Part 8
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, or _two_ times 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’t _add_ any 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.
1 2 4 8 16 32 3 3 5 9 17 33 5 6 6 10 18 34 7 7 7 11 19 35 9 10 12 12 20 36 11 11 13 13 21 37 13 14 14 14 22 38 15 15 15 15 23 39 17 18 20 24 24 40 19 19 21 25 25 41 21 22 22 26 26 42 23 23 23 27 27 43 25 26 28 28 28 44 27 27 29 29 29 45 29 30 30 30 30 46 31 31 31 31 31 47 33 34 36 40 48 48 35 35 37 41 49 49 37 38 38 42 50 50 39 39 39 43 51 51 41 42 44 44 52 52 43 43 45 45 53 53 45 46 46 46 54 54 47 47 47 47 55 55 49 50 52 56 56 56 51 51 53 57 57 57 53 54 54 58 58 58 55 55 55 59 59 59 57 58 60 60 60 60 59 59 61 61 61 61 61 62 62 62 62 62 63 63 63 63 63 63
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.
“_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.
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.
D contains 9 small squares. A & B do. 16 do. -- 25
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, purchased THE 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 4 _on_. 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:--
Smith, once in 3 times Brown, " " 5 " Jones, " " 10 "
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.
Examination Gems.
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.”
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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.
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 the _6_), 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 between 17/21 and 11/14 of 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.
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.
1 2, 1 3, 3, 1 4, 6, 4, 1 5, 10, 10, 5, 1 6, 15, 20, 15, 6, 1 7, 21, 35, 35, 21, 7, 1 8, 28, 56, 70, 56, 28, 8, 1
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, see DOCTRINE 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.
242. Cut out of a piece of card five pieces similar in shape and proportion to the annexed figures.
1 piece similar to 1 3 pieces " " 2 1 piece " " 3
These five pieces are then to be so joined as to form a cross like that represented by 4.
Irish Counting.
An Irishman who had lately arrived in the colony was employed as handy man at one of our large suburban mansions. The lady of the house, hearing that some midnight thief had walked off with some of her prize poultry, desired Pat to count them as speedily as possible and to inform her how many there were; he accordingly left off cleaning the buggy, and proceeded to enumerate the feathered bipeds. The lady, getting impatient of waiting for him, repaired to the poultry yard, and noticing him chasing a small chicken, enquired, “Pat, whatever are you doing!” when the Irishman replied; “I’ve counted all the chickens except this one; but the little varmint won’t stand still till I count him.”
THE JEW “JEWED.”
243. An old Jew took a diamond cross to a jeweller to have the diamonds re-set, and fearing that the jeweller might be dishonest he counted the diamonds, and found that they numbered 7 in three different ways. Now, the jeweller stole two diamonds, but arranged the remainder so that they counted 7 each way as before. How was it done?
7 6 7 6 5 6 7 4 3 2 1
244. A person wishing to enclose a piece of ground with palisades found that if he set them a foot apart that he should have too few by 150, but if he set them a yard apart he should have too many by 70. How many had he?
245. A mechanic is hired for 60 days on consideration that for each day he works he shall receive 7s. 6d., but for each day he is idle he shall pay 2s. 6d. for his board, and at the end he receives £6. How many days did he work?
246. Take one from nineteen and leave twenty.
THE CAMEL PROBLEM.