Amusements in Mathematics 1 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: Amusements in Mathematics 2 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 ¼d. Half−penny ½d. 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 ¼, ½ and ¾) 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. Amusements in Mathematics 3 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 popular dress and introduced with a less flippant phraseology. I can only refer them to the first word of my title and remind them that we are primarily out to be amused−−not, it is true, without some hope of picking up morsels of knowledge by the way. If the manner is light, I can only say, in the words of Touchstone, that it is "an ill−favoured thing, sir, but my own; a poor humour of mine, sir." As for the question of difficulty, some of the puzzles, especially in the Arithmetical and Algebraical category, are quite easy. Yet some of those examples that look the simplest should not be passed over without a little consideration, for now and again it will be found that there is some more or less subtle pitfall or trap into which the reader may be apt to fall. It is good exercise to cultivate the habit of being very wary over the exact wording of a puzzle. It teaches exactitude and caution. But some of the problems are very hard nuts indeed, and not unworthy of the attention of the advanced mathematician. Readers will doubtless select according to their individual tastes. In many cases only the mere answers are given. This leaves the beginner something to do on his own behalf in working out the method of solution, and saves space that would be wasted from the point of view of the advanced student. On the other hand, in particular cases where it seemed likely to interest, I have given rather extensive solutions and treated problems in a general manner. It will often be found that the notes on one Amusements in Mathematics 4 problem will serve to elucidate a good many others in the book; so that the reader's difficulties will sometimes be found cleared up as he advances. Where it is possible to say a thing in a manner that may be "understanded of the people" generally, I prefer to use this simple phraseology, and so engage the attention and interest of a larger public. The mathematician will in such cases have no difficulty in expressing the matter under consideration in terms of his familiar symbols. I have taken the greatest care in reading the proofs, and trust that any errors that may have crept in are very few. If any such should occur, I can only plead, in the words of Horace, that "good Homer sometimes nods," or, as the bishop put it, "Not even the youngest curate in my diocese is infallible." I have to express my thanks in particular to the proprietors of The Strand Magazine, _Cassell's Magazine_, The Queen, _Tit−Bits_, and The Weekly Dispatch for their courtesy in allowing me to reprint some of the puzzles that have appeared in their pages. THE AUTHORS' CLUB March 25, 1917 CONTENTS PREFACE v ARITHMETICAL AND ALGEBRAICAL PROBLEMS 1 Money Puzzles 1 Age and Kinship Puzzles 6 Clock Puzzles 9 Locomotion and Speed Puzzles 11 Digital Puzzles 13 Various Arithmetical and Algebraical Problems 17 GEOMETRICAL PROBLEMS 27 Dissection Puzzles 27 Greek Cross Puzzles 28 Various Dissection Puzzles 35 Patchwork Puzzles 46 Various Geometrical Puzzles 49 POINTS AND LINES PROBLEMS 56 MOVING COUNTER PROBLEMS 58 UNICURSAL AND ROUTE PROBLEMS 68 COMBINATION AND GROUP PROBLEMS 76 CHESSBOARD PROBLEMS 85 The Chessboard 85 Statical Chess Puzzles 88 The Guarded Chessboard 95 Dynamical Chess Puzzles 96 Various Chess Puzzles 105 MEASURING, WEIGHING, AND PACKING PUZZLES 109 CROSSING RIVER PROBLEMS 112 PROBLEMS CONCERNING GAMES 114 PUZZLE GAMES 117 MAGIC SQUARE PROBLEMS 119 Subtracting, Multiplying, and Dividing Magics 124 Magic Squares of Primes 125 MAZES AND HOW TO THREAD THEM 127 THE PARADOX PARTY 137 UNCLASSIFIED PROBLEMS 142 SOLUTIONS 148 INDEX 253 AMUSEMENTS IN MATHEMATICS. ARITHMETICAL AND ALGEBRAICAL PROBLEMS. "And what was he? Forsooth, a great arithmetician." Othello, I. i. The puzzles in this department are roughly thrown together in classes for the convenience of the reader. Some are very easy, others quite difficult. But they are not arranged in any order of difficulty−−and this is intentional, for it is well that the solver should not be warned that a puzzle is just what it seems to be. It may, therefore, prove to be quite as simple as it looks, or it may contain some pitfall into which, through want of care or over−confidence, we may stumble. Also, the arithmetical and algebraical puzzles are not separated in the manner adopted by some authors, who arbitrarily require certain problems to be solved by one method or the other. The reader is left to make his own choice and determine which puzzles are capable of being solved by him on purely arithmetical lines. MONEY PUZZLES. "Put not your trust in money, but put your money in trust." OLIVER WENDELL HOLMES. Amusements in Mathematics 5 1.−−A POST−OFFICE PERPLEXITY. In every business of life we are occasionally perplexed by some chance question that for the moment staggers us. I quite pitied a young lady in a branch post−office when a gentleman entered and deposited a crown on the counter with this request: "Please give me some twopenny stamps, six times as many penny stamps, and make up the rest of the money in twopence−halfpenny stamps." For a moment she seemed bewildered, then her brain cleared, and with a smile she handed over stamps in exact fulfilment of the order. How long would it have taken you to think it out? 2.−−YOUTHFUL PRECOCITY. The precocity of some youths is surprising. One is disposed to say on occasion, "That boy of yours is a genius, and he is certain to do great things when he grows up;" but past experience has taught us that he invariably becomes quite an ordinary citizen. It is so often the case, on the contrary, that the dull boy becomes a great man. You never can tell. Nature loves to present to us these queer paradoxes. It is well known that those wonderful "lightning calculators," who now and again surprise the world by their feats, lose all their mysterious powers directly they are taught the elementary rules of arithmetic. A boy who was demolishing a choice banana was approached by a young friend, who, regarding him with envious eyes, asked, "How much did you pay for that banana, Fred?" The prompt answer was quite remarkable in its way: "The man what I bought it of receives just half as many sixpences for sixteen dozen dozen bananas as he gives bananas for a fiver." Now, how long will it take the reader to say correctly just how much Fred paid for his rare and refreshing fruit? 3.−−AT A CATTLE MARKET. Three countrymen met at a cattle market. "Look here," said Hodge to Jakes, "I'll give you six of my pigs for one of your horses, and then you'll have twice as many animals here as I've got." "If that's your way of doing business," said Durrant to Hodge, "I'll give you fourteen of my sheep for a horse, and then you'll have three times as many animals as I." "Well, I'll go better than that," said Jakes to Durrant; "I'll give you four cows for a horse, and then you'll have six times as many animals as I've got here." No doubt this was a very primitive way of bartering animals, but it is an interesting little puzzle to discover just how many animals Jakes, Hodge, and Durrant must have taken to the cattle market. 4.−−THE BEANFEAST PUZZLE. A number of men went out together on a bean−feast. There were four parties invited−−namely, 25 cobblers, 20 tailors, 18 hatters, and 12 glovers. They spent altogether £6, 13s. It was found that five cobblers spent as much as four tailors; that twelve tailors spent as much as nine hatters; and that six hatters spent as much as eight glovers. The puzzle is to find out how much each of the four parties spent. 5.−−A QUEER COINCIDENCE. Seven men, whose names were Adams, Baker, Carter, Dobson, Edwards, Francis, and Gudgeon, were recently engaged in play. The name of the particular game is of no consequence. They had agreed that whenever a player won a game he should double the money of each of the other players−−that is, he was to give the players just as much money as they had already in their pockets. They played seven games, and, strange to say, each won a game in turn, in the order in which their names are given. But a more curious coincidence is this−−that when they had finished play each of the seven men had exactly the same amount−−two shillings Amusements in Mathematics 6 and eightpence−−in his pocket. The puzzle is to find out how much money each man had with him before he sat down to play. 6.−−A CHARITABLE BEQUEST. A man left instructions to his executors to distribute once a year exactly fifty−five shillings among the poor of his parish; but they were only to continue the gift so long as they could make it in different ways, always giving eighteenpence each to a number of women and half a crown each to men. During how many years could the charity be administered? Of course, by "different ways" is meant a different number of men and women every time. 7.−−THE WIDOW'S LEGACY. A gentleman who recently died left the sum of £8,000 to be divided among his widow, five sons, and four daughters. He directed that every son should receive three times as much as a daughter, and that every daughter should have twice as much as their mother. What was the widow's share? 8.−−INDISCRIMINATE CHARITY. A charitable gentleman, on his way home one night, was appealed to by three needy persons in succession for assistance. To the first person he gave one penny more than half the money he had in his pocket; to the second person he gave twopence more than half the money he then had in his pocket; and to the third person he handed over threepence more than half of what he had left. On entering his house he had only one penny in his pocket. Now, can you say exactly how much money that gentleman had on him when he started for home? 9.−−THE TWO AEROPLANES. A man recently bought two aeroplanes, but afterwards found that they would not answer the purpose for which he wanted them. So he sold them for £600 each, making a loss of 20 per cent. on one machine and a profit of 20 per cent. on the other. Did he make a profit on the whole transaction, or a loss? And how much? 10.−−BUYING PRESENTS. "Whom do you think I met in town last week, Brother William?" said Uncle Benjamin. "That old skinflint Jorkins. His family had been taking him around buying Christmas presents. He said to me, 'Why cannot the government abolish Christmas, and make the giving of presents punishable by law? I came out this morning with a certain amount of money in my pocket, and I find I have spent just half of it. In fact, if you will believe me, I take home just as many shillings as I had pounds, and half as many pounds as I had shillings. It is monstrous!'" Can you say exactly how much money Jorkins had spent on those presents? 11.−−THE CYCLISTS' FEAST. 'Twas last Bank Holiday, so I've been told, Some cyclists rode abroad in glorious weather. Resting at noon within a tavern old, They all agreed to have a feast together. "Put it all in one bill, mine host," they said, "For every man an equal share will pay." The bill was promptly on the table laid, And four pounds was the reckoning that day. But, sad to state, when they prepared to square, 'Twas found that two had sneaked outside and fled. So, for two shillings more than his due share Each honest man who had remained was bled. They settled later with those rogues, no doubt. How many were they when they first set out? 12.−−A QUEER THING IN MONEY. It will be found that £66, 6s. 6d. equals 15,918 pence. Now, the four 6's added together make 24, and the Amusements in Mathematics 7 figures in 15,918 also add to 24. It is a curious fact that there is only one other sum of money, in pounds, shillings, and pence (all similarly repetitions of one figure), of which the digits shall add up the same as the digits of the amount in pence. What is the other sum of money? 13.−−A NEW MONEY PUZZLE. The largest sum of money that can be written in pounds, shillings, pence, and farthings, using each of the nine digits once and only once, is £98,765, 4s. 3½d. Now, try to discover the smallest sum of money that can be written down under precisely the same conditions. There must be some value given for each denomination−−pounds, shillings, pence, and farthings−−and the nought may not be used. It requires just a little judgment and thought. 14.−−SQUARE MONEY. "This is queer," said McCrank to his friend. "Twopence added to twopence is fourpence, and twopence multiplied by twopence is also fourpence." Of course, he was wrong in thinking you can multiply money by money. The multiplier must be regarded as an abstract number. It is true that two feet multiplied by two feet will make four square feet. Similarly, two pence multiplied by two pence will produce four square pence! And it will perplex the reader to say what a "square penny" is. But we will assume for the purposes of our puzzle that twopence multiplied by twopence is fourpence. Now, what two amounts of money will produce the next smallest possible result, the same in both cases, when added or multiplied in this manner? The two amounts need not be alike, but they must be those that can be paid in current coins of the realm. 15.−−POCKET MONEY. What is the largest sum of money−−all in current silver coins and no four−shilling piece−−that I could have in my pocket without being able to give change for a half−sovereign? 16.−−THE MILLIONAIRE'S PERPLEXITY. Mr. Morgan G. Bloomgarten, the millionaire, known in the States as the Clam King, had, for his sins, more money than he knew what to do with. It bored him. So he determined to persecute some of his poor but happy friends with it. They had never done him any harm, but he resolved to inoculate them with the "source of all evil." He therefore proposed to distribute a million dollars among them and watch them go rapidly to the bad. But he was a man of strange fancies and superstitions, and it was an inviolable rule with him never to make a gift that was not either one dollar or some power of seven−−such as 7, 49, 343, 2,401, which numbers of dollars are produced by simply multiplying sevens together. Another rule of his was that he would never give more than six persons exactly the same sum. Now, how was he to distribute the 1,000,000 dollars? You may distribute the money among as many people as you like, under the conditions given. 17.−−THE PUZZLING MONEY−BOXES. Four brothers−−named John, William, Charles, and Thomas−−had each a money−box. The boxes were all given to them on the same day, and they at once put what money they had into them; only, as the boxes were not very large, they first changed the money into as few coins as possible. After they had done this, they told one another how much money they had saved, and it was found that if John had had 2s. more in his box than at present, if William had had 2s. less, if Charles had had twice as much, and if Thomas had had half as much, they would all have had exactly the same amount. Now, when I add that all four boxes together contained 45s., and that there were only six coins in all in them, it becomes an entertaining puzzle to discover just what coins were in each box. Amusements in Mathematics 8 18.−−THE MARKET WOMEN. A number of market women sold their various products at a certain price per pound (different in every case), and each received the same amount−−2s. 2½d. What is the greatest number of women there could have been? The price per pound in every case must be such as could be paid in current money. 19.−−THE NEW YEAR'S EVE SUPPERS. The proprietor of a small London café has given me some interesting figures. He says that the ladies who come alone to his place for refreshment spend each on an average eighteenpence, that the unaccompanied men spend half a crown each, and that when a gentleman brings in a lady he spends half a guinea. On New Year's Eve he supplied suppers to twenty−five persons, and took five pounds in all. Now, assuming his averages to have held good in every case, how was his company made up on that occasion? Of course, only single gentlemen, single ladies, and pairs (a lady and gentleman) can be supposed to have been present, as we are not considering larger parties. 20.−−BEEF AND SAUSAGES. "A neighbour of mine," said Aunt Jane, "bought a certain quantity of beef at two shillings a pound, and the same quantity of sausages at eighteenpence a pound. I pointed out to her that if she had divided the same money equally between beef and sausages she would have gained two pounds in the total weight. Can you tell me exactly how much she spent?" "Of course, it is no business of mine," said Mrs. Sunniborne; "but a lady who could pay such prices must be somewhat inexperienced in domestic economy." "I quite agree, my dear," Aunt Jane replied, "but you see that is not the precise point under discussion, any more than the name and morals of the tradesman." 21.−−A DEAL IN APPLES. I paid a man a shilling for some apples, but they were so small that I made him throw in two extra apples. I find that made them cost just a penny a dozen less than the first price he asked. How many apples did I get for my shilling? 22.−−A DEAL IN EGGS. A man went recently into a dairyman's shop to buy eggs. He wanted them of various qualities. The salesman had new−laid eggs at the high price of fivepence each, fresh eggs at one penny each, eggs at a halfpenny each, and eggs for electioneering purposes at a greatly reduced figure, but as there was no election on at the time the buyer had no use for the last. However, he bought some of each of the three other kinds and obtained exactly one hundred eggs for eight and fourpence. Now, as he brought away exactly the same number of eggs of two of the three qualities, it is an interesting puzzle to determine just how many he bought at each price. 23.−−THE CHRISTMAS−BOXES. Some years ago a man told me he had spent one hundred English silver coins in Christmas−boxes, giving every person the same amount, and it cost him exactly £1, 10s. 1d. Can you tell just how many persons received the present, and how he could have managed the distribution? That odd penny looks queer, but it is all right. 24.−−A SHOPPING PERPLEXITY. Amusements in Mathematics 9 Two ladies went into a shop where, through some curious eccentricity, no change was given, and made purchases amounting together to less than five shillings. "Do you know," said one lady, "I find I shall require no fewer than six current coins of the realm to pay for what I have bought." The other lady considered a moment, and then exclaimed: "By a peculiar coincidence, I am exactly in the same dilemma." "Then we will pay the two bills together." But, to their astonishment, they still required six coins. What is the smallest possible amount of their purchases−−both different? 25.−−CHINESE MONEY. The Chinese are a curious people, and have strange inverted ways of doing things. It is said that they use a saw with an upward pressure instead of a downward one, that they plane a deal board by pulling the tool toward them instead of pushing it, and that in building a house they first construct the roof and, having raised that into position, proceed to work downwards. In money the currency of the country consists of taels of fluctuating value. The tael became thinner and thinner until 2,000 of them piled together made less than three inches in height. The common cash consists of brass coins of varying thicknesses, with a round, square, or triangular hole in the centre, as in our illustration. [Illustration] These are strung on wires like buttons. Supposing that eleven coins with round holes are worth fifteen ching−changs, that eleven with square holes are worth sixteen ching−changs, and that eleven with triangular holes are worth seventeen ching−changs, how can a Chinaman give me change for half a crown, using no coins other than the three mentioned? A ching−chang is worth exactly twopence and four−fifteenths of a ching−chang. 26.−−THE JUNIOR CLERK'S PUZZLE. Two youths, bearing the pleasant names of Moggs and Snoggs, were employed as junior clerks by a merchant in Mincing Lane. They were both engaged at the same salary−−that is, commencing at the rate of £50 a year, payable half−yearly. Moggs had a yearly rise of £10, and Snoggs was offered the same, only he asked, for reasons that do not concern our puzzle, that he might take his rise at £2, 10s. half−yearly, to which his employer (not, perhaps, unnaturally!) had no objection. Now we come to the real point of the puzzle. Moggs put regularly into the Post Office Savings Bank a certain proportion of his salary, while Snoggs saved twice as great a proportion of his, and at the end of five years they had together saved £268, 15s. How much had each saved? The question of interest can be ignored. 27.−−GIVING CHANGE. Every one is familiar with the difficulties that frequently arise over the giving of change, and how the assistance of a third person with a few coins in his pocket will sometimes help us to set the matter right. Here is an example. An Englishman went into a shop in New York and bought goods at a cost of thirty−four cents. The only money he had was a dollar, a three−cent piece, and a two−cent piece. The tradesman had only a half−dollar and a quarter−dollar. But another customer happened to be present, and when asked to help produced two dimes, a five−cent piece, a two−cent piece, and a one−cent piece. How did the tradesman manage to give change? For the benefit of those readers who are not familiar with the American coinage, it is only necessary to say that a dollar is a hundred cents and a dime ten cents. A puzzle of this kind should rarely cause any difficulty if attacked in a proper manner. 28.−−DEFECTIVE OBSERVATION. Our observation of little things is frequently defective, and our memories very liable to lapse. A certain judge Amusements in Mathematics 10 recently remarked in a case that he had no recollection whatever of putting the wedding−ring on his wife's finger. Can you correctly answer these questions without having the coins in sight? On which side of a penny is the date given? Some people are so unobservant that, although they are handling the coin nearly every day of their lives, they are at a loss to answer this simple question. If I lay a penny flat on the table, how many other pennies can I place around it, every one also lying flat on the table, so that they all touch the first one? The geometrician will, of course, give the answer at once, and not need to make any experiment. He will also know that, since all circles are similar, the same answer will necessarily apply to any coin. The next question is a most interesting one to ask a company, each person writing down his answer on a slip of paper, so that no one shall be helped by the answers of others. What is the greatest number of three−penny−pieces that may be laid flat on the surface of a half−crown, so that no piece lies on another or overlaps the surface of the half−crown? It is amazing what a variety of different answers one gets to this question. Very few people will be found to give the correct number. Of course the answer must be given without looking at the coins. 29.−−THE BROKEN COINS. A man had three coins−−a sovereign, a shilling, and a penny−−and he found that exactly the same fraction of each coin had been broken away. Now, assuming that the original intrinsic value of these coins was the same as their nominal value−−that is, that the sovereign was worth a pound, the shilling worth a shilling, and the penny worth a penny−−what proportion of each coin has been lost if the value of the three remaining fragments is exactly one pound? 30.−−TWO QUESTIONS IN PROBABILITIES. There is perhaps no class of puzzle over which people so frequently blunder as that which involves what is called the theory of probabilities. I will give two simple examples of the sort of puzzle I mean. They are really quite easy, and yet many persons are tripped up by them. A friend recently produced five pennies and said to me: "In throwing these five pennies at the same time, what are the chances that at least four of the coins will turn up either all heads or all tails?" His own solution was quite wrong, but the correct answer ought not to be hard to discover. Another person got a wrong answer to the following little puzzle which I heard him propound: "A man placed three sovereigns and one shilling in a bag. How much should be paid for permission to draw one coin from it?" It is, of course, understood that you are as likely to draw any one of the four coins as another. 31.−−DOMESTIC ECONOMY. Young Mrs. Perkins, of Putney, writes to me as follows: "I should be very glad if you could give me the answer to a little sum that has been worrying me a good deal lately. Here it is: We have only been married a short time, and now, at the end of two years from the time when we set up housekeeping, my husband tells me that he finds we have spent a third of his yearly income in rent, rates, and taxes, one−half in domestic expenses, and one−ninth in other ways. He has a balance of £190 remaining in the bank. I know this last, because he accidentally left out his pass−book the other day, and I peeped into it. Don't you think that a husband ought to give his wife his entire confidence in his money matters? Well, I do; and−−will you believe it?−−he has never told me what his income really is, and I want, very naturally, to find out. Can you tell me what it is from the figures I have given you?" Yes; the answer can certainly be given from the figures contained in Mrs. Perkins's letter. And my readers, if not warned, will be practically unanimous in declaring the income to be−−something absurdly in excess of the correct answer! 32.−−THE EXCURSION TICKET PUZZLE. When the big flaming placards were exhibited at the little provincial railway station, announcing that the