Section 2 of 27 - Table of Contents
36.--BUYING APPLES.
As the purchase of apples in small quantities has always presented
considerable difficulties, I think it well to offer a few remarks on
this subject. We all know the story of the smart boy who, on being told
by the old woman that she was selling her apples at four for threepence,
said: "Let me see! Four for threepence; that's three for twopence, two
for a penny, one for nothing--I'll take _one_!"
There are similar cases of perplexity. For example, a boy once picked up
a penny apple from a stall, but when he learnt that the woman's pears
were the same price he exchanged it, and was about to walk off. "Stop!"
said the woman. "You haven't paid me for the pear!" "No," said the boy,
"of course not. I gave you the apple for it." "But you didn't pay for
the apple!" "Bless the woman! You don't expect me to pay for the apple
and the pear too!" And before the poor creature could get out of the
tangle the boy had disappeared.
Then, again, we have the case of the man who gave a boy sixpence and
promised to repeat the gift as soon as the youngster had made it into
ninepence. Five minutes later the boy returned. "I have made it into
ninepence," he said, at the same time handing his benefactor threepence.
"How do you make that out?" he was asked. "I bought threepennyworth of
apples." "But that does not make it into ninepence!" "I should rather
think it did," was the boy's reply. "The apple woman has threepence,
hasn't she? Very well, I have threepennyworth of apples, and I have just
given you the other threepence. What's that but ninepence?"
I cite these cases just to show that the small boy really stands in need
of a little instruction in the art of buying apples. So I will give a
simple poser dealing with this branch of commerce.
An old woman had apples of three sizes for sale--one a penny, two a
penny, and three a penny. Of course two of the second size and three of
the third size were respectively equal to one apple of the largest size.
Now, a gentleman who had an equal number of boys and girls gave his
children sevenpence to be spent amongst them all on these apples. The
puzzle is to give each child an equal distribution of apples. How was
the sevenpence spent, and how many children were there?
37.--BUYING CHESTNUTS.
Though the following little puzzle deals with the purchase of chestnuts,
it is not itself of the "chestnut" type. It is quite new. At first sight
it has certainly the appearance of being of the "nonsense puzzle"
character, but it is all right when properly considered.
A man went to a shop to buy chestnuts. He said he wanted a pennyworth,
and was given five chestnuts. "It is not enough; I ought to have a
sixth," he remarked! "But if I give you one chestnut more." the shopman
replied, "you will have five too many." Now, strange to say, they were
both right. How many chestnuts should the buyer receive for half a
crown?
38.--THE BICYCLE THIEF.
Here is a little tangle that is perpetually cropping up in various
guises. A cyclist bought a bicycle for L15 and gave in payment a cheque
for L25. The seller went to a neighbouring shopkeeper and got him to
change the cheque for him, and the cyclist, having received his L10
change, mounted the machine and disappeared. The cheque proved to be
valueless, and the salesman was requested by his neighbour to refund the
amount he had received. To do this, he was compelled to borrow the L25
from a friend, as the cyclist forgot to leave his address, and could not
be found. Now, as the bicycle cost the salesman L11, how much money did
he lose altogether?
39.--THE COSTERMONGER'S PUZZLE.
"How much did yer pay for them oranges, Bill?"
"I ain't a-goin' to tell yer, Jim. But I beat the old cove down
fourpence a hundred."
"What good did that do yer?"
"Well, it meant five more oranges on every ten shillin's-worth."
Now, what price did Bill actually pay for the oranges? There is only one
rate that will fit in with his statements.
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 one-quarter 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 171/2 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."
"And this reminds me," went on the Parson, "of a man in my parish who
married the sister of his widow. This man--"
"Stop a moment, sir," said the Professor. "Married the sister of his
widow? Do you marry dead men in your parish?"
"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-in-law, 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?
55.--A MIXED PEDIGREE.
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-in-law, 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--"
"Phew! I'm sorry to interrupt you again, Wilson, but we can't get on
like this. Is it true that the machine would only carry two?"
"Of course. I said at the start that it only carried two."
"Then what in the name of aerostation do you mean by saying that there
were three persons on board?" shouted Mr. Stubbs.
"Who said there were three?"
"You have told us that Parker, your uncle, and your nephew went up on
this blessed flying-machine."
"That's right."
"And the thing would only carry two!"
"Right again."
"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.
CLOCK PUZZLES.
"Look at the clock!"
_Ingoldsby Legends_.
In considering a few puzzles concerning clocks and watches, and the
times recorded by their hands under given conditions, it is well that a
particular convention should always be kept in mind. It is frequently
the case that a solution requires the assumption that the hands can
actually record a time involving a minute fraction of a second. Such a
time, of course, cannot be really indicated. Is the puzzle, therefore,
impossible of solution? The conclusion deduced from a logical syllogism
depends for its truth on the two premises assumed, and it is the same in
mathematics. Certain things are antecedently assumed, and the answer
depends entirely on the truth of those assumptions.
"If two horses," says Lagrange, "can pull a load of a certain weight, it
is natural to suppose that four horses could pull a load of double that
weight, six horses a load of three times that weight. Yet, strictly
speaking, such is not the case. For the inference is based on the
assumption that the four horses pull alike in amount and direction,
which in practice can scarcely ever be the case. It so happens that we
are frequently led in our reckonings to results which diverge widely
from reality. But the fault is not the fault of mathematics; for
mathematics always gives back to us exactly what we have put into it.
The ratio was constant according to that supposition. The result is
founded upon that supposition. If the supposition is false the result is
necessarily false."
If one man can reap a field in six days, we say two men will reap it in
three days, and three men will do the work in two days. We here assume,
as in the case of Lagrange's horses, that all the men are exactly
equally capable of work. But we assume even more than this. For when
three men get together they may waste time in gossip or play; or, on the
other hand, a spirit of rivalry may spur them on to greater diligence.
We may assume any conditions we like in a problem, provided they be
clearly expressed and understood, and the answer will be in accordance
with those conditions.
57.--WHAT WAS THE TIME?
"I say, Rackbrane, what is the time?" an acquaintance asked our friend
the professor the other day. The answer was certainly curious.
"If you add one quarter of the time from noon till now to half the time
from now till noon to-morrow, you will get the time exactly."
What was the time of day when the professor spoke?
58.--A TIME PUZZLE.
How many minutes is it until six o'clock if fifty minutes ago it was
four times as many minutes past three o'clock?
59.--A PUZZLING WATCH.
A friend pulled out his watch and said, "This watch of mine does not
keep perfect time; I must have it seen to. I have noticed that the
minute hand and the hour hand are exactly together every sixty-five
minutes." Does that watch gain or lose, and how much per hour?
60.--THE WAPSHAW'S WHARF MYSTERY.
There was a great commotion in Lower Thames Street on the morning of
January 12, 1887. When the early members of the staff arrived at
Wapshaw's Wharf they found that the safe had been broken open, a
considerable sum of money removed, and the offices left in great
disorder. The night watchman was nowhere to be found, but nobody who had
been acquainted with him for one moment suspected him to be guilty of
the robbery. In this belief the proprietors were confirmed when, later
in the day, they were informed that the poor fellow's body had been
picked up by the River Police. Certain marks of violence pointed to the
fact that he had been brutally attacked and thrown into the river. A
watch found in his pocket had stopped, as is invariably the case in such
circumstances, and this was a valuable clue to the time of the outrage.
But a very stupid officer (and we invariably find one or two stupid
individuals in the most intelligent bodies of men) had actually amused
himself by turning the hands round and round, trying to set the watch
going again. After he had been severely reprimanded for this serious
indiscretion, he was asked whether he could remember the time that was
indicated by the watch when found. He replied that he could not, but he
recollected that the hour hand and minute hand were exactly together,
one above the other, and the second hand had just passed the forty-ninth
second. More than this he could not remember.
What was the exact time at which the watchman's watch stopped? The watch
is, of course, assumed to have been an accurate one.
61.--CHANGING PLACES.
[Illustration]
The above clock face indicates a little before 42 minutes past 4. The
hands will again point at exactly the same spots a little after 23
minutes past 8. In fact, the hands will have changed places. How many
times do the hands of a clock change places between three o'clock p.m.
and midnight? And out of all the pairs of times indicated by these
changes, what is the exact time when the minute hand will be nearest to
the point IX?
62.--THE CLUB CLOCK.
One of the big clocks in the Cogitators' Club was found the other night
to have stopped just when, as will be seen in the illustration, the
second hand was exactly midway between the other two hands. One of the
members proposed to some of his friends that they should tell him the
exact time when (if the clock had not stopped) the second hand would
next again have been midway between the minute hand and the hour hand.
Can you find the correct time that it would happen?
[Illustration]
63.--THE STOP-WATCH.
[Illustration]
We have here a stop-watch with three hands. The second hand, which
travels once round the face in a minute, is the one with the little ring
at its end near the centre. Our dial indicates the exact time when its
owner stopped the watch. You will notice that the three hands are nearly
equidistant. The hour and minute hands point to spots that are exactly a
third of the circumference apart, but the second hand is a little too
advanced. An exact equidistance for the three hands is not possible.
Now, we want to know what the time will be when the three hands are next
at exactly the same distances as shown from one another. Can you state
the time?
64.--THE THREE CLOCKS.
On Friday, April 1, 1898, three new clocks were all set going precisely
at the same time--twelve noon. At noon on the following day it was found
that clock A had kept perfect time, that clock B had gained exactly one
minute, and that clock C had lost exactly one minute. Now, supposing
that the clocks B and C had not been regulated, but all three allowed to
go on as they had begun, and that they maintained the same rates of
progress without stopping, on what date and at what time of day would
all three pairs of hands again point at the same moment at twelve
o'clock?
65.--THE RAILWAY STATION CLOCK.
A clock hangs on the wall of a railway station, 71 ft. 9 in. long and 10
ft. 4 in. high. Those are the dimensions of the wall, not of the clock!
While waiting for a train we noticed that the hands of the clock were
pointing in opposite directions, and were parallel to one of the
diagonals of the wall. What was the exact time?
66.--THE VILLAGE SIMPLETON.
A facetious individual who was taking a long walk in the country came
upon a yokel sitting on a stile. As the gentleman was not quite sure of
his road, he thought he would make inquiries of the local inhabitant;
but at the first glance he jumped too hastily to the conclusion that he
had dropped on the village idiot. He therefore decided to test the
fellow's intelligence by first putting to him the simplest question he
could think of, which was, "What day of the week is this, my good man?"
The following is the smart answer that he received:--
"When the day after to-morrow is yesterday, to-day will be as far from
Sunday as to-day was from Sunday when the day before yesterday was
to-morrow."
Can the reader say what day of the week it was? It is pretty evident
that the countryman was not such a fool as he looked. The gentleman went
on his road a puzzled but a wiser man.
LOCOMOTION AND SPEED PUZZLES.
"The race is not to the swift."--_Ecclesiastes_ ix. II.
67.--AVERAGE SPEED.
In a recent motor ride it was found that we had gone at the rate of ten
miles an hour, but we did the return journey over the same route, owing
to the roads being more clear of traffic, at fifteen miles an hour. What
was our average speed? Do not be too hasty in your answer to this simple
little question, or it is pretty certain that you will be wrong.
68.--THE TWO TRAINS.
I put this little question to a stationmaster, and his correct answer
was so prompt that I am convinced there is no necessity to seek talented
railway officials in America or elsewhere.
Two trains start at the same time, one from London to Liverpool, the
other from Liverpool to London. If they arrive at their destinations one
hour and four hours respectively after passing one another, how much
faster is one train running than the other?
69.--THE THREE VILLAGES.
I set out the other day to ride in a motor-car from Acrefield to
Butterford, but by mistake I took the road going _via_ Cheesebury, which
is nearer Acrefield than Butterford, and is twelve miles to the left of
the direct road I should have travelled. After arriving at Butterford I
found that I had gone thirty-five miles. What are the three distances
between these villages, each being a whole number of miles? I may
mention that the three roads are quite straight.
70.--DRAWING HER PENSION.
"Speaking of odd figures," said a gentleman who occupies some post in a
Government office, "one of the queerest characters I know is an old lame
widow who climbs up a hill every week to draw her pension at the village
post office. She crawls up at the rate of a mile and a half an hour and
comes down at the rate of four and a half miles an hour, so that it
takes her just six hours to make the double journey. Can any of you tell
me how far it is from the bottom of the hill to the top?"
[Illustration]
71.--SIR EDWYN DE TUDOR.
In the illustration we have a sketch of Sir Edwyn de Tudor going to
rescue his lady-love, the fair Isabella, who was held a captive by a
neighbouring wicked baron. Sir Edwyn calculated that if he rode fifteen
miles an hour he would arrive at the castle an hour too soon, while if
he rode ten miles an hour he would get there just an hour too late. Now,
it was of the first importance that he should arrive at the exact time
appointed, in order that the rescue that he had planned should be a
success, and the time of the tryst was five o'clock, when the captive
lady would be taking her afternoon tea. The puzzle is to discover
exactly how far Sir Edwyn de Tudor had to ride.
72.--THE HYDROPLANE QUESTION.
The inhabitants of Slocomb-on-Sea were greatly excited over the visit of
a certain flying man. All the town turned out to see the flight of the
wonderful hydroplane, and, of course, Dobson and his family were there.
Master Tommy was in good form, and informed his father that Englishmen
made better airmen than Scotsmen and Irishmen because they are not so
heavy. "How do you make that out?" asked Mr. Dobson. "Well, you see,"
Tommy replied, "it is true that in Ireland there are men of Cork and in
Scotland men of Ayr, which is better still, but in England there are
lightermen." Unfortunately it had to be explained to Mrs. Dobson, and
this took the edge off the thing. The hydroplane flight was from Slocomb
to the neighbouring watering-place Poodleville--five miles distant. But
there was a strong wind, which so helped the airman that he made the
outward journey in the short time of ten minutes, though it took him an
hour to get back to the starting point at Slocomb, with the wind dead
against him. Now, how long would the ten miles have taken him if there
had been a perfect calm? Of course, the hydroplane's engine worked
uniformly throughout.
73.--DONKEY RIDING.
During a visit to the seaside Tommy and Evangeline insisted on having a
donkey race over the mile course on the sands. Mr. Dobson and some of
his friends whom he had met on the beach acted as judges, but, as the
donkeys were familiar acquaintances and declined to part company the
whole way, a dead heat was unavoidable. However, the judges, being
stationed at different points on the course, which was marked off in
quarter-miles, noted the following results:--The first three-quarters
were run in six and three-quarter minutes, the first half-mile took the
same time as the second half, and the third quarter was run in exactly
the same time as the last quarter. From these results Mr. Dobson amused
himself in discovering just how long it took those two donkeys to run
the whole mile. Can you give the answer?
74.--THE BASKET OF POTATOES.
A man had a basket containing fifty potatoes. He proposed to his son, as
a little recreation, that he should place these potatoes on the ground
in a straight line. The distance between the first and second potatoes
was to be one yard, between the second and third three yards, between
the third and fourth five yards, between the fourth and fifth seven
yards, and so on--an increase of two yards for every successive potato
laid down. Then the boy was to pick them up and put them in the basket
one at a time, the basket being placed beside the first potato. How far
would the boy have to travel to accomplish the feat of picking them all
up? We will not consider the journey involved in placing the potatoes,
so that he starts from the basket with them all laid out.
75.--THE PASSENGER'S FARE.
At first sight you would hardly think there was matter for dispute in
the question involved in the following little incident, yet it took the
two persons concerned some little time to come to an agreement. Mr.
Smithers hired a motor-car to take him from Addleford to Clinkerville
and back again for L3. At Bakenham, just midway, he picked up an
acquaintance, Mr. Tompkins, and agreed to take him on to Clinkerville
and bring him back to Bakenham on the return journey. How much should he
have charged the passenger? That is the question. What was a reasonable
fare for Mr. Tompkins?
DIGITAL PUZZLES.
"Nine worthies were they called."
DRYDEN: _The Flower and the Leaf._
I give these puzzles, dealing with the nine digits, a class to
themselves, because I have always thought that they deserve more
consideration than they usually receive. Beyond the mere trick of
"casting out nines," very little seems to be generally known of the laws
involved in these problems, and yet an acquaintance with the properties
of the digits often supplies, among other uses, a certain number of
arithmetical checks that are of real value in the saving of labour. Let
me give just one example--the first that occurs to me.
If the reader were required to determine whether or not
15,763,530,163,289 is a square number, how would he proceed? If the
number had ended with a 2, 3, 7, or 8 in the digits place, of course he
would know that it could not be a square, but there is nothing in its
apparent form to prevent its being one. I suspect that in such a case he
would set to work, with a sigh or a groan, at the laborious task of
extracting the square root. Yet if he had given a little attention to
the study of the digital properties of numbers, he would settle the
question in this simple way. The sum of the digits is 59, the sum of
which is 14, the sum of which is 5 (which I call the "digital root"),
and therefore I know that the number cannot be a square, and for this
reason. The digital root of successive square numbers from 1 upwards is
always 1, 4, 7, or 9, and can never be anything else. In fact, the
series, 1, 4, 9, 7, 7, 9, 4, 1, 9, is repeated into infinity. The
analogous series for triangular numbers is 1, 3, 6, 1, 6, 3, 1, 9, 9. So
here we have a similar negative check, for a number cannot be triangular
(that is, (n squared + n)/2) if its digital root be 2, 4, 5, 7, or 8.
Prev
Next
All
Printer Friendly Version | Send this page to a friend | Discuss this Book
Your email address is safe with us. View our Privacy policy.
 |
 |
|
|
The Adventures of Tom Sawyer Sections: 35 What's this? Table of Contents |
 FictionShort StoriesPoetryPlaysSci FiPhilosophyReligionBiography |
Body Mass
