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Amusements in Mathematics

Project Gutenberg's Amusements in Mathematics, by Henry Ernest Dudeney This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.net

Title: Amusements in Mathematics

Author: Henry Ernest Dudeney

Release Date: September 17, 2005 [EBook #16713]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK AMUSEMENTS IN MATHEMATICS ***

Produced by Stephen Schulze, Jonathan Ingram and the Online Distributed Proofreading Team at http://www.pgdp.net

[Transcribers note: Many of the puzzles in this book assume a familiarity with the currency of Great Britain in the early 1900s. As this is likely not common knowledge for those outside Britain (and possibly many within,) I am including a chart of relative values.

The most common units used were:

the Penny, abbreviated: d. (from the Roman penny, denarius) the Shilling, abbreviated: s. the Pound, abbreviated: ￡

There was 12 Pennies to a Shilling and 20 Shillings to a Pound, so there was 240 Pennies in a Pound.

To further complicate things, there were many coins which were various fractional values of Pennies, Shillings or Pounds.

Farthing ?d.

Half-penny ?d.

Penny 1d.

Three-penny 3d.

Sixpence (or tanner) 6d.

Shilling (or bob) 1s.

Florin or two shilling piece 2s.

Half-crown (or half-dollar) 2s. 6d.

Double-florin 4s.

Crown (or dollar) 5s.

Half-Sovereign 10s.

Sovereign (or Pound) ￡1 or 20s.

This is by no means a comprehensive list, but it should be adequate to solve the puzzles in this book.

Exponents are represented in this text by ^, e.g. '3 squared' is 3^2.

Numbers with fractional components (other than ?, ? and ?) have a + symbol separating the whole number component from the fraction. It makes the fraction look odd, but yeilds correct solutions no matter how it is interpreted. E.G., 4 and eleven twenty-thirds is 4+11/23, not 411/23 or 4-11/23.

]

AMUSEMENTS IN MATHEMATICS

by

HENRY ERNEST DUDENEY

In Mathematicks he was greater Than Tycho Brahe or Erra Pater: For he, by geometrick scale, Could take the size of pots of ale; Resolve, by sines and tangents, straight, If bread or butter wanted weight; And wisely tell what hour o' th' day The clock does strike by algebra.

BUTLER'S Hudibras.

1917

PREFACE

In issuing this volume of my Mathematical Puzzles, of which some have appeared in periodicals and others are given here for the first time, I must acknowledge the encouragement that I have received from many unknown correspondents, at home and abroad, who have expressed a desire to have the problems in a collected form, with some of the solutions given at greater length than is possible in magazines and newspapers. Though I have included a few old puzzles that have interested the world for generations, where I felt that there was something new to be said about them, the problems are in the main original. It is true that some of these have become widely known through the press, and it is possible that the reader may be glad to know their source.

On the question of Mathematical Puzzles in general there is, perhaps, little more to be said than I have written elsewhere. The history of the subject entails nothing short of the actual story of the beginnings and development of exact thinking in man. The historian must start from the time when man first succeeded in counting his ten fingers and in dividing an apple into two approximately equal parts. Every puzzle that is worthy of consideration can be referred to mathematics and logic. Every man, woman, and child who tries to "reason out" the answer to the simplest puzzle is working, though not of necessity consciously, on mathematical lines. Even those puzzles that we have no way of attacking except by haphazard attempts can be brought under a method of what has been called "glorified trial"--a system of shortening our labours by avoiding or eliminating what our reason tells us is useless. It is, in fact, not easy to say sometimes where the "empirical" begins and where it ends.

When a man says, "I have never solved a puzzle in my life," it is difficult to know exactly what he means, for every intelligent individual is doing it every day. The unfortunate inmates of our lunatic asylums are sent there expressly because they cannot solve puzzles--because they have lost their powers of reason. If there were no puzzles to solve, there would be no questions to ask; and if there were no questions to be asked, what a world it would be! We should all be equally omniscient, and conversation would be useless and idle.

It is possible that some few exceedingly sober-minded mathematicians, who are impatient of any terminology in their favourite science but the academic, and who object to the elusive x and y appearing under any other names, will have wished that various problems had been presented in a less

Amusements in Mathematics

Project Gutenberg's Amusements in Mathematics, by Henry Ernest Dudeney This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.net

Title: Amusements in Mathematics

Author: Henry Ernest Dudeney

Release Date: September 17, 2005 [EBook #16713]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK AMUSEMENTS IN MATHEMATICS ***

Produced by Stephen Schulze, Jonathan Ingram and the Online Distributed Proofreading Team at http://www.pgdp.net

[Transcribers note: Many of the puzzles in this book assume a familiarity with the currency of Great Britain in the early 1900s. As this is likely not common knowledge for those outside Britain (and possibly many within,) I am including a chart of relative values.

The most common units used were:

the Penny, abbreviated: d. (from the Roman penny, denarius) the Shilling, abbreviated: s. the Pound, abbreviated: ￡

There was 12 Pennies to a Shilling and 20 Shillings to a Pound, so there was 240 Pennies in a Pound.

To further complicate things, there were many coins which were various fractional values of Pennies, Shillings or Pounds.

Farthing ?d.

Half-penny ?d.

Penny 1d.

Three-penny 3d.

Sixpence (or tanner) 6d.

Shilling (or bob) 1s.

Florin or two shilling piece 2s.

Half-crown (or half-dollar) 2s. 6d.

Double-florin 4s.

Crown (or dollar) 5s.

Half-Sovereign 10s.

Sovereign (or Pound) ￡1 or 20s.

This is by no means a comprehensive list, but it should be adequate to solve the puzzles in this book.

Exponents are represented in this text by ^, e.g. '3 squared' is 3^2.

Numbers with fractional components (other than ?, ? and ?) have a + symbol separating the whole number component from the fraction. It makes the fraction look odd, but yeilds correct solutions no matter how it is interpreted. E.G., 4 and eleven twenty-thirds is 4+11/23, not 411/23 or 4-11/23.

]

AMUSEMENTS IN MATHEMATICS

by

HENRY ERNEST DUDENEY

In Mathematicks he was greater Than Tycho Brahe or Erra Pater: For he, by geometrick scale, Could take the size of pots of ale; Resolve, by sines and tangents, straight, If bread or butter wanted weight; And wisely tell what hour o' th' day The clock does strike by algebra.

BUTLER'S Hudibras.

1917

PREFACE

In issuing this volume of my Mathematical Puzzles, of which some have appeared in periodicals and others are given here for the first time, I must acknowledge the encouragement that I have received from many unknown correspondents, at home and abroad, who have expressed a desire to have the problems in a collected form, with some of the solutions given at greater length than is possible in magazines and newspapers. Though I have included a few old puzzles that have interested the world for generations, where I felt that there was something new to be said about them, the problems are in the main original. It is true that some of these have become widely known through the press, and it is possible that the reader may be glad to know their source.

On the question of Mathematical Puzzles in general there is, perhaps, little more to be said than I have written elsewhere. The history of the subject entails nothing short of the actual story of the beginnings and development of exact thinking in man. The historian must start from the time when man first succeeded in counting his ten fingers and in dividing an apple into two approximately equal parts. Every puzzle that is worthy of consideration can be referred to mathematics and logic. Every man, woman, and child who tries to "reason out" the answer to the simplest puzzle is working, though not of necessity consciously, on mathematical lines. Even those puzzles that we have no way of attacking except by haphazard attempts can be brought under a method of what has been called "glorified trial"--a system of shortening our labours by avoiding or eliminating what our reason tells us is useless. It is, in fact, not easy to say sometimes where the "empirical" begins and where it ends.

When a man says, "I have never solved a puzzle in my life," it is difficult to know exactly what he means, for every intelligent individual is doing it every day. The unfortunate inmates of our lunatic asylums are sent there expressly because they cannot solve puzzles--because they have lost their powers of reason. If there were no puzzles to solve, there would be no questions to ask; and if there were no questions to be asked, what a world it would be! We should all be equally omniscient, and conversation would be useless and idle.

It is possible that some few exceedingly sober-minded mathematicians, who are impatient of any terminology in their favourite science but the academic, and who object to the elusive x and y appearing under any other names, will have wished that various problems had been presented in a less