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Publisher:

Dover Publications

Publication Date:

1987

Number of Pages:

448

Format:

Paperback

Edition:

13

Price:

16.95

ISBN:

9780486253572

Category:

General

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by , on ]

Dennis W. Gordon

05/30/2011

This book was originally written solely by W. W. Rouse Ball in 1892 and revised most recently by H. S. M. Coxeter, whose name currently appears as co-author. It has — thanks to the updated material by Coxeter — remained fresh and interesting for over a century while retaining its original nineteenth century charm.

Mathematics has made considerable progress since 1892, thanks in part to the computer, and this powerful tool has facilitated such recent work as the discovery of new prime Mersenne numbers. There is, of course, much more to enjoy in the book, and the classical mathematics of Rouse Ball’s day includes such delights as magic squares, continued fractions, two cool chapters on geometrical recreations, and a chapter on map coloring problems.

The book is rich in lively historical mathematics, original references, and supplemented by a generous supply of nice, clear illustrations. If I had to choose a favorite chapter, my choice would be the one on calculating prodigies, that is, those individuals who perform amazing calculations in their heads, as I have long been fascinated by such mental gymnastics. Rouse Ball thoughtfully reveals some of the clever techniques used in these peculiar calculations.

Upon reading this book you will enjoy some terrific mathematical adventures and easily choose your own special favorite chapter. As an added bonus, since this is a Dover publication the price is very modest.

Let us hope that some of today’s readers will shed light on some of the remaining unsolved problems and continue to revise this wonderful book in future years. Happy reading!

In spite of having studied chemistry (Wayne State University and The University of Kansas) and had a professional career in both academic and industrial research, Dennis’ greatest personal interest in science is mathematics. Now retired, he is a voracious reader, and with his wife Sally, they enjoy traveling in their sports car, bluegrass music, and the wonders of Wisconsin. Dennis may be contacted at denniswmgordon@cs.com

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 LoubÅ re'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 Hanoâ€¹

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"

Approximation by the theory of probability

XIII CALCULATING PRODIGIES

"John Wallis, 1616-1703"

"Buxton, circ. 1707-1772"

"Fuller, 1710-1790; AmpÅ re"

"Gauss, Whately"

"Colburn, 1804-1840"

"Bidder, 1806-1878"

"Mondeux, Mangiamele"

"Dase, 1824-1861"

"Safford, 1836-1901"

"Zamebone, Diamandi, RÂckle"

"Inaudi, 1867"

Types of memory of numbers

Bidder's analysis of methods usesd

Multiplication

Digital method for division and factors

Square roots. Higher roots

Compound interest

Logarithms

Alexander Craig Aitken

XIV CRYPTOGRAPHY AND CRYPTANALYSIS

Cryptographic systems

Transposition systems

Columnar transposition

Digraphs and trigraphs

Comparision of several messages

The grille

Substitution systems

Tables of frequency

Polyalphabetic systems

The VigenÅ re square

The Playfair cipher

Code

Determination of cryptographic system

A few final remarks

Addendum: References for further study

INDEX

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 LoubÅ re'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 Hanoâ€¹

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"

Approximation by the theory of probability

XIII CALCULATING PRODIGIES

"John Wallis, 1616-1703"

"Buxton, circ. 1707-1772"

"Fuller, 1710-1790; AmpÅ re"

"Gauss, Whately"

"Colburn, 1804-1840"

"Bidder, 1806-1878"

"Mondeux, Mangiamele"

"Dase, 1824-1861"

"Safford, 1836-1901"

"Zamebone, Diamandi, RÂckle"

"Inaudi, 1867"

Types of memory of numbers

Bidder's analysis of methods usesd

Multiplication

Digital method for division and factors

Square roots. Higher roots

Compound interest

Logarithms

Alexander Craig Aitken

XIV CRYPTOGRAPHY AND CRYPTANALYSIS

Cryptographic systems

Transposition systems

Columnar transposition

Digraphs and trigraphs

Comparision of several messages

The grille

Substitution systems

Tables of frequency

Polyalphabetic systems

The VigenÅ re square

The Playfair cipher

Code

Determination of cryptographic system

A few final remarks

Addendum: References for further study

INDEX

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