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Project Gutenberg’s Mathematical Recreations and Essays, by W. W. Rouse Ball 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.org Title: Mathematical Recreations and Essays Author: W. W. Rouse Ball Release Date: October 8, 2008 [EBook #26839] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICAL RECREATIONS *** First Edition, Feb. 1892. Reprinted, May, 1892. Second Edition, 1896. Reprinted, 1905. MATHEMATICAL RECREATIONS AND ESSAYS BY W.W. ROUSE BALL Fellow and Tutor of Trinity College, Cambridge. FOURTH EDITION London: MACMILLAN AND CO., Limited NEW YORK: THE MACMILLAN COMPANY  [All rights reserved.] Produced by Joshua Hutchinson, David Starner, David Wilson and the Online Distributed Proofreading Team at http://www.pgdp.net Transcriber’s notes Most of the open questions discussed by the author were settled during the twentieth century. The author’s footnotes are labelled using printer’s marks * ; footnotes showing where corrections to the text have been made are labelled numerically 1 . Minor typographical corrections are documented in the L A T E X source. This document is designed for two-sided printing. Consequently, the many hyperlinked cross-references are not visually distinguished. The document can be recompiled for more comfortable on-screen viewing: see comments in source L A T E X co de. PREFACE TO THE FIRST EDITION. The following pages contain an account of certain mathematical recreations, problems, and speculations of past and present times. I hasten to add that the conclusions are of no practical use, and most of the results are not new. If therefore the reader proceeds further he is at least forewarned. At the same time I think I may assert that many of the diversions— particularly those in the latter half of the book—are interesting, not a few are associated with the names of distinguished mathematicians, while hitherto several of the memoirs quoted have not been easily ac- cessible to English readers. The book is divided into two parts, but in both parts I have in- cluded questions which involve advanced mathematics. The first part consists of seven chapters, in which are included var- ious problems and amusements of the kind usually called mathematical recreations. The questions discussed in the first of these chapters are connected with arithmetic; those in the second with geometry; and those in the third relate to mechanics. The fourth chapter contains an account of some miscellaneous problems which involve both num- ber and situation; the fifth chapter contains a concise account of magic squares; and the sixth and seventh chapters deal with some unicursal iii iv PREFACE problems. Several of the questions mentioned in the first three chap- ters are of a somewhat trivial character, and had they been treated in any standard English work to which I could have referred the reader, I should have pointed them out. In the absence of such a work, I thought it best to insert them and trust to the judicious reader to omit them altogether or to skim them as he feels inclined. The second part consists of five chapters, which are mostly histori- cal. They deal respectively with three classical problems in geometry— namely, the duplication of the cube, the trisection of an angle, and the quadrature of the circle —astrology, the hypotheses as to the nature of space and mass, and a means of measuring time. I have inserted detailed references, as far as I know, as to the sources of the various questions and solutions given; also, wherever I have given only the result of a theorem, I have tried to indicate authorities where a proof may be found. In general, unless it is stated otherwise, I have taken the references direct from the original works; but, in spite of considerable time spent in verifying them, I dare not suppose that they are free from all errors or misprints. I shall be grateful for notices of additions or corrections which may occur to any of my readers. W.W. ROUSE BALL Trinity College, Cambridge. February, 1892. NOTE TO THE FOURTH EDITION. In this edition I have inserted in the earlier chapters descriptions of several additional Recreations involving elementary mathematics, and I have added in the second part chapters on the History of the Mathe- matical Tripos at Cambridge, Mersenne’s Numbers, and Cryptography and Ciphers. It is with some hesitation that I include in the book the chapters on Astrology and Ciphers, for these subjects are only remotely connected with Mathematics, but to afford myself some latitude I have altered the title of the second part to Miscellaneous Essays and Problems. W.W.R.B. Trinity College, Cambridge. 13 May, 1905. v TABLE OF CONTENTS. PART I. Mathematical Recreations. Chapter I. Some Arithmetical Questions. PAGE Elementary Questions on Numbers (Miscellaneous) . . . . . . 4 Arithmetical Fallacies . . . . . . . . . . . . . . . . . . . . . . 20 Bachet’s Weights Problem . . . . . . . . . . . . . . . . . . . . 27 Problems in Higher Arithmetic . . . . . . . . . . . . . . . . . 29 Fermat’s Theorem on Binary Powers . . . . . . . . . . . . 31 Fermat’s Last Theorem . . . . . . . . . . . . . . . . . . . . 32 Chapter II. Some Geometrical Questions. Geometrical Fallacies . . . . . . . . . . . . . . . . . . . . . . . 35 Geometrical Paradoxes . . . . . . . . . . . . . . . . . . . . . . 42 Colouring Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Physical Geography . . . . . . . . . . . . . . . . . . . . . . . 46 Statical Games of Position . . . . . . . . . . . . . . . . . . . . 48 Three-in-a-row. Extension to p-in-a-row . . . . . . . . . 48 Tesselation. Cross-Fours . . . . . . . . . . . . . . . . . . 50 Colour-Cube Problem . . . . . . . . . . . . . . . . . . . . 51 vi TABLE OF CONTE NTS. vii PAGE Dynamical Games of Position . . . . . . . . . . . . . . . . . . 52 Shunting Problems . . . . . . . . . . . . . . . . . . . . . . 53 Ferry-Boat Problems . . . . . . . . . . . . . . . . . . . . . 55 Geodesic Problems . . . . . . . . . . . . . . . . . . . . . . 57 Problems with Counters placed in a row . . . . . . . . . . 58 Problems on a Chess-board with Counters or Pawns . . . . 60 Guarini’s Problem . . . . . . . . . . . . . . . . . . . . . . 63 Geometrical Puzzles (rods, strings, &c.) . . . . . . . . . . . . 64 Paradromic Rings . . . . . . . . . . . . . . . . . . . . . . . . . 64 Chapter III. Some Mechanical Questions. Paradoxes on Motion . . . . . . . . . . . . . . . . . . . . . . . 67 Force, Inertia, Centrifugal Force . . . . . . . . . . . . . . . . . 70 Work, Stability of Equilibrium, &c. . . . . . . . . . . . . . . . 72 Perpetual Motion . . . . . . . . . . . . . . . . . . . . . . . . . 75 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Sailing quicker than the Wind . . . . . . . . . . . . . . . . . . 79 Boat moved by a rope inside the boat . . . . . . . . . . . . . 81 Results dependent on Hauksbee’s Law . . . . . . . . . . . . . 82 Cut on a tennis-ball. Spin on a cricket-ball . . . . . . . 83 Flight of Birds . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Curiosa Physica . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Chapter IV. Some Miscellaneous Questions. The Fifteen Puzzle . . . . . . . . . . . . . . . . . . . . . . . . 88 The Tower of Hano¨ı . . . . . . . . . . . . . . . . . . . . . . . 91 Chinese Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 The Eight Queens Problem . . . . . . . . . . . . . . . . . . . 97 Other Problems with Queens and Chess-pieces . . . . . . . . . 102 The Fifteen School-Girls Problem . . . . . . . . . . . . . . . . 103 viii TABLE OF CO NTENTS. PAGE Problems connected with a pack of cards . . . . . . . . . . . . 109 Monge on shuffling a pack of cards . . . . . . . . . . . . . 109 Arrangement by rows and columns . . . . . . . . . . . . . 111 Determination of one out of 1 2 n(n + 1) given couples . . . . 113 Gergonne’s Pile Problem . . . . . . . . . . . . . . . . . . . 115 The Mouse Trap. Treize . . . . . . . . . . . . . . . . . . 119 Chapter V. Magic Squares. Notes on the History of Magic Squares . . . . . . . . . . . . . 122 Construction of Odd Magic Squares . . . . . . . . . . . . . . . 123 Method of De la Loub`ere . . . . . . . . . . . . . . . . . . . 124 Method of Bachet . . . . . . . . . . . . . . . . . . . . . . . 125 Method of De la Hire . . . . . . . . . . . . . . . . . . . . . 126 Construction of Eve n Magic Squares . . . . . . . . . . . . . . 128 First Method . . . . . . . . . . . . . . . . . . . . . . . . . 129 Method of De la Hire and Labosne . . . . . . . . . . . . . 132 Composite Magic Squares . . . . . . . . . . . . . . . . . . . . 134 Bordered Magic Squares . . . . . . . . . . . . . . . . . . . . . 135 Hyper-Magic Squares . . . . . . . . . . . . . . . . . . . . . . . 136 Pan-diagonal or Nasik Squares . . . . . . . . . . . . . . . . 136 Doubly Magic Squares . . . . . . . . . . . . . . . . . . . . 137 Magic Pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Magic Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Card Square . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Euler’s Officers Problem . . . . . . . . . . . . . . . . . . . 140 Domino Squares . . . . . . . . . . . . . . . . . . . . . . . . 141 Coin Squares . . . . . . . . . . . . . . . . . . . . . . . . . 141 Chapter VI. Unicursal Problems. Euler’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Euler’s Theorems . . . . . . . . . . . . . . . . . . . . . . . 145 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 [...]... 236 236 Astrology Two branches: natal and horary astrology Rules for casting and reading a horoscope Houses and their significations Planets and their significations Zodiacal signs and their significations Knowledge that rules were worthless Notable instances of horoscopy Lilly’s prediction of the Great Fire and Plague Flamsteed’s guess ... vol xlviii, p 234 6 ARITHMETICAL RECREATIONS [CH I this case by assigning proper values to a and c.] Thus x and y are both known, and therefore the number selected, namely 9x + y, is known Fifth Method * Ask any one to select a number less than 60 (i) Request him to divide it by 3 and mention the remainder; suppose it to be a (ii) Request him to divide it by 4, and mention the remainder; suppose... ARITHMETICAL RECREATIONS [CH I A to give 3 counters to B; and then you would ask B to give to A a number of counters equal to twice the number then in A’s possession; after this was done you would know that B had 3(2 + 1), that is, 9 left This trick (as also some of the following problems) may be performed equally well with one person, in which case A may stand for his right hand and B for his left hand Third... provided they were prime to one another and one of them * † Bachet, problem ix, p 107 Bachet, problem xi, p 113 10 ARITHMETICAL RECREATIONS [CH I was not itself a prime Let the numbers be m and n, and suppose that n is exactly divisible by p Ask A to select one of these numbers, and B to take the other Choose a number prime to p, say q Ask A to multiply his number by q, and B to multiply his number by p... represents the product of that digit and a power of ten, and the number is equal to the sum of these products For example, 2017 signifies (2×103 )+(0×102 )+(1×10)+7; that is, the 2 represents 2 thousands, i.e the product of 2 and 103 , the 0 represents 0 hundreds, i.e the product of 0 and 102 ; the 1 represents 1 ten, i.e the product of 1 and 10, and the 7 represents 7 units Thus every digit has a local... rearranged, and ask that the top card be counted as the (x + 1)th, the next as the (x + 2)th, and so on, in which case the card originally chosen will be the (y + m + 1)th Now y and m can be chosen as we please, and may be varied every time the trick is performed; thus any one unskilled in arithmetic will not readily detect the modus operandi Fourth Example* Place a card on the table, and on it place... that there are two numbers, one even and the other odd, and that a person A is asked to select one of them, and that another person B takes the other It is desired to know whether A selected the even or the odd number Ask A to multiply his number by 2 (or any even number) and B to multiply his by 3 (or any odd number) Request them to add the two products together and tell you the sum If it is even,... Oughtred in his Mathematicall Recreations (translated from or founded on van Etten’s work of 1633), London, 1653, problem xxxiv; and by Ozanam, part i, chapter x CH I] ELEMENTARY TRICKS AND PROBLEMS 11 Second Example* Similarly, if three numbers, say, a, b, c, are chosen, then, if each of them is less than ten, they can be found by the following rule (i) Take one of the numbers, say, a, and multiply... on a table in some numerical order, and the nth thing is selected by a spectator Then the first (19 − m) taps are immaterial, the (20 − m)th tap must be on the mth thing and be reckoned by the spectator as the (n + 20 − m)th, the (20 − m + 1)th tap must be on the (m−1)th thing and be reckoned as the (n+20−m+1)th, and finally the (20 − n)th tap will be on the nth thing and is reckoned as the 20th tap Third... multiply it by 10, and to add any number he pleases, a, which is less than 10 (ii) Request him to divide the result of step (i) by 3, and to mention the remainder, say, b (iii) Request him to multiply the quotient obtained in step (ii) by 10, and to add any number he pleases, c, which is less than 10 (iv) Request him to divide the result of step (iii) by 3, and to mention the remainder, say d, and the third . Project Gutenberg’s Mathematical Recreations and Essays, by W. W. Rouse Ball This eBook is for the use of anyone anywhere at no cost and with almost. the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Mathematical Recreations and Essays Author: W.