Chapter IV: Part 4

180. A man and a boy being paid for certain days’ work, the man received 27s., and the boy, who had been absent 3 days out of the time, received 12s. Had the man, instead of the boy, been absent the 3 days they would both have claimed an equal sum. Find out the wages of each per day.

181. The extremes of an arithmetical series are 21 and 497, and the number of terms is 41. What is the common difference?

182. A wine which contains 7½ per cent. of spirit is frozen, and the ice which contains no spirit being removed the proportion of spirit in the wine is increased by 8¾ per cent. How much water in the shape of ice was removed from 504 gallons of the mixture?

THE SHARP SELECTOR.

183. A selector rented a farm, and agreed to give his landlord two-fifths of the produce, but prior to the time of dividing the corn the selector used 45 bushels. When the general division was made it was proposed to give to the landlord 18 bushels from the heap in lieu of the share of the 45 bushels which the tenant had used, and then to begin and divide the remainder as though none had been used. Would this method have been correct?

A GOOD “AD.”

A member of a certain firm appeared in a law court with a complaint that his partner would sell goods at less than cost price, and he desired to have him restrained. The defendant utterly denied the charge, and the case was adjourned for a fortnight. As the plaintiff went out of court he exclaimed in a tragic tone: “Then the sacrifice must still go on!” and “I’ll be ruined!” The story was noised abroad, and the result was that the shop was besieged by customers every day. There the case ended, for at the end of the fortnight the plaintiff failed to appear in court, having accomplished his purpose--advertisement.

184. I give 3 sovereigns for 2 dozen wine at different rates per dozen, and by selling the cheaper kind at a profit of 15 per cent. and the dearer at a loss of 8 per cent. I obtain a uniform price for both. What did each dozen cost me?

185. I have in my garden a shrub that grows 12 inches every day, but during the night it withers off to half the height that it was at the end of the previous day. How much short of 2 feet will it be at the end of a year?

TIT-FOR-TAT.

186. A farmer puts a 3 lb. stone in a keg of butter worth 11d. a pound. The merchant cheats him out of 1 lb. on the weight, and then does him out of 1s. 11d. on calico, tobacco, and a shovel. Who is ahead, and how much?

187. Trains leave London and Edinburgh (400 miles apart) at the same time and meet after 5 hours; the train which leaves London travels 8 miles an hour faster than that which leaves Edinburgh. At what rate did the former travel, and at what speed must the latter travel after they have met, in order that they both may reach their destinations at the same time?

“GOOD ENOUGH!”

“Will you give me a glass of beer, please?” asked a rather seedy-looking fellow with an old but well-brushed coat and almost too shiny a hat. It was produced by the barmaid, frothing over the edge of the tumbler.

“Thank you,” said the recipient, as he placed it to his lips. Having finished it in a swallow, he smacked his lips and said, “That is very good beer--_very_! Whose is it?”

“Why, that Perkins’s----”

“Ah! Perkins’s, is it! Well, give us another glass.”

It was done; and holding it up to the light and looking through it, the connoisseur said:--

“’Pon my word, it is grand beer--clear as Madeira! What a fine color! I must have some more of that; give me another glass.”

The glass was filled again, but before putting it to his lips the imbiber said:--

“_Whose_ beer did you say this was?”

“Perkins’s,” emphatically replied the barmaid.

The contents of the glass was exhausted, as also the vocabulary of praise, and it only remained for the appreciative gentleman to say, as he wiped his mouth and went towards the door:--

“Perkins’s beer, is it! I know Perkins very well; I shall see him soon, and will settle with him for three long glasses of his incomparable brew. Good morning.”

A Conspiracy.

188. Three gentlemen are going over a ferry with their three servants, who conspire to rob them if they can get one gentleman to two of them, or two to three, on either side of the ferry. They have a boat that will only carry two at once, and either a gentleman or a servant must bring back the boat each time a cargo of them goes over. How can the gentlemen get over with all their servants so as to avoid an attack?

189. Find two numbers whose product is equal to the difference of their squares, and the sum of their squares equal to the difference of their cubes?

190. Divide 1400 into such parts as shall have the same ratio as the cubes of the first four natural numbers.

This was the tempting notice lately exhibited in the window of a dealer in cheap shirts: “They won’t last long at this price!”

POSTING THE LEDGER.

The well known author of several works on account-keeping, Mr. Yaldwyn, tells a rather good thing which actually occurred in New Zealand some time back. Mr. Yaldwyn was at the time engaged examining the books in one of the offices in a country town, and enquired from one of the clerks standing near if the ledger were posted. The person appealed to answered that “he didn’t know,” whereupon Mr. Y. said that he required it done, and with as little delay as possible. A few minutes later the same individual came rushing in and informed him that the ledger was “posted.” Such a piece of “lightning book-keeping” so surprised Mr. Y. that he further questioned the man, who replied “You said you wanted the ledger posted, and, begorra, I posted it.” It then dawned upon Mr. Yaldwyn that the clerk, who was an Irishman, had actually _posted_ the book in the post office!

THEY MANAGED IT.

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 a _nouse_.” “No,” said the second, “It’s a _nut_.” The third exclaimed “You’re both wrong; it’s a _nin_!”

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.”

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.

19199 19199 368601601
----- x ----- = --------- and so on.
960 960 921600

Many adopt the following method:--

£20 x £20 = £400

£ s d
400 0 0
¼d x ¼d = 1/16 less 1/16
----------------
£399 19 11-15/16 Ans.

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 are _abstract_ or _concrete_--3 is abstract, £3 is concrete.

Two abstract numbers can be multiplied together--as, 4 times 3 = 12.

Proof: 3
3
3
3
--
12

One abstract number and one concrete number can be multiplied together--as 2s. multiplied by 3 = 6s.

Proof: 2s.
2s.
2s.
---
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_

15
____
10)300
----
30 Ans. 30s.

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.”

| | | | | | | | | | | | | | | | | | | |

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.

247. An Arab Sheik, when departing this life, left the whole of his property to his three sons. The property consisted of 17 camels, and in dividing it the following proportions were to be observed:--

The oldest son was to have one-half of the camels, the second son one-third, and the youngest son one-ninth; but it was provided that the camels were not, on any account, to be injured, but to be divided as they were--living--between the three sons.

Thereupon, a great argument ensued. The eldest son claimed 8½ camels. The second insisted upon receiving 5⅔ of a camel; while the youngest son would not be comforted with less than 1-8/9 of a camel. The Cadi (or Judge) happened to appear on the scene. To him the matter was explained. Without a moment’s hesitation he gave his decision--a decision by which the claims of all three contestants were fully satisfied.

How did the Cadi settle this knotty question?

248. A grocer has 6 weights--each one twice as much as the one before it in size. If he weighed the first five against the largest, it (the largest) would only be 2 lbs. heavier than the combined weights of the rest. What are the weights?

249. A squatter said to a new manager, whom he wished to test in arithmetic: “I have as many pigs as I have cattle and horses, and if I had twice as many horses I should then have as many horses as cattle, and I should also have 13 more cattle and horses than pigs.” How many of each had he?

250. A gentleman a garden had, five score[2] long and four score broad;
A walk of equal width half round he made, which took up half the
ground--
You skilful in Geometry, tell us how wide the walk must be.

[2] Feet.

251. Two boys, meeting at a farmhouse, had a mug of milk set down to them; the one, being very thirsty, drank till he could see the centre of the bottom of the mug; the other drank the rest. Now, if we suppose that the milk cost 4½d., and that the mug measured 4 inches diameter at the top and bottom, and 6 inches in depth, what would each boy have to pay in proportion to the milk he drank?

Weight-for-Age Problem.

252. There are 6 children seated at a table whose total ages amount to 39 years. Tom, who is only half the age of Jack (the oldest) is seated at the top, with Bob--who is a year older than him--next; whilst Fred, who is four-fifths the age of Jack, is at the foot with James, who is 1 year younger than Jack, next, him; the youngest, who is a baby, is one-eighth the age of her brother Fred. Find the ages of each, and weight of Fred, and by placing him third from the top his initial and surname. You must express the ages in words, and use the initial letters.

253. A flagstaff there was whose height I would know,
The sun shining clear straight to work I did go.
The length of the shadow, upon level ground,
Just sixty-five feet, when measured I found;
A pole I had there just five feet in length--
The length of its shadow was four feet one-tenth
How high was the flagstaff I gladly would know;
And it is the thing you’re desired to show.

254. Put 4 figures together to equal 30, and the same figures to equal 40.

255. A Salvation Army captain took up a collection, his lieutenant took up another; if what the captain took up was squared and the lieutenant’s added the sum would be 11d.; if what the lieutenant took up was squared and the captain’s added the sum would be 7d. What was the amount of the collection?

256. Find a number which, if multiplied by 17, gives a product consisting only of 3’s.

THE “FOWL” PROBLEM.

257. If a hen and a half lay an egg and a half in a day and a half, how many eggs will 6 hens lay in 7 days?

258. Tom and Bill work 5 days each. Tom has as much and half as much per day as Bill. The total amount of their wages for the 5 days is £1 17s. 6d. What are their respective wages per day?

259. How many ¼ inch cubes can be cut out of a 2½ inch cube?

260.

miles. furl. po. yds. ft. in.
From 1 0 0 0 0 0
Subtract 7 39 5 1 5
------------------------------

THE SQUARE PUZZLE.

261. A man has a square of land, out of which he reserves one-fourth (as shown in the diagram) for himself. The remainder he wishes to divide among his four sons so that each will have an equal share and in similar shape with his brother. How can he divide it?

Although this is a very old puzzle it is often the cause of much amusement.

GENEROUS.

262. A gentleman, having a certain number of shillings in his possession, made up his mind to visit 17 different barracks and treat the soldiers, and he did so in the following manner:--On going into the first barracks, he gave the sentry one shilling and then spent half of his shillings in the canteen amongst the soldiers, and on coming out of barracks again he gave the sentry another shilling; he repeated the same until he had finished with the seventeenth barracks, and had no more shillings left. How many had he when he commenced?

263. What part of 3 is a third part of 2?

264. Make 91 less by adding two figures to it.

265. If a church bell takes two seconds to strike the hour at 2 o’clock, how many seconds will it take to strike 3 o’clock?

THIS CATCHES EVERYBODY.

Ask a friend how many penny stamps make a dozen? He will reply, “Why, twelve, of course.” Then ask again, “Well, how many half-penny ones?” He is almost sure to reply, “Twenty-four.”

Before he settles his account with nature, man charges the debit of his profit and loss account to Fate, but the credit he takes to himself.

THE PUZZLE ABOUT THE “PROFITS.”

Perhaps there is no form of commercial calculation so confusing and so little understood as that of mercantile profits. It might surprise many to state, nevertheless it is perfectly true, that it is impossible to buy goods and sell them to show a profit as great as 100 per cent.

The correct method to calculate profit is to reckon on the _return_--the price received for the goods sold--_not on the cost price_, and as it is impossible to sell goods at 100 per cent. discount, so also goods cannot be sold to show that percentage of profit, unless they actually cost nothing.

Some time ago, in New Zealand, a well-known boot manufacturer had a “GREAT DISCOUNT SALE.“ He had large posters displayed on the windows of his shops, and advertisements in the newspapers, announcing the fact that 5s. in the £ would be allowed as discount to all customers. The profit he usually obtained in the ordinary way of trade was 25 per cent., and having had a good season, he was prepared to sell off the balance of his stock at cost price. The selling price of his goods was marked in plain figures. A pair of boots which cost him 8s. was marked 10s., thus showing a profit of 2s., which he considered to be 25 per cent. (2s. being a quarter of 8s.) Instructions were issued to all his employees engaged in selling to deduct a quarter from the marked price, the result being that a pair of boots which cost 8s., and marked 10s., was being sold at 7s. 6d. (2s. 6d., the quarter of the marked price being deducted from 10s.) Although he imagined he was getting 25 per cent. profit, he was in reality receiving only 20 per cent. It was not long before the posters were altered, announcing that 4s. in the £ would be allowed to his customers.

The following question was asked some little time ago;--If a chemist sold a bottle of medicine for 2s. 6d., which cost him 2½d., what percentage would be his profit?

Many work out the problem and answer 1100 per cent., but this answer is incorrect. He received 2s. 6d. for that which cost him 2½d., accordingly there was a profit of 2s. 3½d. We must now find out what percentage is the latter amount of the selling price, 2s. 6d., and we discover that it is 91⅔ per cent.

266. A pork butcher buys at auction £100 worth of bacon at 4d. per lb. and sells it at 8d. per lb.; also £100 worth at 8d. per lb., which he sells for 4d. per lb. Does he lose or gain? And if so how much.

“THE JUMPING FROG.”

267. A frog, sitting on one end of a log eight feet long, starts to jump into a pond at the opposite end. With his first jump he clears half the distance, the second jump half the remaining distance, and so on. How many jumps does he take before entering the pond?

OBLONG PUZZLE.

268. Cut out of a piece of cardboard fourteen pieces of the same shape as those shown in the diagram--the same number of pieces as is there represented--and then form an oblong with them.

269. If a man can load a cart in five minutes, and a friend can load it in two and a half minutes, how long will it take them both to load it, both working together?

270. A gentleman on being asked how old he was, said that if he did not count Mondays and Thursdays he would be 35. What was his actual age?

TOO SMART FOR DAD.

“Pa,” said a boy from school, “How many peas are in a pint?” “How can anybody tell that, foolish boy?” “I can every time. There is just one ‘p’ in pint the world over.” He was sent off to bed early.

SIMPLE PROPORTION.

271. If it takes three minutes to boil one egg, how long will it take to boil two?

“PUNCH’S” MONEY VAGARIES.

The early Italians used cattle as a currency instead of coin (thus a bull equals 5s.) and a person would send for change for a thousand pound bullock, when he would receive 200 five pound sheep. If he wanted _very_ small change there would be a few lambs amongst them. The inconvenience of keeping a flock of sheep at one‘s bankers’, or paying in a short-horned heifer to one’s private account led to the introduction of _bullion_.

As to the unhealthy custom of _sweating sovereigns_, it may be well to recollect that Charles I., the earliest Sovereign, who was sweated to such an extent that his immediate successor, Charles II., became one of the lightest Sovereigns ever known in England.

Formerly every gold watch weighed so many _carats_, from which it became usual to call a silver watch a turnip.

The Romans were in the habit of tossing their coins in the presence of their legions, and if a piece of money went higher than the top of their Ensign’s flag it was presumed to be “above the standard.”

“MARCH ON! MARCH ON!”

272. An army 25 miles long starts on a journey of 50 miles, just as an orderly at the rear starts to deliver a message to the General in front. The orderly, travelling at a uniform speed, delivers his message and returns to the rear, arriving just as the army finishes the journey. How many miles does the orderly travel?

“WITH A LONG, LONG PULL.”

273. If eight men are engaged in a tug-o’-war, four pulling against four, on a continuous rope, and each man is exerting a force of 100 lbs., what strain is there at the centre of the rope?

“FIND OUT.”

274. A gentleman in a train with a boy got into conversation with a stranger, who asked him the lad’s age. The boy quickly replied, “This gentleman, who is my uncle, is twice as old as me, but the sum of the figures in my age are twice the sum of those in his.” What was the age of each?