  If the number 37 be multiplied by 3, or any multiple of 3 up to 27, the product is expressed by three similar digits. Thus—
The products succeed each other in the order of the digits read downwards, 1, 2, 3, etc., these being multiplied by 3 (their number of places) reproduce the multiplicand of 37.
If it be multiplied by multiples of 3, beyond 27, this peculiarity is continued, except that the extreme figures taken together represent the multiple of 3 that is used as a multiplier. Thus—
The number 73 (which is 37 inverted) multiplied by each of the numbers of arithmetical progression 3, 6, 9, 12, 15, etc., produces products terminating (unit’s place) by one of the ten different figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. These figures will be found in the reverse order to that of the progression, 73 × 3 produces 9, by 6 produces 8, and 9 produces 7, and so on. Another number which falls under some mysterious law of series is 142,857, which, multiplied by 1, 2, 3, 4, 5, or 6 gives the same figures in the same order, beginning differently; but if multiplied by 7, gives all 9’s.
Multiplied by 8, it gives 1,142,856, the first figure added to the last makes the original number—142,857. The vulgar fraction 1/7 = ·142,857. The following number, 526315789473684210, if multiplied as above, will, in the product, present the same peculiarities, as also will the number 3448275862068965517241379310.
Taking the same multiplicand and multiplying by 27 (half 54) the product is 26,666,666,667, all 6’s except the extremes, which read the original multiplier (27). If 72 be used as a multiplier, a similar series of progression is produced. 6. In stables five, can you contrive to put in horses twenty— In each stable an odd horse, and not a stable empty? “THREE THREES ARE TEN.” This little trick often puzzles many:— Place three matches, coins, or other articles on the table, and by picking each one up and placing it back three times, counting each time to finish with number 10, instead of 9. Pick up the first match and return it to the table saying 1; the same with the second and third, saying 2 and 3; repeat this counting 4; but the fifth match must be held in the hand, saying at the time it is picked up, 5; the other two are also picked up and held in hand, making 6 and 7; the three matches are then returned to the table as 8, 9, and 10. If done quickly few are able to see through it. 7. A man bought a colt for a certain sum and sold him 2 years afterwards for £50 14s., gaining thereby as much per cent. per annum compound interest as it had cost him. What was the original price? Do Figures Lie? “Figures cannot lie,” is a very old saying. Nevertheless, we can all be deceived by them. Perhaps one of the best instances of them leading us astray is the following:— An employer engaged two young men, A and B, and agreed to pay them wages at the rate of £100 per annum. A enquires if there is to be a “rise,” and is answered by the employer, “Yes, I will increase your wages £5 every six months.” “Oh! that is very small; it’s only £10 per year,” replied A. “Well,” said the employer, “I will double it, and give you a rise of £20 per year.” A accepts the situation on those terms. B, in making his choice, prefers the £5 every six months. At the first glance, it would appear that A’s position was the better. Now, let us see how much each receives up to the end of four years:—
A spieler at a Country Show amused the people with the following game:—He had 6 large dice, each of which was marked only on one face—the first with 1, the second 2, and so on to the sixth, which was marked 6. He held in his hand a bundle of notes, and offered to stake £100 to £1 if, in throwing these six dice, the six marked faces should come up only once, and the person attempting it to have 20 throws. Though the proposal of the spieler does not on the first view appear very disadvantageous to those who wagered with him, it is certain there were a great many chances against them. The six dice can come up 46,656 different ways, only one of which would give the marked faces; the odds, therefore, in doing this in one throw would be 46,655 to 1 against, but, as the player was allowed 20 throws, the probability of his succeeding would be— 20 To play an equal game, therefore, the spieler should have engaged to return 2332 times the money deposited. TREBLE RULE OF THREE. If 70 dogs with 5 legs each catch 90 rabbits with 3 legs each in 25 minutes, how many legs must 80 rabbits have to get away from 50 dogs with 2 legs each in half an hour? 8. Suppose a greyhound makes 27 springs whilst a hare makes 25, and the springs are equal: if the hare is 50 springs before the hound at the start, in how many springs will the hound overtake the hare? The first Arithmetic in English was written by Tonstal, Bishop of London, and printed by Pinson in 1552. Two persons playing dominoes 10 hours a day and making 4 moves a minute could continue 118,000 years without exhausting all the combinations of the game. A schoolmaster wrote the word “dozen” on the blackboard, and asked the pupils to each write a sentence containing the word. He was somewhat taken aback to find on one of the slates the following unique sentence: “I dozen know my lesson.” 9. I have a piece of ground, which is neither square nor round, But an octagon, and this I have laid out In a novel way, though plain in appearance, and retain Three posts in each compartment; but I doubt Whether you discover how I apportioned it, e’en tho’ I inform you ’tis divided into four. But, if you solve it right, ’twill afford you much delight And repay you for the trouble, I am sure. _ At an examination in arithmetic, a little boy was asked “what two and two made?” Answer—“Four.” “Two and four?” Answer—“Six.” “Two and six?” Answer—“Half-a-crown.” 10. A certain gentleman dying left his executor the sum of £3,000 to be disposed of in the following manner, viz.:—To give to his son £1,000, to his wife £1,000, to his sister £1,000, and to his sister’s son £1,000, to his mother’s grandson £1,000, to his own father and mother £1,000, and to his wife’s own father and mother £1,000—required, the scheme of kindred. COPY OF LETTER FROM FIRM TO COMMERCIAL TRAVELLER. Mr. Einstein, Townsville, Ve hav receved your letter on the 18th mit expense agount and round list. vat ve vants is orders, ve haf plenty maps in Sydney vrom vich to make up round lists also big families to make expenses. Mr. Einstein ve find in going through your expenses agount 10s. for pilliards please don’t buy no more pilliards for us. vat ve vants is orders, also ve do see 30s. for a Horse and Buggy, vere is de horse and vot haf you done mit de Buggy the rest on your expenses agount vas nix but drinks—vy don’t you suck ice. ve sended you to day two boxes cigars, 1 costed 6/- and the oder 3/6 you can smoke the 6/- box, but gif de oders to your gustomers, ve send you also samples of a necktie vat costed us 28/- gross, sell dem for 30/- dozen if you can’t get 30/- take 8/6, vat ve vants is orders. The neckties is a novelty as ve hav dem in stock for seven years and ain’d sold none. My brother Louis says you should stop in Rockhampton. His cousin Marks livs dere. Louis says you should sell Marks a good bill; dry him mit de neckties first, and sell mostly for cash, he is Louis’s cousin. Ve only giv credit to dem gustomers vat pays cash. Don’t date any more bills ahead, as the days are longer in the summer as in the vinter. Don’t show Marks any of the good sellers, and finaly remember Mr. Einstein mit us veder you do bisness or you do nothings at all vat ve vants is orders. P.S.—Keep the expenses down. 11. Two fathers and two sons went into a hotel to have drinks, which amounted to one shilling. They each spent the same amount. How much did each pay? 12. In a cricket match, a side of 11 men made a certain number of runs. One obtained one-eighth of the number, each of two others one-tenth, and each of three others one-twentieth. The rest made up among them 126 (the remainder of the score), and four of the last scored five times as many as the others. What was the whole number of runs, and the score of each man? BRAINS v. BRAWN. _ Schoolmaster—“What is meant by mental occupation?” Pupil—“One in which we use our minds.” Schoolmaster—“And a manual occupation?” Pupil—“One in which we use our hands.” Schoolmaster—“Now, which of these occupations is mine. Come, now; what do I use most in teaching you?” Pupil (quickly)—“Your cane, sir!” MAGIC ADDITION. To write the answer of an addition sum, when only one line has been written. _ Tell a person to write down a row of figures. Now, this row will constitute the main body of the answer. Tell him to write another row beneath it; you now write a row also, matching his second row in pairs of 9’s he writes one more row, and you again supply another in the same manner. Your addition sum will now consist of five lines, four of which are paired; the first line, or key line, being the answer to the sum. From the unit figure in the key line deduct the number of pairs of 9’s—in this instance two—and place the remainder, 6, as the unit figure of the answer, then write in order the rest of the figures in the key line, annexing the 2 to the extreme left; this will constitute the complete answer. It, of course, is not necessary to adhere to two pairs of 9’s; there may be three, four, or even more; but the total number of lines, including the key line, must be odd, and the number of pairs must be deducted from the unit figure of the key line, and this same number be written down at the extreme left. The number of figures in each line should always be the same. As the location of the key line may be changed if necessary, the artifice could not easily be detected. Punctuation was first used in literature in the year 1520. Before that time wordsandsentenceswereputtogetherlikethis. _ 13. Smith and Brown meet a dairymaid with a pail containing milk. Smith maintains that it is exactly half full; Brown that it is not. The result is a wager. They have no instrument of any kind, nor can they procure one by means of which to decide the wager; nevertheless they manage to find out accurately, and without assistance, whether the pail is half-full or not. How is it done?—It should be added that the pail is true in every direction. A HINT FOR TAILORS. “There, stand in that position, please, and look straight at that notice while I take your measure.” Customer reads the notice— NUMBER 9. If two numbers divisible by 9 be added together the sum of the figures in the amount will be either 9 or a number divisible by 9.
If one number divisible by 9 be subtracted from another number divisible by 9, the remainder will be either a 9 or a number divisible by 9.
If one number divisible by 9 be multiplied by another number divisible by 9, the product will be divisible by 9.
In the above examples it is worth noting that the figures in each answer added together continually produce 9. (1) 90 = 9 (2) 54 = 9 (3) 1458 = 18 = 9 (4) 135 = 9 Also, if these answers be multiplied by any number whatever, a similar result will be produced. Example: 135 x 8 = 1080 = 9 If any row of two or more figures be reversed and subtracted from itself, the figures composing the remainder will, when added, be a multiple of 9, and if added together continually will result in 9.
Tell a person to write a row of figures, then to add them together, and to subtract the total from the row first written, then to cross out any one of the figures in the answer, and to add the remaining figures in the answer together, omitting the figure crossed out; if the total be now told, it is easy to discover the figure crossed out.
It should be observed that the figures of the answer to the subtraction when added together equal 27—a multiple of 9; this, of course, is always the case. Now, suppose that 7 was the figure crossed out, then the sum of the figures in the answer (omitting 7) would be 20; this number being told by the person, it is easily seen that 7 must have been crossed out, as that figure is required to complete the multiple 27. If after the figure has been crossed out, the remaining figures total a multiple of 9, it is evident that either a cipher or a 9 must have been the figure erased. Multiply the digits—omitting 8—by any multiple of 9, and the product will consist of that multiple,
If a figure with a number of ciphers attached to it be divided by 9, the quotient will be composed of that figure only repeated as many times as there are ciphers in the dividend; with the same figure as the remainder.
EXCUSES. “Miss Brown,—You must stop teach my Lizzie fisical torture. She needs reading and figgers more an that. If I want her to do jumpin I kin make her jump.” “Please let Willie home at 3 o’clock. I take him out for a little pleasure, to see his father’s grave.” “Dear Teecher,—Please excuse John for staying home—he had the meesels to oblige his father.” “Dear Miss——, Please excuse my boy scratching hisself, he’s got a new flannel shirt on.” “A country schoolmaster received from a small boy a slip of paper which was supposed to contain an excuse for the non-attendance of the boy’s brother. He examined the paper, and saw thereon: “Kepatomtogoataturing.” Unable to understand, the small boy explained to the master that his big brother had been “kept at home to go taturing”—that is, to dig potatoes. “Tommy,” said the school teacher, “you must get your father to give you an excuse the next time you stay away from school.” “That’s no use, teacher. Dad’s no good at making excuses; mother bowls him out every time.” HARVESTING. 14. A and B engage to reap a field for 90s. A could reap it in 9 days by himself; they promised to complete it in five days; they found, however, that they were obliged to call in C (an inferior workman) to assist them the last two days, in consequence of which B received 3s. 9d. less than he otherwise would have done. In what time could B and C reap the field alone? 15. A man has a triangular block of land, the largest side being 136 chains, and each of the other sides 68 chains. What is the value of the grass on it, at the rate of £2 an acre? A school inspector in the North of Ireland was once examining a geography class, and asked the question: “What is a lake?” He was much amused when a little fellow, evidently a true gem of the emerald isle, answered: “It’s a hole in a can, sur.” Canvasser—“I’ve got some signs that I’m selling to shopkeepers all day long. Everybody buys ’em. Here’s one—“If You Don’t See What You Want, Ask For It.”
16. The number of soldiers placed at a review is such that they could be formed into 4 hollow squares, each 4 deep, and contain 24 men in the front rank more than when formed into a solid square. Find the whole number. In the counting-house of an Irishman the following notice is exhibited in a conspicuous place: “Persons having no business in this office will please get it done as soon as possible and leave.” 17. Upon a piece of cardboard draw The three designs you see— I should have said of each shape four— Which when cut out will be, If joined correctly, that which you Are striving to unfold— An octagon, familiar to My friends both young and old. _ “I was induced to-day, by the importunity of your traveller,” wrote an up-country store-keeper to a Brisbane firm, “to give him an order; but, as I did it merely to get rid of him in a civil manner, and to prevent my losing any more time, I must ask you to cancel the same.” A CATCH IN EUCHRE. 18. What card in the game of euchre is always trumps and yet never turned up? This often puzzles many. RELIGIOUS RECKONING.—(The New Jerusalem.) Revelations xxi. (15)—“And he that talked with me had a golden rule to measure the city and the gates thereof and the wall thereof; (16) “And the city lieth four square, and the length is as large as the breadth, and he measured the city with the reed twelve thousand furlongs. The length and the breadth and the height of it are equal.” 12,000 furlongs = 7,920,000 feet, which cubed = 496793088000000000000 cubic feet; half of this we will reserve for the Throne and Court of Heaven, and half the balance for streets, &c., leaving a remainder of 124198272000000000000 cubic feet. Divide this by 4096 (the cubic feet in a room 16 feet square) and there will be 3032184375 000000 rooms. Suppose that the world always did and always will contain 990,000,000 inhabitants, and that a generation lasts 33? years, making in all 2,970,000,000 every century, and that the world will stand 100,000 years, totalling 2,970,000,000,000 inhabitants; then suppose there were 100 worlds equal to this in number of inhabitants and duration of years, making a total of 297,000,000,000,000 persons. There would then be more than 100 rooms 16 feet square for each person. 19. A man had a certain number of £’s, which he divided among 4 men. To the first he gave a part, to the second one-third of what was left after the first’s share, to the third he gave five-eighths of what was left, and to the fourth the balance, which equalled two-fifths of the first man’s share. How much money did he have, and how much did each receive, none receiving as much as £20? ROWING AGAINST TIME. 20. In a time race, one boat is rowed over the course at an average pace of 4 yards per second, another moves over the first half of the course at the rate of 3½ yards per second, and over the last half at 4½ yards per second, reaching the winning post 15 seconds later than the first. Find time taken by each. STOCK-BREEDING. 21. A farmer, being asked what number of animals he kept, answered: “They’re all horses but two, all sheep but two, and all pigs but two.” How many had he? A QUIBBLE. 22. What is the difference between twice one hundred and five, and twice one hundred, and ten? 23. The product of two numbers is six times their sum, and the sum of their squares is 325. What are the numbers? THE PUZZLE ABOUT THE “PER CENTS.” There are many persons engaged in business who often become badly mixed when they attempt to handle the subject of per centages. The ascending scale is easy enough: 5 added to 20 is a gain of 25%; given any sum of figures the doubling of it is an addition of 100%. But the moment the change is a decreasing calculation the inexperienced mathematician betrays himself, and even the expert is apt to stumble or go astray. An advance from 20 to 25 is an increase of 25%; but the reverse of this, that is, a decline from 25 to 20 is a decrease of only 20%. There are many persons, otherwise intelligent, who cannot see why the reduction of 100 to 50 is not a decrease of 100%, if an advance from 50 to 100 is an increase of 100%. The other day, an article of merchandise which had been purchased at 10 pence a pound was resold at 30 pence a pound—an advance of 200%. Whereupon, a writer in chronicling the sale said that at the beginning of the recent depression several invoices of the same class of goods which had cost over 30 pence per pound had been finally sold at 10 pence per pound—a loss of over 200%! Of course there cannot be a decrease or loss of more than 100%, because this wipes out the whole investment and makes the price nothing. An advance from 10 to 30 is a gain of 200%; but a decline of 30 to 10 is a loss of only 66?%. A very deserving trader was ruined by his miscalculations respecting mercantile discounts. The article he manufactured he at first supplied to retail dealers at a large profit of about 30%. He afterwards confined his trade almost exclusively to large wholesale houses, to whom he charged the same price, but allowed a discount of 20%, believing that he was still realising 10% for his own profit. His trade was very extensive, and it was not till after some years that he discovered the fact that in place of making 10% profit, as he imagined, by this mode of making his sales he was realising only 4%. To £100 value of goods he added 30%, and invoiced them at £130. At the end of each month, in the settlement of accounts amounting to some thousands of pounds with individual houses, he deducted 20%, or £26 on each £130, leaving £104, value of goods at prime cost, instead of £110, as he all along expected. 24. Divide 75 into two parts so that three times the greater may exceed seven times the less by 15. 25. What number is that which, being divided by 7 and the quotient diminished by 10, three times the remainder shall be 24? N.B. “Trust men and they will trust you,” said Emerson. “Trust men and they will bust you,” says the business man. _ 26. Two years ago to Hobart-town A certain number of folk came down. The square root of half of them got married, And then in Hobart no longer tarried; Eight-ninths of all went away as well (This is a story sad to tell): The square root of four now live here in woe! How many came here two years ago? PECULIARITIES OF SQUARES. The following is well worth examining:—
27. How many inches are there in the diagonal of a cubic foot? and how many square inches in a superficies made by a plane through two opposite edges of the cube? Father (who has helped his son in his arithmetic at home)—“What did the teacher remark when you showed him your sums?” Johnny—“He said I was getting more stupid every day.” A “CATCH.” 28. 2 plus 2 = 4 Find another number that when added to itself the sum will equal its square. _ 29. A man went to market with 3 baskets of oranges, which he sold at 6d. per dozen; after paying 2s. for refreshments and his coach fare, he had remaining 7s. The contents of the first and second baskets were equal to four times the first, and the contents of the first and half the third were together equal to the second; if he had sold the second and third baskets at 4d per dozen, he would have made as much money as he had now remaining. What was the coach fare? 30. A farmer has a triangular paddock, the sides of which are 900, 750, and 600 links; he requires to cut off 3 roods and 28 perches therefrom by a straight fence parallel to its least side. What distance must be taken on the largest and intermediate sides? THE SOVEREIGNS OF ENGLAND. By the aid of the following, the order of the kings and queens of England may be easily remembered:— First William the Norman, then William, his son; Henry, Stephen, and Henry, then Richard and John. Next Henry the Third, Edwards, one, two, and three; And again after Richard three Henrys we see. Two Edwards, third Richard, if rightly I guess, Two Henrys, sixth Edward, Queens Mary and Bess; Then Jamie the Scot, then Charles, whom they slew; Then followed Cromwell, another Charles, too. Next James, called the Second, ascended the Throne, Then William and Mary together came on. Then Anne, four Georges, and fourth William past, Succeeded Victoria, the youngest and last. 31. Take from 33 the fourth, fifth, and tenth parts of a certain number, and the remainder is 0. What is the number? A WALKING MATCH. 32. T bets D he can walk 7 miles to his 6 for any time or distance; so they agree to walk a certain distance, starting from opposite points. T starts from point M to walk to N. D starts from N and walks to M. They both started at the same moment, and met at a spot 10 miles nearer to N than M. T arrives at N in 8 hours, and D arrives at M in 12½ hours after meeting. Who wins the wager? How far from M to N? And find the pace at which each walked? THE ALPHABET. The total number of different combinations of the 26 letters of the alphabet is 403291461126605635584000000. All the inhabitants on the globe could not together, in a thousand million years, write out all the combinations, supposing that each wrote 40 pages daily, each page containing 40 different combinations of the letters. “10 INTO 9 MUST GO.” _ 33. Ten weary footsore travellers, all in a woeful plight, Sought shelter at a wayside inn one dark and stormy night. “Nine rooms-no more,” the landlord said, “have I to offer you; To each of eight a single bed, but the ninth must serve for two.” A din arose; the troubled host could only scratch his head, For of those tired men no two would occupy one bed. The puzzled host was soon at ease (he was a clever man), And so, to please his guests, devised this most ingenious plan. Bobby (just from school)—“Mamma, I’ve got through the promisecue-us examples, an’ I’m into dismal fractures.” 34. Find the expense of flooring a circular skating rink 30 feet in diameter at 2s. 3d. per square foot, leaving in the centre a space for a band kiosk in the shape of a regular hexagon, each side of which measures 24 inches. 35. Gold can be hammered so thin that a grain will make 56 square inches for leaf gilding. How many such leaves will make an inch thick if the weight of a cubic foot of gold is 12 cwt. 95 lbs.? School Inspector: “What part of speech is the word “am”? Smart Cockney Youth: “What? the ‘’am’ what you eat, sir, or the ’am‘ what you is?” MIND-READING WITH CARDS. Hand the pack (a full one) to be shuffled by as many spectators as wish; then propose that someone takes the pack in his hand and secretly chooses a card, not removing it, but noticing at what number it stands counting from the bottom; he then returns the pack to you. Now you have to tell what number the card is from the top. You ask any one of the spectators to choose any number between 40 and 50, and whatever number is chosen the card will appear at that number in the pack. Let us suppose the number chosen is 48. You then say that it is not necessary for you to even see the cards, which will give you a good excuse for holding them under the table, or behind your back. Now subtract the number chosen, 48, from 52, which gives remainder 4, count off that many cards from the top, and place them at the bottom. You next say to the gentleman who chooses the card, that “it is now number 48, according to the general desire, would you please let us know at what number it originally stood?” Suppose he answers 7. Then, in order to save time, you commence counting from the top at that number, dealing off the cards one by one, calling the first card 7, the next 8, and so on. When you reach 48, it will be the card the gentleman had chosen. It is not necessary to limit the choice of position to between 40 and 50, but it is better for two reasons. First, that the number chosen be higher than that at which the card first stood, also the higher the number chosen, the fewer cards are there to slip from the top to the bottom. _ 36. Divide a St. George cross, by two straight cuts, into four pieces, so that the pieces, when put together, will form a square. PARSING. “What part of speech is ‘kiss’?” asked the High School teacher. “A conjunction,” replied one of the smart girls. “Wrong,” said the teacher, severely. “Next girl.” “A noun,” put in a demure maiden. “What kind of a noun?” continued the teacher. “Well—er—it is both common and proper,” answered the shy girl, and she was promoted to the head of the class. “QUICK.” Teacher (to class)—“What is velocity?” Bright Youth—“Velocity is what a person puts a hot plate down with.” OFFICE RULES. I. Gentlemen entering this Office will please leave the door wide open. II. Those having no business will please call often, remain as long as possible, take a chair, make themselves comfortable, and gossip with the Clerks. III. Gentlemen are requested to smoke, and expectorate on the floor, especially during Office Hours; Cigars and Newspapers supplied. IV. The Money in this Office is not intended for business purposes—by no means—it is solely to lend. Please note this. V. A Supply of Cash is always provided to Cash Cheques for all comers, and relieve Bank Clerks of their legitimate duties. Stamped cheque forms given gratis. VI. Talk loud and whistle, especially when we are engaged; if this has not the desired effect, sing. VII. The Clerks receive visits from their friends and their relatives; please don’t interrupt them with business matters when so engaged. VIII. Gentlemen will please examine our letters, and jot down the Names and Addresses of our Customers, particularly if they are in the same profession. IX. As we are always glad to see old friends, it will be particularly refreshing to receive visits and renewal of orders from any former Customer who has passed through the Bankruptcy Court, and paid us not more than Sixpence in the Pound. A Warm welcome may be relied on. X. Having no occupation for our Office Boy, he is entirely at the service of callers. XI. Our Telephone is always at the disposal of anyone desirous of using it. XII. The following are kept at this Office for Public Convenience:— XIII. Should you find our principals engaged, do not hesitate to interrupt them. No business can possibly be of greater importance than yours. XIV. If you have the opportunity of overhearing any conversation, do not hesitate to listen. You may gain information which may be useful in the event of disputes arising. XV. In case you wish to inspect our premises, kindly do so during wet weather, and carry your umbrella with you. We admire the effect on the floor; it gives an air of comfort to the establishment. (The Umbrella Stand is only for ornament, and on no account to be used). P.S.—Our hours for listening to Commercial Travellers, Beggars, Hawkers, and Advertising Men are all day. We attend to our Business at Night only. A NEW WAY OF PUTTING IT. “Dirty days hath September, April, June and November; From January up to May, The rain it raineth every day. All the rest have thirty-one, Without a blessed gleam of sun; And if any of them had two and thirty, They’d be just as wet and twice as dirty.” Does the top of a carriage wheel move faster than the bottom? This question seems absurd. That the top moves faster, however, is perfectly correct; for if not it would simply move round in the same place: in a wheel on a fixed axle the bottom moves backward as fast as the top moves forward; but in a wheel that is going forward, drawn by a progressive axle, the bottom does not go back at all, but remains almost stationary until it is its turn to rise and go forward. 37. A General, arranging his army in a solid square, finds he has 284 men to spare, but on increasing the sides of the square by one man, he wants 25 men to complete the square. How many men has he? “STEWING.” 38. A student reads two lines more of “Virgil” each day than he did the day before, and finds that, having read a certain quantity in 18 days, he will read at this rate the same quantity in the next 14 days. How much will he read in the whole time? 39. Two bootmakers who lived in the town of B., thrown out of employment, resolved to go to G., a town 24 miles north from B., where there is a large factory; one of them went straight on to G., but the other went first to C., a small township west of B., and then went direct to G., his whole journey being 45 miles. What is the distance from C. to G.? 40. A tree which grows each year 1 inch less than the previous year, grew a yard in the first year; the value of the tree at any time is equal to the number of pence in the cube of the number of yards of its height. What is the value of the tree when done growing? THIS OFTEN “STICKS” PEOPLE UP. 41. What two odd numbers multiplied together make 7? MAGIC SQUARES. A Magic Square is a series of figures arranged in the equal divisions of a square in such a manner that the figures in each row when added up, whether horizontally, vertically, or diagonally, form exactly the same sum. _ They have been called “Magic” because the ancients ascribed to them great virtues, and because this arrangement of numbers formed the basis and principle of their talismans. Archimedes devoted a great amount of attention to them, which has caused a great many to speak of them as “the squares of Archimedes.” They may be either odd or even. When the former, the following method will be found valuable:— With the digits from 1 to 25 form a square so that the numbers when added up horizontally, vertically, or diagonally will amount to 65. Method.—Imagine an exterior line of squares above the magic square you wish to form, and another on the right hand of it. These two imaginary lines are shown in the diagram. 1st. In placing the numbers in the square, we must go in the ascending diagonal direction from left to right, any number which, by pursuing this direction, would fall into the exterior line must be carried along that line of squares, whether vertical or horizontal, to the last square. Thus, 1 having been placed in the centre of the top row, 2 would fall into the exterior square above the fourth vertical line; then ascending diagonally 3 falls into the square diagonally from 2, but 4 falls out of it to the end of a horizontal line, and it must be carried along that line to the extreme left and there placed. Resuming our diagonal ascension to the right we place 5 where the reader sees it, and would place 6 in the middle of the top row, but as we find 1 is already there we look for the direction to 2nd. That when in ascending diagonally we come to a square already occupied, we must place the number which, according to the 1st rule should go into that occupied square directly under the last number placed: thus, in ascending with 4, 5, 6, the 6 must be placed under the 5, because the square next to 5 in diagonal direction is occupied. A Promising Sign—I O U. HOW TO FIND THE TOTAL OF A ROW OF Rule.—Multiply half the sum of the extremes by the square root of the greatest extreme. Referring to the example given above, we see that the extremes 1 and 25 added equal 26—half of which is 13; this multiplied by 5 (the square root of 25) gives 65 as the total for each row. Again, in the next question, the two extremes 1 and 81 equal 82, half of this sum is 41, which multiplied by 9 (the square root of 81) gives 369 as the total for each row. 42. Arrange the figures from 1 to 81 in a square that when added up horizontally, vertically, or diagonally the sum will be 369. HOW THEY WORKED IT. Mick and Pat, working in the country some distance from a hotel, arranged with the landlord to take to their hut a small keg of rum. They were unable to pay for the liquor at the time, having only one threepenny piece between them; but Mick proposed that every time he had a drink he would give Pat threepence, and Pat also agreed to pay Mick for his drinks, the cash thus gathered to be brought to the publican when the keg was empty. This proposal was accepted by the publican, the keg of rum handed over to the two Irishmen, who immediately started on their journey. They had not proceeded very far before their burden made them thirsty. Mick is the first to pull up with: “Hold on, Pat, I think I’ll have a drink.” “Begorra,” replied Pat, “you’ll have to pay me for it then.” Mick hands the 3d. to Pat before having a good “pull.” Pat now being the possessor of the price of a drink, slakes his thirst by paying Mick 3d. for it. This form of payment is kept up till the rum has disappeared. On their next visit to the hotel, the 3d piece is handed to the landlord as being payment, according to terms of agreement adopted by him. 43. Arrange the figure’s from 1 to 9 in a square, so that they will add up to 15, horizontally, vertically, or diagonally. _ 45. A man sold a horse for £35 and half as much as he gave for it, and gained thereby 10 guineas. What did he pay for the horse? THE DISHONEST SERVANT. _ 46. A gentleman having bought 28 bottles of wine, and suspecting his servant of tampering with the contents of the wine cellar, caused these bottles to be arranged in a bin in such a way as to count 9 bottles on each side. Nothwithstanding this precaution, the servant in two successive visits stole 8 bottles—4 each time—re-arranging the bottles each time so that they still counted 9 on a side. How did he do it? Father—“You are very backward in your arithmetic. When I was your age I was doing cube roots.” Boy—“What’s them?” Father—“What! You don’t know what they are? My! my! that’s terrible! There, give me your pencil. Now, we take, say, 28764289, and find the cube root. First, you divide—no, you point off—no—let me see?—um—yes—no—don’t stand there grinning like a Cheshire cat; go upstairs and stay in your bedroom for an hour.” A “TAKE-DOWN” WITH CARDS. This is a card trick which depends upon a certain “key,” the possessor of which will always have the advantage over his uninstructed adversary. It is played with the first six of each suit—the four aces in one row, next row the deuces, threes, fours, fives and sixes. The object now will be to turn down cards alternately, and endeavour to make thirty-one points by so turning without over-running that number. The chief point is to count so as to end with the following numbers: 3, 10, 17 or 24. For instance, we will suppose it your privilege to commence the count; you would commence with 3, and your adversary would add 6, which would make 9; it would be then your policy to add 1 and make 10; then, no matter what number he adds he cannot prevent you making 17, which gives you the command of the trick. We will suppose he adds 6 and make 16; then you add 1 and make 17; then he to add 6 and make 23, you add 1 and make 24; then he cannot add any number to make 31, as the highest number he can add is 6, which would only count 30, so that you can easily add the remaining 1 and make 31. If your adversary is not wary, you may safely turn indifferent numbers at the beginning, trusting to his ignorance to let you count 17 or 24; but, as his knowledge increases, he will soon learn that 24 is a critical number, and to play for it accordingly. If both players know the trick, the first to play must be the winner, as he is sure to begin with a 3, which commands the game. ON AN OFFICE DOOR IN GOULBURN. A baptism in Hades’ depths, As hot as boiling tar, Awaits the man who quits this room And leaves the door ajar. But he who softly shuts the door Shall dwell among the blest— Where the wicked cease from troubling And the weary are at rest. 47. There are 5 eggs on a dish; divide them amongst 5 persons so that each will get 1 egg and yet 1 still remain on the dish. 48. If a goose weighs 10 lbs. and a half of its own weight, what is the weight of the goose? THE GEOMETRICAL WONDER AND ARITHMETICAL ABSURDITY. _ 5 × 13 = 65 square inches. Take a piece of cardboard 13 inches long and 5 wide, thus giving a surface of 65 inches. Cut this strip diagonally, giving two pieces in the shape of a triangle, and measure exactly 5 inches from the larger end of each strip and cut in two pieces. Take these strips and put them into the shape of an exact square, and it will appear to be just 8 inches each way, or 64 inches—a loss of one square inch of superficial measurement with no diminution of surface. 49. If we buy 20 sheep for 20 shillings, and give 2s. for wethers, 1s. 6d. for ewes, and 4d. for lambs, how many of each must we buy? 50. A sets out from a place and travels 5 miles an hour. B sets out 4½ hours after A and travels in the same direction 3 miles in the first hour, 3½ miles the second hour, 4 miles the third hour, and so on. In how many hours will B overtake A? OFTEN ASKED. 51. What is the difference between 4 square miles and 4 miles square? TO TELL THE NUMBER THOUGHT OF ON A CLOCK. _ Ask a person to think of any number on the dial of a clock; you then point, promiscuously at the various numbers, telling the person to add the number of times you point to the number he thought of, and when the total reaches 20, you will be pointing at the number he selected. For instance, suppose he selected the number 5. You point indifferently 7 times at the various numbers, but the 8th time your pointer must be at XII., his addition will then be 13 (for 5 and 8 added equal 13), the next at XI., his addition then 14, next at X., and so on. When he calls 20, you will be pointing at the number he thought of—5. A very amusing experiment is to ask a person to write down the figures around the dial of a clock. Nearly all know that the figures are generally the Roman numerals; but, in writing them down, when they come to the four, it is very often written IV. instead of IIII. It is said that a certain king, being unable to find any other fault in a clock that had been constructed for him, declared that the figure four should be represented by four strokes (IIII) instead of IV. In vain did the clock-maker point out the mistake, for his majesty adhered obstinately to his own opinion, and angrily ordered the alteration to be made. This was done, and the precedent thus formed has been followed by clockmakers ever since. 52. At dinner table: one great grandfather, 2 grandfathers, 1 grandmother, 3 fathers, 2 mothers, 4 children, 3 grandchildren, 1 great grandchild, 3 sisters, 1 brother, 2 husbands, 2 wives, 1 mother-in-law, 1 father-in-law, 2 brothers-in-law, 3 sisters-in-law, 2 uncles, 3 aunts, 1 nephew, 2 nieces, and 2 cousins. How many persons? “Can February March?” he asked. “No, but April May,” was the reply. “Look here, old man, you are out of June.” “Don’t July about it.” “It is not often one gets the better of your August personage.” “Ha! now you have me Noctober.” And then there was work for the coroner. PANCAKE DAY. _ _ 53. On Shrove Tuesday last, I’ll tell you what pass’d In a neighbouring gentleman’s kitchen, Where pancakes were making, with eggs, and with bacon As good as e’er cut off a flitchen. The cook-maid she makes four lusty pancakes For William her favourite gardener, “Pray be quick with that four,” cries Jack, “and make more, For William won’t let me go partner.” Being sparing of lard, the pan’s bottom she marr’d In making the last of Will’s four; So she said, “Pr’ythee, John, run and borrow a pan, Or else I can’t make any more.” Jack soon got a pan, but found by his span That the first was more wide than the latter, This being a foot o’er, whereas that before Was three inches more and a quarter. Jack cries, “Don’t me cozen, but make half a dozen. For the pan is much less than before;” Says Will, “For a crown (and I’ll put the cash down) Your six will be more than my four.” “Tis done,” says brisk Jack, and his crown he did stake, So both of them sent for a gauger; The dimensions he takes, of all their pancakes, To determine this important wager. He found, by his stick, they were equally thick, So one of Will’s cakes he did take, Which he straight cut in twain, twelve one-fifth A Great Prophet—100 per cent. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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