"How very, very sad!" exclaimed Clara; and the eyes of the gentle girl filled with tears as she spoke,
"Sad--but very curious when you come to look at it arithmetically," was her aunt's less romantic reply. "Some of them have lost an arm in their country's service, some a leg, some an ear, some an eye----"
"And some, perhaps, all!" Clara murmured dreamily, as they passed the long rows of weather-beaten heroes basking in the sun. "Did you notice that very old one, with a red face, who was drawing a map in the dust with his wooden leg, and all the others watching? I think it was a plan of a battle----"
"The Battle of Trafalgar, no doubt," her aunt interrupted briskly. "Hardly that, I think," Clara ventured to say. "You see, in that case, he couldn't well be alive----"
"Couldn't well be alive!" the old lady contemptuously repeated. "He's as lively as you and me put together! Why, if drawing a map in the dust--with one's wooden leg--doesn't prove one to be alive, perhaps you'll kindly mention what does prove it!"
Clara did not see her way out of it. Logic had never been her forte.
"To return to the arithmetic," Mad Mathesis resumed --the eccentric old lady never let slip an opportunity of driving her niece into a calculation--"what percentage do you suppose must have lost all four--a leg, an arm, an eye, and an ear?"
"How can I tell?" gasped the terrified girl. She knew well what was coming.
"You ca'n't, of course, without data," her aunt replied: "but I'm just going to give you "
"Give her a Chelsea bun, miss! That's what most young ladies like best!" The voice was rich and musical, and the speaker dexterously whipped back the snowy cloth that covered his basket, and disclosed a tempting array of the familiar square buns, joined together in rows, richly egged and browned and glistening in the sun.
"No, sir! I shall give her nothing so indigestible! Be off!" The old lady waved her parasol threateningly: but nothing seemed to disturb the good humour of the jolly old man, who marched on, chanting his melodious refrain:
"Far too indigestible, my love!" said the old lady. Percentages will agree with you ever so much better!"
Clara sighed, and there was a hungry look in her eyes as she watched the basket lessening in the distance; but she meekly listened to the relentless old lady, who at once proceeded to count off the data on her fingers.
"Say that 70 per cent have lost an eye--75 per cent an ear--80 per cent an arm--85 per cent a leg--that'll do it beautifully. Now, my dear, what percentage, at least, must have lost all four?"
No more conversation occurred unless a smothered exclamation of, "Piping hot!" which escaped from Clara's lips as the basket vanished round a corner could be counted as such--until they reached the old Chelsea mansion, where Clara's father was then staying, with his three sons and their old tutor.
Balbus, Lambert, and Hugh had entered the house only a few minutes before them. They had been out walking, and Hugh had been propounding a difficulty which had reduced Lambert to the depths of gloom, and had even puzzled Balbus.
"It changes from Wednesday to Thursday at midnight, doesn't it?" Hugh had begun.
"Sometimes," said Balbus cautiously.
"Always," said Lambert decisively.
"Sometimes," Balbus gently insisted. "Six midnights out of seven, it changes to some other name."
"I meant, of course," Hugh corrected, "when it does change from Wednesday to Thursday, it does it at midnight--and only at midnight."
"Surely," said Balbus. Lambert was silent.
"Well, now, suppose it's midnight here in Chelsea. Then it's Wednesday west of Chelsea (say in Ireland or America), where midnight hasn't arrived yet: and it's Thursday east of Chelsea (say in Germany or Russia), where midnight has just passed by?"
"Surely," Balbus said again. Even Lambert nodded this time.
"But it isn't midnight anywhere else; so it ca'n't be changing from one day to another anywhere else. And yet, if Ireland and America and so on call it Wednesday, and Germany and Russia and so on call it Thursday, there must be some place--not Chelsea--that has different days on the two sides of it. And the worst of it is, people there get their days in the wrong order: they've Wednesday east of them, and Thursday west--just as if their day had changed from Thursday to Wednesday!"
"I've heard that puzzle before!" cried Lambert. "And I'll tell you the explanation. When a ship goes round world from east to west, we know that it loses a day in its reckoning: so that when it gets home and calls its day Wednesday , it finds people here calling it Thursday, because we've had one more midnight than the ship has had. And when you go the other way round you gain a day."
"I know all that," said Hugh, in reply to this not lucid explanation: "but it doesn't help me, because the ship hasn't proper days. One way round, you get more than twenty-four hours to the day, and the other way you get less: so of course the names get wrong: but people that live on in one place always get twenty-four hours to the day."
"I suppose there is such a place," Balbus said, meditatively, "though I never heard of it, And the people must find it queer, as Hugh says, to have the old day east of them, and the new one west: because, when midnight comes round to them, with the new day in front of it and the old one behind it, one doesn't see exactly what happens. I must think it over."
So they had entered the house in the state I have described--Balbus puzzled, and Lambert buried in gloomy thought.
"Yes, m'm, Master is at home, m'm," said the stately old butler. (N.B.--It is only a butler of experience who can manage a series of three M's together, without any interjacent vowels.) "And the ole party is a-waiting for you in the libery."
"I don't like his calling your father an old party," Mad Mathesis whispered to her niece, as they crossed the hall. And Clara had only just time to whisper in reply, "He meant the whole party," before they were ushered into the library, and the sight of the five solemn faces there assembled chilled her into silence.
Her father sat at the head of the table, and mutely signed to the ladies to take the two vacant chairs, one on each side of him. His three sons and Balbus completed the party. Writing materials had been arranged round the table, after the fashion of a ghostly banquet: the butler had evidently bestowed much thought on the grim device. Sheets of quarto paper, each flanked by a pen on One side and a pencil on the other, represented the plates --penwipers did duty for rolls of bread--while ink-bottles stood in the places usually occupied by wine-glasses. The piece de resistance was a large green baize bag, which gave forth, as the old man restlessly lifted it from side to side, a charming jingle, as of innumerable golden guineas,
"Sister, daughter, sons and Balbus----" the old man began, so nervously that Balbus put in a gentle "Hear, hear!" while Hugh drummed on the table with his fists. This disconcerted the unpractised orator. "Sister--" he began again, then paused a moment, moved the bag to the other side, and went on with a rush, "I mean--this being--a critical occasion--more or less--being the year, when one of my sons comes of age----" he paused again, in some confusion, having evidently got into the middle of his speech sooner than he intended: but it was too late, to go back. "Hear, hear!" cried Balbus. "Quite so," said the old gentleman, recovering his self-possession a little: "when first I began this annual custom--my friend Balbus will correct me if I am wrong--" (Hugh whispered, "With a strap!" but nobody heard him except Lambert, who only frowned and shook his head at him) "--this annual custom of giving each of my sons as many guineas, as would represent his age--it was a critical time--so Balbus informed me--as the ages of two of you were together equal to that of the third--so on that occasion I made a speech---- He paused so long that Balbus thought it well to come to the rescue with the words, "It was a most----" but the old man checked him with a warning look: "yes, made a speech," he repeated. "A few years after that, Balbus pointed out--I say pointed out--" ("Hear, hear!" cried Balbus. "Quite so," said the grateful old man.) "--that it was another critical occasion. The ages of two of you were together double that of the third. So I made another speech--another speech. And now again it's a critical occasion--so Balbus says--and I am making-----" (here Mad Mathesis pointedly referred to her watch) "all the haste I can!" the old man cried, with wonderful presence of mind. "Indeed, sister, I'm coming to the point now! The number of years that have passed since that first occasion is just two-thirds of the numbers of guineas I then gave you. Now, my boys, calculate your ages from the data, and you shall have the money!"
"But we know our ages!" cried Hugh.
"Silence, sir!" thundered the old man, rising to his full height (he was exactly five-foot five) in his indignation. "I say you must use the data only! You mustn't even assume which it is that comes of age!" He clutched the bag as he spoke, and with tottering steps (it was about as much as he could do to carry it) he left the room.
"And you shall have a similar cadeau" the old lady whispered to her niece, "when you've calculated that percentage!" And she followed her brother.
Nothing could exceed the solemnity with which the old couple had risen from the table, and yet was it as it a grin with which the father turned away from his unhappy sons? Could it be--could it be a wink with which the aunt abandoned her despairing niece? And were those-- were those sounds of suppressed chuckling which floated into the room, just before Balbus (who had followed them out) closed the door? Surely not: and yet the butler told the cook--but no-- that was merely idle gossip, and I will not repeat it.
The shades of evening granted their unuttered petition, and "closed not o'er" them (for the butler brought in the lamp): the same obliging shades left them a "lonely bark" (the wail of a dog, in the back-yard, baying the moon) for "a while": but neither "morn, alas", nor any other epoch, seemed likely to "restore" them--to that peace of Mind which had once been theirs ere ever these problems had swooped upon them, and crushed them with a load of unfathomable mystery!
"It's hardly fair," muttered Hugh, "to give us such a jumble as this to work out!"
"Fair?" Clara echoed bitterly. "Well!"
And to all my readers I can but repeat the last words of gentle Clara:
FARE-WELL!
Problem.--If 70 per cent have lost an eye, 75 per cent an ear, 80 per cent an arm, 85 per cent a leg: what percentage, at least, must have lost all four? § 1. The Chelsea Pensioners
Answer.--Ten.
Solution.--(I adopt that of Polar Star, as being better than my own.) Adding the wounds together, we get 70+75+80+85=310, among 100 men; which gives 3 to each, and 4 to 10 men. Therefore the least percentage is 10.
Nineteen answers have been received. One is "5" but, as no working is given with it, it must, in accordance with the rule, remain "a deed without a name". Janet makes it "35 7/10". I am sorry she has misunderstood the question, and has supposed that those who had lost an ear were 75 per cent of those who had lost an eye; and so on. Of course, on this supposition, the percentages must all be multiplied together. This she has done correctly, but I can give her no honours, as I do not think the question will fairly bear her interpretation. Three Score and Ten makes it "19 3/8" Her solution has given me--I will not say "many anxious days and sleepless nights", for I wish to be strictly truthful, but--some trouble in making any sense at all of it. She makes the number of "pensioners wounded once" to be 310 ("per cent," I supposd): dividing by 4, she gets 77 1/2 as "average percentage": again dividing by 4, she gets 19 3/8 as "percentage wounded four times". Does she suppose wounds of different kinds to "absorb" each other, so to speak! Then, no doubt, the data are equivalent to 77 pensioners with one wound each and a half-pensioner with a half-wound. And does she then suppose these concentrated wounds to be transferable, so that 3/4 of these unfortunates can obtain perfect health by handing over their wounds to the remaining 1/4? Granting these suppositions, her answer is right; or rather if the question had been, "A road is covered with one inch of gravel, along 77 1/2 per cent of it. How much of it could be covered 4 inches deep with the same material?" her answer would have been right. But alas, that wasn't the question! Delta makes some most amazing assumptions: "let every one who has not lost an eye have lost an ear," "let every one who has not lost both eyes and ears have lost an arm." Her ideas of a battlefield are grim indeed. Fancy a warrior who would continue fighting after losing both eyes, both ears, and both arms! This is a case which she (or "it "?) evidently considers possible.
Next come eight writers who have made the unwarrantable assumption that, because 70 per cent have lost an eye, therefore 30 per cent have not lost one, so that they have both eyes. This is illogical. If you give me a bag containing 100 sovereigns, and if in an hour I come to you (my face not beaming with gratitude nearly so much as when I received the bag) to say, "I am sorry to tell you that 70 of these sovereigns are bad," do I thereby guarantee the other 30 to be good? Perhaps I have not tested them yet. The sides of this illogical octagon are as follows, in alphabetical order: Algernon Bray, Dinah Mite, G. S. C., Jane E., J. D. W., Magpie (who makes the delightful remark, "Therefore 90 per cent have two of something," recalling to one's memory that fortunate monarch with whom Xerxes was so much pleased that "he gave him ten of everything"!), S. S G., and Tokio.
Bradshaw of the Future and T. R. do the question in a piecemeal fashion--on the principle that the 70 per cent and the 75 per cent, though commenced at opposite ends of the 100, must overlap by at least 45 per cent; and so on. This is quite correct working, but not, I think, quite the best way of doing it.
The other five competitors will, I hope, feel themselves sufficiently glorified by being placed in the first class, without my composing a Triumphal Ode for each!
Class List. I.
Old Cat. Polar Straw. Old Hen. Simple Susan. White Sugar. II.
Bradshaw of the Future. T.R. III.
Algernon Bray. J. D. W. Dinah Mite. Magpie. G. S. C. Jane E. S. S. G. Tokio.
§ 2. Change of Day
I must postpone, sine die, the geographical problem-- partly because I have not yet received the statistics I am hoping for, and partly because I am myself so entirely puzzled by it; and when an examiner is himself dimly hovering between a second class and a third, how is he to decide the position of others?
Problem.--At first, two of the ages are together equal to the third. A few years afterwards, two of them are together double of the third. When the number of years since the first occasion is two-thirds of the sum of the ages On that occasion, one age is 21. What are the other two? § 3. The Son's Ages
Answer.--15 and 18.
Solution. Let the ages at first be x, y, (x + y) Now, if a+b=2c, then (a - n) + (b -n)=2(c - n), whatever be the value of n. Hence the second relationship, if ever true, was always true. Hence it was true at first. But it cannot be true that x and y are together double of (x +y). Hence it must be true of (x +y), together with x or y; and it does not matter which we take. We assume, then, (x +y) +x = 2y,. i.e. y = 2x. Hence the three ages were, at first, x, 2x, 3x, and the number of years since that time is two-thirds of 6x, i.e. is 4x. Hence the present ages are 5x, 6x, 7x. The ages are clearly integers, since this is only "the year when one of my sons comes of age". Hence 7x=21, x=3, and the other ages are 15, 18.
Eighteen answers have been received. One of the writers merely asserts that the first occasion was 12 years ago, that the ages were then 9, 6, and 3; and that on the second occasion they were 14, 11, and 8! As a Roman father, I ought to withhold the name of the rash writer; but respect for age makes me break the rule: it is Three Score and Ten. Jane E. also asserts that the ages at first were 9, 6, 3: then she calculates the present ages, leaving the second occasion unnoticed. Old Hen is nearly as bad; she "tried various numbers till I found one that fitted all the conditions"; but merely scratching up the earth, and pecking about, is not the way to solve a problem, O venerable bird! And close after Old Hen prowls, with hungry eyes, Old Cat, who calmly assumes, to begin with, that the son who comes of age is the eldest. Eat your bird, Puss, for you will get nothing from me!
There are yet two zeroes to dispose of. Minerva assumes that, on every occasion, a son comes of age; and that it is only such a son who is "tipped with gold" Is it wise thus to interpret, "Now, my boys, calculate your ages, and you shall have the money" ? Bradshaw of the Future says "let" the ages at first be 9, 6, 3, then assumes that the second occasion was 6 years afterwards, and on these baseless assumptions brings out the right answers. Guide future travelers, an thou wilt; thou art no Bradshaw for this Age!
Of those who win honours, the merely "honourable" are two. Dinah Mite ascertains (rightly) the relationship between the three ages at first, but then assumes one of them to be "6", thus making the rest of her solution tentative. M. F. C. does the algebra all right up to the conclusion that the present ages are 5z, 6z, and 7z; it then assumes, without giving any reason, that 7z=21.
Of the more honourable, Delta attempts a novelty--to discover which son comes of age by elimination: it assumes, successively, that it is the middle one, and that it is the youngest; and in each case it apparently brings out an absurdity. Still, as the proof contains the following bit of algebra: "63=7x+4y; .'. 21 =x +4/7 of y," I trust it will admit that its proof is not quite conclusive. The rest of its work is good. Magpie betrays the deplorable tendency of her tribe--to appropriate any stray conclusion she comes across, without having any strict logical right to it. Assuming A, B, C, as the ages at first, and E as the number of the years that have elapsed since then, she finds (rightly) the 3 equations, 2A=B, C=B + A, D = 2B. She then says, "Supposing that A=1, then B=2, C=3, and D=4. Therefore for A, B, C, D, four numbers are wanted which shall be to each other as 1:2:3:4." It is in the "therefore" that I detect the unconscientiousness of this bird. The conclusion is true, but this is only because the equations are "homogeneous" (i.e. having one "unknown" in each term), a fact which I strongly suspect had not been grasped--I beg pardon, clawed--by her. Were I to lay this little pitfall: "A+1 =B, B+1 =C; supposing A = 1, then B =2, and C =3. Therefore for A, B, C, three numbers are wanted which shall be to one another as 1:2:3," would you not flutter down into it, O Magpie! as amiably as a Dove? Simple Susan is anything but simple to me. After ascertaining that the 3 ages at first are as 3:2:1, she says, "Then, as two-thirds of their sum, added to one of them, =21, the sum cannot exceed 30, and consequently the highest cannot exceed 15." I suppose her (mental) argument is something like this: "Two-thirds of sum, + one age, =21; .'. sum, +3 halves of one age, =31 1/2. But 3 halves of one age cannot be less than 1 1/2 [Here I perceive that Simple Susan would on no account present a guinea to a newborn baby!]; hence the sum cannot exceed 30." This is ingenious, but her proof, after that, is (as she candidly admits) "clumsy and round-about". She finds that there are 5 possible sets of ages, and eliminates four of them. Suppose that, instead of 5, there had been 5 million possible sets! Would Simple Susan have courageously ordered in the necessary gallon of ink and ream of paper?
The solution sent in by C. R. is, like that of Simple Susan, partly tentative, and so does not rise higher than being Clumsily Right.
Among those who have earned the highest honours, Algernon Bray solves the problem quite correctly, but adds that there is nothing to exclude the supposition that all the ages were fractional. This would make the number of answers infinite. Let me meekly protest that I never intended my readers to devote the rest of their lives to writing out answers! E. M. Rix points out that, if fractional ages be admissible, any one of the three sons might be the one "come of age but she rightly rejects this supposition on the ground that it would make the problem indeterminate. White Sugar is the only one who has detected an oversight of mine: I had forgotten the possibility (which of course ought to be allowed for) that the son who came of age that year, need not have done so by that day, so that he might be only 20. This gives a second solution, viz., 20, 24, 28. Well said, pure Crystal! Verily, thy "fair discourse hath been as sugar"!
Class List. I.
Algernon Bray. S. S. G. An Old Fogey. Tokio. E. M. Rix. T. R. G. S. C. White Sugar. II.
C. R. Magpie. Delta. Simple Susan. III.
Dinah Mite. M. F. C. I have received more than one remonstrance on my assertion, in the Chelsea Pensioners' problem, that it was illogical to assume, from the datum, "70 per cent have lost an eye," that 30 per cent have not. Algernon Bry states, as a paralel case, "Suppose Tommy's father gives him 4 apples, and he eats one of them, how many has he left?" and says, "I think we are justified in answering, 3." I think so too. There is no "must" here, and data are evidently meant to fix the answer exactly: but, if the question were set me, "How many must he have left?" I should understand the data to be that his father gave him 4 at least, but may have given him more.
I take this opportunity of thanking those who have sent, along with their answers to the Tenth Knot, regrets that there are no more Knots to come, or petitions that I should recall my resolution to bring them to an end. I am most grateful for their kind words; but I think it wisest to end what, at best, was but a lame attempt. "The stretched metre of an antique song" is beyond my compass; and my puppets were neither distinctly in my life (like those I now address), nor yet (like Alice and the Mock Turtle) distinctly out of it. Yet let me at least fancy, as I lay down the pen, that I carry with me into my silent life, dear reader, a farewell smile from your unseen face, "and a kindly farewell pressure from your unfelt hand! And so, good night! Parting is such sweet sorrow, that I Shall say "good night!" till it be morrow.
