191.Billy and Tommy, two aboriginals, killed a kangaroo in the bush, and began quarrelling over the weight of the animal. They had no proper means of weighing it, but, knowing their own weights, Billy 130 lbs. and Tommy 190 lbs., they placed a log of wood across a stump so that it balanced with one on each end. They then exchanged places, and, the lighter man taking the kangaroo on his knees, the log again balanced. What was the weight of the kangaroo?
192.A son asked his father how old he was, and received the following answer: “Your age is now one quarter of mine, but five years ago it was only one-fifth.” How old is the father?
193.Place three sixes together so as to make seven.
THE PASSING TRAINS PUZZLE.
194.If through passenger trains running to and from New York and San Francisco daily start at the same hour from each place (difference of longitude not being considered) and take the same time—seven days—for the trip, how many such trains coming in an opposite direction will a train leaving New York meet before it arrives at San Francisco?
THE SCHOOL-TEACHER “CAUGHT.”
Two of our Public Schools were engaged playing a football match one afternoon. The head master of one of them had generously given the boys a half-holiday; but the gentleman who held the same capacity in the other school, not being an ardent admirer of Australia’s national game, refused to do so. When school assembled in the afternoon, a boy volunteered to ask the master for the desired holiday. When the question was put, he firmly answered, “No, no!” whereupon the bright youth called out: “Hurrah! we have our holiday; two negatives make an affirmative.” The teacher was so pleased at the boy’s sharpness that he dismissed the school right away.
195.A man arrives at the railway station nearest to his home 1½ hours before the time at which he had ordered his carriage to meet him. He sets out at once to walk at the rate of four miles an hour, and, meeting his carriage when it had travelled eight miles, reaches home exactly one hour earlier than he had originally expected. How far was his house from the station, and at what rate was his carriage driven?
“OFF THE TRACK.”
196.A man starts to walk from a town, A, to a town B, a distance by road of 16 miles, at the rate of 4 miles an hour. There is a point C on the road, at which the road to B leads away to the right, and another road at right-angles to this latter goes to the left, “to no place in particular.” The unwary traveller gets on to this left hand road, and is walking for 2¼ hours since he left A, before he finds out his mistake, and he resolves not to go back to the junction, which is five miles away, but makes straight across the bush to B, and strikes it exactly. How long did it take to go from A to B?
GAMBLING.
197.Three friends, A, B, and C, sit down to play cards. As a result of the first game, A lost to each of B and C as much money as they started to play with; the result of the second game B lost similarly to each of A and C; and in the third, C lost similarly to each of A and B;—and they then had 24s. each. What had they each at first?
This Sticks Them Up.
198.A, who is a dealer in horses, sells one to B for £55. B very soon discovers that he does not require the animal, and sells him back to A for £50. Now, A is not long in finding another customer for the horse: he sells it to C for £60. How much money does A make out of this transaction?
This question has been the cause of endless discussion and argument.
It might be as well to state that when A first sold the horse to B he neither made nor lost any money by the deal.
SCRIPTURAL FINANCE.
199.What is the earliest banking transaction mentioned in the Bible? The answer generally given to this is, “The check which Pharaoh received on the banks of the Red Sea, crossed by Moses & Co.” There is still an earlier instance: see if you can find it out.
200.How much tea at 6s. per lb. must be mixed with 12 lbs. at 3s. 8d. per lb. so that the mixture may be worth 4s. 4d. per lb.?
201.Place 17 little sticks—matches, for instance—making six equal squares, as in the margin, then remove five sticks and leave three perfect squares of the same size.
FOR THE JEWELLER.
202.How much gold of 21 and 23 carats must be mixed with 30 oz of 20 carats, so that the mixture may be 22 carats?
LONDON GRAMMAR.
Three cockneys, being out one evening in a dense fog, came up to a building that they thus described. The first said, “There’s anouse.” “No,” said the second, “It’s anut.” The third exclaimed “You’re both wrong; it’s anin!”
203.A draper sold 12 yards of cloth at 20s. per yard, and lost 10 per cent. What was the prime cost?
204.A jockey, on a horse galloping at the rate of 18 miles an hour on the Flemington racecourse, passes in 30 minutes over the diameter and curve of a semi-circle. What area does he enclose by the ride?
205.How many trees 20 feet apart cover an acre?
“Multiplication is vexation,Division is as bad.The rule of three, it puzzles me,And fractions drive me mad.”
“Multiplication is vexation,Division is as bad.The rule of three, it puzzles me,And fractions drive me mad.”
MULTIPLY £19 19s. 11¾d. BY £19 19s. 11¾d.
This very old question is continually cropping up, and will continue to do so as long as men are able to reckon. The answer generally given is £399 19s. 2d. and a fraction, and the method of working it out as follows:—
£19 19s. 11¾d. = 19199 farthings.
Many adopt the following method:—
£20 x £20 = £400
It would be possible to adopt other methods, each of which would give a different result.
Properly speaking,this sum cannot be done.
Multiplication is merely a contracted form of addition: it means taking a number or quantity a certain number of times. Every multiplication can be proved by addition. All numbers areabstractorconcrete—3 is abstract, £3 is concrete.
Two abstract numbers can be multiplied together—as, 4 times 3 = 12.
One abstract number and one concrete number can be multiplied together—as 2s. multiplied by 3 = 6s.
Two concrete numbers cannot be multiplied together.
In the example just given, 2s. multiplied by 3, we see it simply means to write down 2s. three times, and by addition we discover the answer to be 6s. Suppose the reader lent a friend 2s. on Monday, 2s. on Tuesday, and 2s. on Wednesday, he has lent 2s. three times, making 6s. lent in all.
Now, we will attempt to multiply 2s. by 3s., but it is impossible to comprehend how many times is 3s. times. The answer to 2s. x 3s. usually given is 6s. On the same lines, we multiply 9d. by 10d., and our answer is—90d., that is 7s. 6d.—a greater product than 2s. multiplied by 3s.
Although it is stated that two concrete numbers cannot be multiplied together, it should be borne in mind that we can multiply yards, feet, and inches, by yards, feet, and inches (length by breadth), which will result in square or cubic measure: 12 inches make 1 foot, and 3 feet make one yard, 144 square inches make 1 square foot, &c. 12 pence make 1 shilling, but how many square pence make 1 square shilling?
The argument generally brought forward in favour of the performance of this problem is, that when the Rule of Three is applied to financial questions (such as interests, &c.) money is multiplied by money.
Example.—If the interest on £10 is 15s., what is the interest on £20?
As £10 : £20 :: 15s. :x
The multiplication in the above is in appearance only, for all we get in the Rule of Three is the ratio between the sums of money and this ratio is an abstract number, and not concrete. On examination we find the ratio between £10 and £20; that the latter is double, ortwotimes as much as the former, and not £2 times more than it.
We extend a general invitation to all our readers who hold a different opinion to multiply three pints of Dewar’s Whisky by 6 quarts of soda-water, but in case they might plead inability to perform this little feat, on conscientious grounds, we will extend the invitation to three cups of tea by six spoonfuls of sugar. And if any of them have a few pounds (say £10) in the Savings Bank we would advise “Don’taddany more deposits, but wait till you have £2, then proceed to the bank and multiply the £10 by the £2, and prove to the teller that you have £20 to your account. Be careful to take no less a sum than £2, or the result might be a little surprising, for if you take only £1, the teller might argue after he has received your sovereign that “ten ones are ten,” and then your £10 would remain the same.”
206.What is the difference between six dozen dozen and half a dozen dozen?
A TELL-TALE TABLE.
There is a good deal of amusement in the following table. It will enable you to tell how old the young ladies are. Ask a young lady to tell you in which column or columns her age is found, add together the figures at the top of the columns in which she says her age is, and you have the secret. Suppose a young lady is 19. You will find that number in the first, second and fifth columns; add the first figures of these columns—1, 2 and 16—and you get the age.
COIN PUZZLE.
207.Place four florins alternately with four pennies, and in four moves, moving two adjacent coins each time, bring the florins together and the pence together. When finished there must be no spaces between the coins.
208.If 2 be added to the numerator of a certain fraction, it is made equal to one-fifth, whilst if 2 be taken from the denominator it becomes equal to one-sixth. Find the fraction.
EUCLID.—The Famous Forty-Seventh.
Fig. 1.
Fig. 1.
“In any right-angled triangle, the square which is described upon the side opposite to the right-angle is equal to the squares described upon the sides which contain the right-angle.”
Here is a simple way of proving this proposition. Although perhaps not exactly scholastic, it is none the less interesting.
Draw an exact square, whose sides measure 7 in.; then divide it into 49 square inches. Having done this, cut the figure in following the big lines as shown by Fig 1. It will be observed that C is a complete square, and that A and B will form a square: but as D is 1 in. short of being a square, it is necessary to cut a square inch and add it on.
Fig. 2.
Fig. 2.
Then construct a right-angled triangle as shown by Figure 2.
We then see that the sum of the two small squares is equivalent to the large square.
And as we see that C has 25 small squares, it is thus proved that the sum of the squares upon the sides which contain the right angle are equal to the squares upon the side opposite the right angle.
Q.E.D.
THE GREAT FISH PROBLEM.
209.There is a fish the head of which is 9 in. long, the tail is as long as the head and half the back, and the back is as long as the head and tail together. What is the length of the fish?
210.How may 100 be expressed with four nines?
211.Two shepherds, A and B, meeting on the road, began talking of the number of sheep each had, when A said to B, “Give me one of your sheep, and I will have as many as you.” “Oh, no!” replied B; “give me one of yours, and I will have as many again as you.” How many sheep had each?
A BRICK PUZZLE.
One for Builders, Contractors, &c.
212.Suppose the measurements of a brick to be:—Length, 9 in.; breadth, 4½ in.; depth, 3 in. How many “stretchers, headers and closures” can be cut out of one, and what would be the face area of same?
For the benefit of the uninitiated we might say that
“stretcher” = length of brick x depth“header” = breadth "“closure” = half-breadth "
213.A woman has a basket of 150 eggs; for every 1½ goose egg she has 2½ duck eggs and 3½ hen eggs. How many of each had she?
The Great Chess Problem.
THE KNIGHT MOVE.
214.Move the Knight over all the 64 squares of the chess board so as to successively cover each square and, of course, not enter any square twice. This problem has always proved to be an interesting one. Mathematicians throughout all ages have devoted a good deal of time to it. To chess players it should be especially attractive.
215.If 3 times a certain number be taken from 7 times the same number the remainder will be 8. What is the number?
216.Divide £27 among 3 persons, A, B and C, so that B may have twice as much as A, and C 3 times as much as B.
ANSWER THIS.
217.Suppose it were possible for a man in Sydney to start on Sunday noon, January 1st, and travel westward with the sun, so that it might be in his meridian all the time, he would arrive at Sydney next day at noon, Monday, Jan. 2nd. Now, it was Sunday noon when he started, it was noon with him all the way round, and is Monday noon when he returns. The question is, at what point did it change from Sunday to Monday?
218.Start with 1 and keep on doubling for eight times, thus giving nine numbers, and arrange them in a square that when multiplied together, horizontally, vertically, or diagonally, the product of each row will be the cube of the number which must go in the centre of the square.
The happiest year in a man’s life is 40; for then he can XL.
Bound to Win!
219.A certain gentleman, who was employed in one of our city offices, purchasedThe Doctrine of Chance, which he studied in his spare time, with the result that he sent in his resignation to the head of the firm in order to try his luck on the racecourse.
At the first meeting he attended, there were only three horses in a race. His brother bookmakers were crying out the odds—
“Two to 1 bar one.”
The odds on this latter horse which was “barred” he discovered to be 6 to 4on. He determined to give far more liberal odds, and called out—
“Even money, 2 to 1, and 3 to 1.”
How could he give such odds, and yet win £1,no matter which horse wins the race?
AN INCH OF RAIN.
How many people really consider what is contained in the expression? Calculated, it amounts to this:—An acre is equal to 6,272,640 square inches; an inch deep of water on this area will be as many cubic inches of water, which, at 277·274 inches to the gallon, is 22622·5 gallons. The quantity weighs 226,225 lbs. Thus, an “inch of rain” is over 100 tons of water to the acre.
Extract from a small boy’s first essay:—“Man has two hans. One is the rite han an one is the left han. The rite han is fur ritin, and the left han is fur leftin. Both hans at once is fur stummik ake.”
220.Find the side of a square whose area is equal to twice the sum of its sides?
“THE EVIDENCE YOU NOW GIVE, &c., &c.”
221.Smith, Brown, and Jones were witnesses in a law case. The first-named gentleman swore that a certain thing occurred; Brown, on being called, confirmed Smith’s statement, but Jones denied it. They are known to tell the truth as follows:—
What is the probability that the statement is true?
When a man attains the age of 90 years, he may be termed XC-dingly old.
A school examination room might not to a casual observer seem to be a very likely place to find entertainment. However, the answers often given by pupils are sometimes excruciatingly funny, as is proved by the following:—
Definitions.
Function.—“When a fellow feels in a funk.”
Quotation.—“The answer to a division sum.”
Civil War.—“When each side gives way a little.”
The Four Seasons.—“Pepper, mustard, salt and vinegar.”
Alias.—“Means otherwise—he was tall, but she was alias.”
Compurgation.—“When he was going to have anything done to him, and if he could get anyone to say, ‘not innocent,’ he was let off.”
The Equator.—“Means the sun. Suppose we draw a straight line and the sun goes up to the top, then it is day, and when it comes down it is night.”
Precession.—“(1) When things happen before they take place. (2) The arrival of the equator in the plane of the ecliptic before it is due.”
Demagogue.—“A vessel that holds beer, wine, gin, whisky, or any other intoxicating liquor.”
Chimera.—“A thing used to take likenesses with.”
Watershed.—“A place in which boats are stored in winter.”
Gender.—“Is the way whereby we tell what sex a man is.”
Cynical.—“A cynical lump of sugar is one pointed at the top.”
Immaculate.—“State of those who have passed the entrance examination at the University.”
Frantic.—“Means wild. I picked up some frantic flowers.”
Nutritious.—“Something to eat that aint got no taste to it.”
Repugnant.—“One who repugs.”
Memory.—“The thing you forget with.”
History.
“Without the uses of History everything goes to the bottom. It is a most interesting study when you know something about it.”
“Oliver Cromwell was a man who was put into prison for his interference in Ireland. When he was in prison he wrote ‘The Pilgrim’s Progress,’ and married a lady called Mrs. O’Shea.”
“Wolsey was a famous General who fought in the Crimean war, and who, after being decapitated several times, said to Cromwell, ‘Ah, if I had only served you as you have served me, I would not have been deserted in my old age.’ He was the founder of the Wesleyan Chapel, and was afterwards called Lord Wellington. A monument was erected to him in Hyde Park, but it has been taken down lately.”
“Perkin Warbeck raised a rebellion in the reign of Henry VIII. He said he was the son of a Prince, but he was really the son of respectable people.”
Which do you consider the greater General, Cæsar or Hannibal? “If we consider who Cæsar and Hannibal were, the age in which they lived, and the kind of men they commanded, and then ask ourselves which was the greater, we shall be obliged to reply in the affirmative.”
Why was it that his great discovery was not properly appreciated until after Columbus was dead? “Because he did not advertise.”
What were the slaves and servants of the King called in England? “Serfs, vassals, and vaselines.”
Divinity.
Parable.—“A heavenly story with no earthly meaning.”
“Esau was a man who wrote fables, and who sold the copyright to a publisher for a bottle of potash.”
What is Divine right? “The liberty to do what you like in church.”
What is a Papal bull? “A sort of cow, only larger, and does not give milk.”
“Titus was a Roman Emperor, supposed to have written the Epistle to the Hebrews. His other name was Oates.”
Explain the difference between the religious belief of the Jews and Samaritans? “The Jews believed in the synagogue, and had their Sunday on a Saturday; but the Samaritans believed in the Church of England and worshipped in groves of oak; therefore the Jews had no dealings with the Samaritans.”
Give two instances in the Bible where an animal spoke? “(1) Balaam’s ass. (2) When the whale said unto Jonah, ‘Almost thou persuadest me to be a Christian.’”
Mathematics.
A Problem.—“Something you can’t find out.”
Hypotenuse.—“A certain thing is given to you, or it means let it be granted that such and such a thing is equal or unequal to something else.”
“If there are no units in a number you have to fill it up with all zeros.”
“Units of any order are expressed by writing in the place of the order.”
“A factor is sometimes a faction.”
“If fractions have a common denominator, find the difference in the denominator.”
“Interest on interest is confound interest.”
Grammar.
“Grammar is the way you speak in 9 different parts of speech; it is an art divided in 4 quarters—tortology is one, and sintax one more.”
An Abstract Noun.—“Something you can think of, but not touch—a red-hot poker.”
An Article.—“That wich begins words and sentences.”
A Pronoun “is when you don’t want to say a noun, and so you say a pronoun.”
“A Adjective is the colour of a noun, a black dog is a adjective.”
“Adjectives of more than one syllable are repaired by adding some more syllables.”
“Nouns are the names of everything that is common and has a proper name.”
Verb.—“To go for a swim is a verb what you do.”
“Adverbs are verbs that end with a lie and distinguish words. It is used to mortify a noun, and is a person, place, or thing, sometimes it is turned into a noun and then becomes a noun or pronoun.”
“Preposition means when you say anything of anything.”
“Conjunction means what joins things together; ‘—and 2 men shook hands.’”
“Nouns denoting male and female and things without sex is neuter. ‘The cow jumped over the fence’ is a transitif nuter verb because fence isen’t the name of anything and has no sex.”
Interjection.—“Words which you use when you sing out.”
“Gender is how you tell what sex a man is.”
Which Hand is It In?
A person having in one hand a piece of gold, and in the other a piece of silver, you may tell in which hand he has the gold, and in which the silver, by the following method:—
Some even number (such as 8) must be given to the gold, and an odd number (such as 3) must be given to the silver; after which, tell the person to multiply the number in the right hand by any even number whatever, and that in the left hand by an odd number; then bid him add together the two products, and if the whole sum be odd, the gold will be in the right hand and the silver in the left; if the sum be even, the contrary will be the case.
To conceal the artifice better, it will be sufficient to ask whether the sum of the two products can be halved without a remainder—for in that case the total will be even, and in the contrary case odd.
222.Which is the heavier, and by how much—a pound of gold or a pound of feathers; an ounce of gold or an ounce of feathers?
223.Plant an orchard of 21 trees, so that there shall be 9 straight rows with 5 trees in each row, the outline to be a regular geometrical figure.
SETTLING UP.
224.A person paid a debt of £5 with sovereigns and half-crowns. Now, there were half the number of sovereigns that there were half-crowns. How many were there of each?
A “CATCH.”
| | | | | | | | | | | | | | | | | | | |
225.How can you rub out 20 marks on a slate, have only five rubs, and rub out every time an odd one?
226.
From six take nine, from nine take ten,From forty take fifty, and six will remain.
From six take nine, from nine take ten,From forty take fifty, and six will remain.
227.A man and his wife lived in wedlock, one-third of his age and one-fourth of hers. Now, the man was eight years older than his wife at marriage, and she survived him 20 years. How old were they when married?
TO PROVE THAT YOU HAVE ELEVEN FINGERS.
Count all the fingers of the two hands, then commence to count backwards on one hand, saying, “10, 9, 8, 7, 6” (with emphasis on the6), and hold up the other hand saying, “and 5 makes 11.” This simple deception has often puzzled many.
228.A man travelled a certain journey at the rate of four miles an hour, and returned at the rate of three miles an hour. He took 21 hours in going and returning. What was the total distance gone over?
229.From what height above the earth will a person see one-third of its surface?
230.The difference between17⁄21and11⁄14of a certain sum is £10. What is the sum?
231.What decimal fraction is a second of a day?
232.Two trains are running on parallel lines in the same direction at rates respectively 45 miles and 35 miles an hour; the length of the first is 17 yds. 2 ft., and of the second 70 yds. 1 ft. How long will the one be in passing the other?
233.
Suppose a bushel to be exactly round,And the depth, when measured, eight inches be found;If the breadth 18·789 inches you discover,This bushel is legal all England over:But a workman would make one of another frame,Seven inches and a half the depth of the same;Now say of what length must the diameter be,That it may with the former in measure agree.
Suppose a bushel to be exactly round,And the depth, when measured, eight inches be found;If the breadth 18·789 inches you discover,This bushel is legal all England over:But a workman would make one of another frame,Seven inches and a half the depth of the same;Now say of what length must the diameter be,That it may with the former in measure agree.
WORTH TRYING.
A well known writer on mathematics, and a member of the Academy of Science, Paris, says that the most skilful calculator could not in less than a month find within a unit the cube root of 696536483318640035073641037.
A PROBLEM THAT WORRIED THE ANCIENTS.
Many profound works have been written on the following famous problem:—
“When a man says ‘I lie,’ does he lie, or does he not? If he lies he speaks the truth; if he speaks the truth he lies.”
Several philosophers studied themselves to death in vain attempts to solve it. Reader, have a “go” at it.
THE CABINET MAKER’S PUZZLE.
234.A cabinet maker has a circular piece of veneering with which he has to veneer the tops of two oval stools; but it so happens that the area of the stools, exclusive of the hand-holes in the centre and that of the circular piece, are the same. How must he cut his veneer so as to be exactly sufficient for his purpose?
THE ARITHMETICAL TRIANGLE.
Write down the numbers 1, 2, 3, &c., as far as you please in a column. On the right hand of 2 place 1, add them together and place 3 under the 1; the 3 added to 3 = 6, which place under the 3, and so on; this gives the second column. The third is found from the second in a similar way. By the triangle we can determine how many combinations can be made, taking any number at a time out of a larger number. For instance, a group of 8 gentlemen agreed that they should visit the Crystal Palace 3 at a time, and that the visits should be continued daily as long as a different three could be selected. In how many days were the possible combinations of 3 out of 8 completed?
Method: Look down the first column till you come to 8, then see what number is horizontal with it in the third column, viz., 56. (For the method usually adopted for working out calculations like the above, seeDoctrine of Chance.)
235.Why is a pound note more valuable than a sovereign?
KEEPING UP STYLE.
236.A certain hotelkeeper was never at a loss to produce a large appearance with small means. In the dining-room were three tables, between which he could divide 21 bottles of wine, of which 7 only were full, 7 half-full, and 7 apparently just emptied, and in such a manner that each table had the same number of bottles and the same quantity of wine. How did he manage it?
A DOMINO TRICK.
Ask the company to arrange the whole set of dominoes whilst you are absent in any way they please, subject, however, to domino rules—a 6 placed next to a 6, a 5 to a 5, and so on. You now return and state that you can tell, without seeing them, what the numbers are at either end of the chain. The secret lies in the fact that the complete set of 28 dominoes, arranged as above-mentioned, forms a circle or endless chain. If arranged in a line the two end numbers will be found to be the same, and may be brought together, completing the circle. You privately abstract one domino (not a double), thus causing a break in the chain. The numbers left at the ends of the line will then be the same as those of the “missing link” (say the 3-5 or 6-2.) The trick may be repeated, but you must not forget to exchange the stolen domino for another.
237. A busman not having room in his stables for eight of his horses increased his stable by one half, and then had room for eight more than his whole number. How many horses had he?
AN ANCIENT QUESTION.
238. “Tell us, illustrious Pythagoras how many pupils frequent thy school?” “One-half,” replied the philosopher, “study mathematics, one fourth natural philosophy, one-seventh observe silence, and there are 3 females besides.” How many had he?
EVADING THE QUESTION.
239. A lady being asked her age, and not wishing to give a direct answer, said, “I have nine children, and three years elapsed between the birth of each of them. The eldest was born when I was 19 years old, and the youngest now is exactly 19.” How old was she?
A ’CENTAGE “CATCH.”
240. A man sells a diamond for £60; the number expressing the profit per cent. is equal to half the number expressing the cost. What was the cost?
241. Having 5½ hours to spare, how far may I go out by a coach at the rate of 8 miles an hour so that I may be back in time, walking at the rate of three miles an hour?
The Cross Puzzle.