordered squares
    Number of squares of a given order
    Symmetrical and pandiagonal squares
      Generalization of De la Loubère's rule
      Arnoux's method
      Margossian's method
    Magic squares of non-consecutive numbers
      Magic squares of primes
    Doubly-magic and trebly-magic squares
    Other magic problems
      Magic domino squares
      Cubic and octahedral dice
      Interlocked hexagons
    Magic cubes
VIII MAP-COLOURING PROBLEMS
    The four-colour conjecture
      The Petersen graph
      Reduction to a standard map
      Minimum number of districts for possible failure
      Equivalent problem in the theory of numbers
    Unbounded surfaces
    Dual maps
    Maps on various surfaces
    "Pits, peaks, and passes"
    Colouring the icosahedron
IX UNICURSAL PROBLEMS
    Euler's problem
    Number of ways of describing a unicursal figure
    Mazes
    Trees
    The Hamiltonian game
    Dragon designs
X COMBINATORIAL DESIGNS
    A projective plane
    Incidence matrices
    An Hadamard matrix
    An error-corrrecting code
    A block design
    Steiner triple systems
    Finite geometries
    Kirkman's school-girl problem
    Latin squares
    The cube and the simplex
    Hadamard matrices
    Picture transmission
    Equiangular lines in 3-space
    Lines in higher-dimensional space
    C-matrices
    Projective planes
XI MISCELLANEOUS
    The fifteen puzzle
    The Tower of Hanoï
    Chinese rings
    Problems connected with a pack of cards
    Shuffling a pack
    Arrangements by rows and columns
    Bachet's problem with pairs of cards
    Gergonne's pile problem
    The window reader
    The mouse trap. Treize
XII THREE CLASSICAL GEOMETRICAL PROBLEMS
    The duplication of the cube
      "Solutions by Hippocrates, Archytas, Plato, Menaechmus, Apollonius, and Diocles"
      "Solutions by Vieta, Descartes, Gregory of St. Vincent, and Newton"
    The trisection of an angle
      "Solutions by Pappus, Descartes, Newton, Clairaut, and Chasles"
    The quadrature of the circle
      Origin of symbo p
      Geometrical methods of approximation to the numerical value of p
        "Results of Egyptians, Babylonians, Jews"
        Results of Archimedes and other Greek writers
        "Results of European writers, 1200-1630"
    Theorems of Wallis and Brouncker
    "Results of European writers, 1699-1873"
    Ap