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# Mathematical Recreations and Essays

## Paperback

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## Overview

"*The* classic work on recreational math in English."—Martin Gardner

For nearly a century, this sparkling classic has provided stimulating hours of entertainment to the mathematically inclined. The problems posed here often involve fundamental mathematical methods and notions, but their chief appeal is their capacity to tease and delight. In these pages you will find scores of "recreations" to amuse you and to challenge your problem-solving faculties—often to the limit.

Now in its 13th edition, *Mathematical Recreations and Essays* has been thoroughly revised and updated over the decades since its first publication in 1892. This latest edition retains all the remarkable character of the original, but the terminology and treatment of some problems have been updated and new material has been added.

Among the challenges in store for you: Arithmetical and geometrical recreations; Polyhedra; Chess-board recreations; Magic squares; Map-coloring problems; Unicursal problems; Cryptography and cryptanalysis; Calculating prodigies; … and more.

You'll even find problems which mathematical ingenuity can solve but the computer cannot. No knowledge of calculus or analytic geometry is necessary to enjoy these games and puzzles. With basic mathematical skills and the desire to meet a challenge you can put yourself to the test and win.

"A must to add to your mathematics library."—*The Mathematics Teacher*

## Product Details

ISBN-13: | 9780486253572 |
---|---|

Publisher: | Dover Publications |

Publication date: | 05/06/2010 |

Series: | Dover Recreational Math Series |

Pages: | 464 |

Sales rank: | 881,967 |

Product dimensions: | 5.41(w) x 8.47(h) x 0.92(d) |

## About the Author

**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

## Read an Excerpt

**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

## First Chapter

**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

## Reading Group Guide

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

## Interviews

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

## Recipe

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