Amusements in Mathematics by Henry Ernest Dudeney - HTML preview
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AGE AND KINSHIP PUZZLES."The days of our years are threescore years and ten." —Psalm xc. 10.
For centuries it has been a favourite method of propounding arithmetical puzzles to pose them in the form of questions as to the age of an individual. They generally lend themselves to very easy solution by the use of algebra, though often the difficulty lies in stating them correctly. They may be made very complex and may demand considerable ingenuity, but no general laws can well be laid down for their solution. The solver must use his own sagacity. As for puzzles in relationship or kinship, it is quite curious how bewildering many people find these things. Even in ordinary conversation, some statement as to relationship, which is quite clear in the mind of the speaker, will immediately tie the brains of other people into knots. Such expressions as "He is my uncle's son-in-law's sister" convey absolutely nothing to some people without a detailed and laboured explanation. In such cases the best course is to sketch a brief genealogical table, when the eye comes immediately to the assistance of the brain. In these days, when we have a growing lack of respect for pedigrees, most people have got out of the habit of rapidly drawing such tables, which is to be regretted, as they would save a lot of time and brain racking on occasions.40.—MAMMA'S AGE. Tommy: "How old are you, mamma?" Mamma: "Let me think, Tommy. Well, our three ages add up to exactly seventy years." Tommy: "That's a lot, isn't it? And how old are you, papa?" Papa: "Just six times as old as you, my son." Tommy: "Shall I ever be half as old as you, papa?" Papa: "Yes, Tommy; and when that happens our three ages will add up to exactly twice as much as to-day." Tommy: "And supposing I was born before you, papa; and supposing mamma had forgot all about it, and hadn't been at home when I came; and supposing——" Mamma: "Supposing, Tommy, we talk about bed. Come along, darling. You'll have a headache."
Now, if Tommy had been some years older he might have calculated the exact ages of his parents from the information they had given him. Can you find out the exact age of mamma?41.—THEIR AGES.
"My husband's age," remarked a lady the other day, "is represented by the figures of my own age reversed. He is my senior, and the difference between our ages is one-eleventh of their sum."42.—THE FAMILY AGES.
When the Smileys recently received a visit from the favourite uncle, the fond parents had all the five children brought into his presence. First came Billie and little Gertrude, and the uncle was informed that the boy was exactly twice as old as the girl. Then Henrietta arrived, and it was pointed out that the combined ages of herself and Gertrude equalled twice the age of Billie. Then Charlie came running in, and somebody remarked that now the combined ages of the two boys were exactly twice the combined ages of the two girls. The uncle was expressing his astonishment at these coincidences when Janet came in. "Ah! uncle," she exclaimed, "you have actually arrived on my twenty-first birthday!" To this Mr. Smiley added the final staggerer: "Yes, and now the combined ages of the three girls are exactly equal to twice the combined ages of the two boys." Can you give the age of each child?43.—MRS. TIMPKINS'S AGE. Edwin: "Do you know, when the Timpkinses married eighteen years ago Timpkins was three times as old as his wife, and to-day he is just twice as old as she?" Angelina: "Then how old was Mrs. Timpkins on the wedding day?" Can you answer Angelina's question? 44—A CENSUS PUZZLE.
Mr. and Mrs. Jorkins have fifteen children, all born at intervals of one year and a half. Miss Ada Jorkins, the eldest, had an objection to state her age to the census man, but she admitted that she was just seven times older than little Johnnie, the youngest of all. What was Ada's age? Do not too hastily assume that you have solved this little poser. You may find that you have made a bad blunder!45.—MOTHER AND DAUGHTER. "Mother, I wish you would give me a bicycle," said a girl of twelve the other day. "I do not think you are old enough yet, my dear," was the reply. "When I am only three times as old as you are you shall have one." Now, the mother's age is forty-five years. When may the young lady expect to receive her present? 46.—MARY AND MARMADUKE. Marmaduke: "Do you know, dear, that in seven years' time our combined ages will be sixty-three years?" Mary: "Is that really so? And yet it is a fact that when you were my present age you were twice as old as I was then. I worked it out last night." Now, what are the ages of Mary and Marmaduke?47—ROVER'S AGE. "Now, then, Tommy, how old is Rover?" Mildred's young man asked her brother. "Well, five years ago," was the youngster's reply, "sister was four times older than the dog, but now she is only three times as old." Can you tell Rover's age?48.—CONCERNING TOMMY'S AGE.
Tommy Smart was recently sent to a new school. On the first day of his arrival the teacher asked him his age, and this was his curious reply: "Well, you see, it is like this. At the time I was born—I forget the year—my only sister, Ann, happened to be just onequarter the age of mother, and she is now one-third the age of father." "That's all very well," said the teacher, "but what I want is not the age of your sister Ann, but your own age." "I was just coming to that," Tommy answered; "I am just a quarter of mother's present age, and in four years' time I shall be a quarter the age of father. Isn't that funny?"This was all the information that the teacher could get out of Tommy Smart. Could you have told, from these facts, what was his precise age? It is certainly a little puzzling. 49.—NEXT-DOOR NEIGHBOURS.
There were two families living next door to one another at Tooting Bec—the Jupps and the Simkins. The united ages of the four Jupps amounted to one hundred years, and the united ages of the Simkins also amounted to the same. It was found in the case of each family that the sum obtained by adding the squares of each of the children's ages to the square of the mother's age equalled the square of the father's age. In the case of the Jupps, however, Julia was one year older than her brother Joe, whereas Sophy Simkin was two years older than her brother Sammy. What was the age of each of the eight individuals?50.—THE BAG OF NUTS.
Three boys were given a bag of nuts as a Christmas present, and it was agreed that they should be divided in proportion to their ages, which together amounted to 17½ years. Now the bag contained 770 nuts, and as often as Herbert took four Robert took three, and as often as Herbert took six Christopher took seven. The puzzle is to find out how many nuts each had, and what were the boys' respective ages.51.—HOW OLD WAS MARY? Here is a funny little age problem, by the late Sam Loyd, which has been very popular in the United States. Can you unravel the mystery?
The combined ages of Mary and Ann are forty-four years, and Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann. How old is Mary? That is all, but can you work it out? If not, ask your friends to help you, and watch the shadow of bewilderment creep over their faces as they attempt to grip the intricacies of the question.52.—QUEER RELATIONSHIPS.
"Speaking of relationships," said the Parson at a certain dinner-party, "our legislators are getting the marriage law into a frightful tangle, Here, for example, is a puzzling case that has come under my notice. Two brothers married two sisters. One man died and the other man's wife also died. Then the survivors married.""The man married his deceased wife's sister under the recent Act?" put in the Lawyer.
"Exactly. And therefore, under the civil law, he is legally married and his child is legitimate. But, you see, the man is the woman's deceased husband's brother, and therefore, also under the civil law, she is not married to him and her child is illegitimate.""He is married to her and she is not married to him!" said the Doctor. "Quite so. And the child is the legitimate son of his father, but the illegitimate son of his mother."
"Undoubtedly 'the law is a hass,'" the Artist exclaimed, "if I may be permitted to say so," he added, with a bow to the Lawyer.
"Certainly," was the reply. "We lawyers try our best to break in the beast to the service of man. Our legislators are responsible for the breed."
"No; but I will explain that later. Well, this man has a sister of his own. Their names are Stephen Brown and Jane Brown. Last week a young fellow turned up whom Stephen introduced to me as his nephew. Naturally, I spoke of Jane as his aunt, but, to my astonishment, the youth corrected me, assuring me that, though he was the nephew of Stephen, he was not the nephew of Jane, the sister of Stephen. This perplexed me a good deal, but it is quite correct."The Lawyer was the first to get at the heart of the mystery. What was his solution? 53.—HEARD ON THE TUBE RAILWAY. First Lady: "And was he related to you, dear?" Second Lady: "Oh, yes. You see, that gentleman's mother was my mother's mother-inlaw, but he is not on speaking terms with my papa." First Lady: "Oh, indeed!" (But you could see that she was not much wiser.) How was the gentleman related to the Second Lady?54.—A FAMILY PARTY.
A certain family party consisted of 1 grandfather, 1 grandmother, 2 fathers, 2 mothers, 4
children, 3 grandchildren, 1 brother, 2 sisters, 2 sons, 2 daughters, 1 father-in-law, 1
mother-in-law, and 1 daughter-in-law. Twenty-three people, you will say. No; there were only seven persons present. Can you show how this might be?
Joseph Bloggs: "I can't follow it, my dear boy. It makes me dizzy!" John Snoggs: "It's very simple. Listen again! You happen to be my father's brother-inlaw, my brother's father-in-law, and also my father-in-law's brother. You see, my father was——"But Mr. Bloggs refused to hear any more. Can the reader show how this extraordinary triple relationship might have come about?56.—WILSON'S POSER. "Speaking of perplexities——" said Mr. Wilson, throwing down a magazine on the table in the commercial room of the Railway Hotel. "Who was speaking of perplexities?" inquired Mr. Stubbs. "Well, then, reading about them, if you want to be exact—it just occurred to me that perhaps you three men may be interested in a little matter connected with myself."
It was Christmas Eve, and the four commercial travellers were spending the holiday at Grassminster. Probably each suspected that the others had no homes, and perhaps each was conscious of the fact that he was in that predicament himself. In any case they seemed to be perfectly comfortable, and as they drew round the cheerful fire the conversation became general."What is the difficulty?" asked Mr. Packhurst.
"There's no difficulty in the matter, when you rightly understand it. It is like this. A man named Parker had a flying-machine that would carry two. He was a venturesome sort of chap—reckless, I should call him—and he had some bother in finding a man willing to risk his life in making an ascent with him. However, an uncle of mine thought he would chance it, and one fine morning he took his seat in the machine and she started off well. When they were up about a thousand feet, my nephew suddenly——""Here, stop, Wilson! What was your nephew doing there? You said your uncle," interrupted Mr. Stubbs. "Did I? Well, it does not matter. My nephew suddenly turned to Parker and said that the engine wasn't running well, so Parker called out to my uncle——" "Look here," broke in Mr. Waterson, "we are getting mixed. Was it your uncle or your nephew? Let's have it one way or the other." "What I said is quite right. Parker called out to my uncle to do something or other, when my nephew——"
"There you are again, Wilson," cried Mr. Stubbs; "once for all, are we to understand that both your uncle and your nephew were on the machine?"
"Certainly. I thought I made that clear. Where was I? Well, my nephew shouted back to Parker——"
"Wilson, I have known you for some time as a truthful man and a temperate man," said Mr. Stubbs, solemnly. "But I am afraid since you took up that new line of goods you have overworked yourself."
"Half a minute, Stubbs," interposed Mr. Waterson. "I see clearly where we all slipped a cog. Of course, Wilson, you meant us to understand that Parker is either your uncle or your nephew. Now we shall be all right if you will just tell us whether Parker is your uncle or nephew.""He is no relation to me whatever."
The three men sighed and looked anxiously at one another. Mr. Stubbs got up from his chair to reach the matches, Mr. Packhurst proceeded to wind up his watch, and Mr. Waterson took up the poker to attend to the fire. It was an awkward moment, for at the season of goodwill nobody wished to tell Mr. Wilson exactly what was in his mind.
"It's curious," said Mr. Wilson, very deliberately, "and it's rather sad, how thick-headed some people are. You don't seem to grip the facts. It never seems to have occurred to either of you that my uncle and my nephew are one and the same man.""What!" exclaimed all three together.
"Yes; David George Linklater is my uncle, and he is also my nephew. Consequently, I am both his uncle and nephew. Queer, isn't it? I'll explain how it comes about." Mr. Wilson put the case so very simply that the three men saw how it might happen without any marriage within the prohibited degrees. Perhaps the reader can work it out for himself.