# MATHEMATICAL

RECREATIONS

AND ESSAYS

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

### CONTENTS

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- UNICURSAL PROBLEMS
- Euler's problem
- Number of ways of describing a unicursal figure
- Mazes
- Trees
- The Hamiltonian game
- Dragon designs

- 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

- 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

- 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

- The duplication of the cube
- 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

- 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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