MATHEMATICAL
RECREATIONS
AND ESSAYS

W.W.Rouse Ball and H.S.M.Coxeter

CONTENTS

  1. ARITHMETICAL 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 problem
    • 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
  2. ARITHMETICAL 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
  3. 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 99
    • Addendum on a solution
  4. 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
  5. 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
  6. CHESS-BOARDRECREATIONS
    • 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
    • Routes on a chess-board
    • Guarini's problem
    • Latin squares
    • Eulerian squares
    • Euler's officers problem
    • Eulerian cubes
  7. MAGIC SQUARES
    • 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 Loubere’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
  8. 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
  9. UNICURSAL PROBLEMS
    • Euler's problem
    • Number of ways of describing a unicursal figure
    • Mazes
    • Trees
    • The Hamiltonian game
    • Dragon designs
  10. COMBINATORIAL DESIGNS
    • A projective plane
    • Incidence matrices
    • An Hadamard matrix
    • An error - correcting 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
  11. MISCELLANEOUS PROBLEMS
    • The fifteen puzzle
    • The Tower of Hanoi
    • 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
  12. 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 symbol it
    • Geometrical methods of approximation to the numerical value of Pi
    • 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
    • Approximations by the theory of probability
  13. CALCULATING PRODIGIES
      • John Wallis, 1616-1703
      • Buxton, circ. 1707-1772
      • Fuller, 1710-1790; Amp,6re
      • Gauss, Whately
      • Colburn,1804-1840
      • Bidder, 1806-1878
      • Mondeux, Mangiamele
      • Dase, 1824-1861
      • Safford, 1836-1901
      • Zamebone, Diamandi, Ruckle
      • Inaudi, 1867-
    • Types of memory of numbers
    • Bidder's analysis of methods used
      • Multiplication
      • Digital method for division and factors
      • Square roots. Higher roots
      • Compound interest
      • Logarithms
    • Alexander Craig Aitken
  14. CRYPTOGRAPHY AND CRYPTANALYSIS
    • Cryptographic systems
    • Transposition systems
    • Columnar transposition
    • Digraphs and trigraphs
    • Comparison of several messages
    • The grille
    • Substitution systems
    • Tables of frequency
    • Polyalphabetic systems
    • The Vigenere square
    • The Playfair cipher
    • Code
    • Determination of cryptographic system
    • A few final remarks
    • Addendum: References for further study

INDEX

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