# MATHEMATICALRECREATIONSAND 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
2. ARITHMETICAL RECREATIONS (continued)
• Arithmetical fallacies
• 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
• Continued fractions and lattice points
• Geometrical dissections
• Cyclotomy
• Compass problems
• The five-disc problem
• Lebesgue's minimal problem
• Kakeya's minimal problem 99
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
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 error - correcting code
• A block design
• Steiner triple systems
• Finite geometries
• Kirkman's school-girl problem
• Latin squares
• The cube and the simplex
• 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 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