A hundred and one by fifty divide,To this let a cypher be duly applied;And when the result you can rightly divine,You find that its value is just one in nine—
A hundred and one by fifty divide,To this let a cypher be duly applied;And when the result you can rightly divine,You find that its value is just one in nine—
A hundred and one by fifty divide,To this let a cypher be duly applied;And when the result you can rightly divine,You find that its value is just one in nine—
is solved by CLIO, one of the nine Muses.
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The man who paid a penny on Monday morning to cross the ferry, spent half of what money he then had left in the town, and paid another penny to recross the ferry, and who repeated this course on each succeeding day, reaching home on Saturday evening with one penny in his pocket, started on Monday with £1 1s. 1d. in hand.
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When the three men agreed to share their mangoes equally after giving one to the monkey, and when each helped himself to a third after giving one to the monkey, without knowing that anyone had been before him, and they finally met together, gave one to the monkey, and divided what still remained, there must have been at least seventy-nine mangoes for division at the first.
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If, after having looked at my watch between 4 and 5, I look again between 7 and 8, and find that the hour and minute-hands have then exactly changed places, it was 3612⁄13minutes past 4 when I first looked. At that time the hour-hand would be pointing to 231⁄13minutes on the dial, and at 231⁄13minutes past 7 the hour hand would be pointing to 3612⁄13minutes.
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The number consisting of 22 figures, ofwhich the last is 7, which is increased exactly sevenfold if this 7 is moved to the first place, is1,014,492,753,623,188,405,797.
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The two sacks of wheat, each 4 feet long and 3 feet in circumference, which the farmer sent to the miller in repayment for one sack 4 feet long and 6 feet in circumference, far from being a satisfactory equivalent, contained but half the quantity of the larger sack, for the area of a circle the diameter of which is double that of another is equal to four times the area of that other.
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The five gamblers, who made the condition that each on losing should pay to the others as much as they then had in hand, and who each lost in turn, and had each £32 in hand at the finish, started with £81, £41, £21, £11, and £6 respectively.
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If we know the square of any number, we can rapidly determine the square of the next number, without multiplication, by adding the two numbers to the known square. Thus if we know that the square of 87 is 7569,
then the square of 88 = 7569 + 87 + 88 = 7744;so too the square of 89 = 7744 + 88 + 89 = 7921;and the square of 90 = 7921 + 89 + 90 = 8100.
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The two numbers which solve theproblem—
Two numbers seek which make eleven,Divide the larger by the less,The quotient is exactly seven,As all who find them will confess—
Two numbers seek which make eleven,Divide the larger by the less,The quotient is exactly seven,As all who find them will confess—
Two numbers seek which make eleven,Divide the larger by the less,The quotient is exactly seven,As all who find them will confess—
are 13⁄8and 95⁄8, for 13⁄8+ 95⁄8= 11, and77⁄8÷11⁄8= 7.
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There must be nine things of each sort, in order that 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9different selections may be made from twenty sorts of things.
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The women who had respectively 33, 29, and 27 apples, and sold the same number for a penny, receiving an equal amount of money, began by selling at the rate of three a penny. The first sold ten pennyworth, the second eight pennyworth, and the third seven pennyworth.
The first had then left three apples, the second five, and the third six. These they sold at one penny each, so that they received on thewhole—
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Thepuzzle—
Take five from five, oh, that is mean!Take five from seven, and this is seen—
Take five from five, oh, that is mean!Take five from seven, and this is seen—
Take five from five, oh, that is mean!Take five from seven, and this is seen—
is solved byfie,seen.
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If a bun and a half cost three halfpence, it is plain that each bun costs a penny, but, by general custom, you buy seven for sixpence.
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The hands of a watch would meet each other twenty-five times in a day, if the minute-hand moved backwards and the hour-hand forwards. They are, of course, together at starting.
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The only way in which half-a-crown can be equally divided between two fathers and two sons, so that a penny is the smallest coin made use of, is to give tenpence each to a grandfather, his son, and his grandson.
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If the number of the revolutions of a bicycle wheel in six seconds is equal to the number of miles an hour at which it is running, the circumference of the wheel is 84⁄5feet.
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The hour that struck was twelve o’clock.
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Sixty years.
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If I jump off a table with a 20lb dumb-bell in my hand there is no pressure upon me from its weight while I am in the air.
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If at a bazaar I paid a shilling on entering each of four tents, and another shilling on leaving it, and spent in each tent half of what was in my pocket, and if my fourth payment on leaving took my last shilling, I started with 45s., spending 22s. in tent 1, 10s. in tent 2, 4s. in tent 3, and 1s. in tent 4, having also paid to the doorkeepers 8s.
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When rain is falling vertically at 5 miles an hour, and I am walking through it at 4 miles an hour, the rain drops will strike the top of my umbrella at right angles if I hold it at an angle of nearly 39 degrees.
As I walk along, meeting the rain, the effect is the same as it would be if I was standing still, and the wind was blowing the rain towards me at the rate of 4 miles an hour.
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When one monkey descends from the top of a tree 100 cubits high, and makes its way to a well 200 yards distant, while another monkey, leaping upwards from the top, descends by the hypotenuse to the well, both passing over an equal space, the second monkey springs 50 cubits into the air.
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The steamboat which springs a leak 105 miles east of Tynemouth Lighthouse, and, putting back, goes at the rate of 10 miles an hour the first hour, but loses ground to the extent in each succeeding hour of one-tenth of her speed in the previous hour, never reaches the lighthouse, but goes down 5 miles short of it.
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Twenty-one hens will lay ninety-eight eggs in a week, if a hen and a-half lays an egg and a-half in a day and a-half. Evidently one egg is laid in a day by a hen and a-half, that is to say three hens lay two eggs in a day. Therefore, twenty-one hens lay fourteen eggs, in a day, or ninety-eight in a week.
Q. E. D. (Quite easily done!)
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If the population of Bristol exceeds by 237 the number of hairs on the head of anyone of its inhabitants that are not bald, at least 474 of them must have the same number of hairs on their heads.
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In tipping his nephew from seven different coins, the uncle may give or retain each, thus disposing of it in two ways, or of all in 2 × 2 × 2 × 2 × 2 × 2 × 2 ways. But as one of these ways would be to retain them all, there are not 128, but only 127 possible variations of the tip.
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The prime number which fulfils the various conditions of the question is 127. Increased by one-third, excluding fractions, it becomes 169, the square of 13. If its first two figures are transposed, and it is increased by one-third, it becomes 289, the square of 17. If its first figure is put last, and it is increased by one-third, it becomes 361, the square of 19. If, finally, its three figures are transposed, and then increased by one-third, it becomes 961, the square of 31.
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Six things can be divided between two boys in 62 ways. They could becarriedby two boys in 64 ways (2 × 2 × 2 × 2 × 2 × 2), but they are notdividedbetween two boys if all are given to one, so that two of the 64 ways must be rejected.
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The highest possible score that the dealer can make at six cribbage, if he is allowed to select the cards, and to determine the order of play, is 78. The dealer and his opponent must each hold 3, 3, 4, 4, the turn-up must be a 5, and crib must have the knave of the suit turned up, and 5, 5, 5. It will amuse many of our readers to test this with the cards.
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The picture frame must be 3 inches in width all round, if it is exactly to equal in area the picture it contains, which measures 18 inches by 12 inches.
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If my mother was 20 when I was born, my sister is two years my junior, and my brother is four years younger still, our ages are 56, 36, 34, and 30.
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The spider in the dockyard, whose thread was drawn from her by a revolving capstan 1 foot in diameter, until 73 feet of it were paid out, after walking for a mile round and round the capstan at the end of the stretched thread in an effort to unwind it all, had, when she stopped in her spiral course, 49 more feet to walk to complete her task.
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The mountebank at a fair, who offered to return any stake a hundredfold to anyone who could turn up all the sequence in twenty throws of dice marked each on one face only with 1, 2, 3, 4, 5, or 6, should in fairness have engaged to return 2332 times the money; for of the 46,656 possible combinations of the faces of the dice, only one can give the six marked faces uppermost. Thus the chance of throwing them all at one throw is expressed by1⁄46656, and in twenty throws by about1⁄2332.
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If 90 groats (each = 4d.) feed twenty cats for three weeks, and five cats consume as much as three dogs, seventy-two hounds can be fed for £39 in a period of ninety-one days.
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When equal wine-glasses, a half and a third full of wine, are filled up with water, and their contents are mixed, and one wine-glass is filled with the mixture, it contains5⁄12wine and7⁄12water.
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The arrangement by which St Peter is said to have secured safety for the fifteen Christians, when half of the vessel’s passengers were thrown overboard in a storm, is asfollows:—
XXXXIIIIIXXIXXXIXIIXXIIIXIIXXI
Each Christian is represented by an X, and if every ninth man is taken until fifteen have been selected, no X becomes a victim.
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If Farmer Southdown’s cow had a fine calf every year, and each of these, and their calves in their turn, at two years old followed this example, the result would be no less than 2584 head in sixteen years.
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The number of the flock was 301. This is found by first taking the least common multiple of 2, 3, 4, 5, 6, which is 60, and then finding the lowest multiple of this, which with 1 added is divisible by 7. This 301 is exactly divisible by 7, but by the smaller numbers there is 1 as remainder.
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The rule for determining easily the number of round bullets in a flat pyramid, with a base line of any length, isthis:—
Add a half to half the number on the base line, and multiply the result by the number on that line. Thus, if there are twelve bullets as afoundation—
12 +1⁄2=13⁄2; and13⁄2×12⁄1= 78.
The same result is reached by multiplying the number on the base line by a number larger by one, and then halving the result.Thus—
12 × 13 = 156, 156 ÷ 2 = 78.
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We can gather from thelines—
Old General HostA battle lost,And reckoned on a hissing,When he saw plainWhat men were slain,And prisoners, and missing.To his dismayHe learned next dayWhat havoc war had wrought;He had, at most,But half his hostPlus ten times three, six, ought.One-eighth were lainOn beds of pain,With hundreds six beside;One-fifth were dead,Captives, or fled,Lost in grim warfare’s tide.Now, if you can,Tell me, my man,What troops the general numbered,When on that nightBefore the fightThe deadly cannon slumbered?
Old General HostA battle lost,And reckoned on a hissing,When he saw plainWhat men were slain,And prisoners, and missing.To his dismayHe learned next dayWhat havoc war had wrought;He had, at most,But half his hostPlus ten times three, six, ought.One-eighth were lainOn beds of pain,With hundreds six beside;One-fifth were dead,Captives, or fled,Lost in grim warfare’s tide.Now, if you can,Tell me, my man,What troops the general numbered,When on that nightBefore the fightThe deadly cannon slumbered?
Old General HostA battle lost,And reckoned on a hissing,When he saw plainWhat men were slain,And prisoners, and missing.
To his dismayHe learned next dayWhat havoc war had wrought;He had, at most,But half his hostPlus ten times three, six, ought.
One-eighth were lainOn beds of pain,With hundreds six beside;One-fifth were dead,Captives, or fled,Lost in grim warfare’s tide.
Now, if you can,Tell me, my man,What troops the general numbered,When on that nightBefore the fightThe deadly cannon slumbered?
that old General Host had an army 24,000 strong.
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When the farmer sent five pieces of chain of 3 links each, to be made into one continuous length, agreeing to pay a penny for each link cut, and a penny for each link joined, the blacksmith, if he worked in the best interest of the farmer, could only charge sixpence: for he could cut asunder one set of 3 links, and use these three single links between the other four sets.
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If, in a parcel of old silver and copper coins, each silver piece is worth as many pence as there are copper coins, and each copper coin is worth as many pence as there are silver coins, there are eighteen silver and six copper coins, when the whole parcel is worth eighteen shillings.
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These are five groups that can be arranged with the numbers 1 to 11 inclusive, so that they are allequal:—
(82- 52+ 1) = (112- 92) = (72- 32) = (62+ 22) = 4(10).
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John Bull, under the conditions given, lived to the age of eighty-four years.
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The two numbers to each of which, or to the halves of which, unity is added, forming in every case a square number, are 48 and 1680.
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The true weight of a cheese that seemed to weigh 16 ℔s. in one scale of a balance with arms of unequal length, and only 9℔s. in the other, is 12℔. This is found by multiplying the 16 by the 9, and finding the square root of the result.
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The two parts into which 100 can be divided, so that if one of them is divided by the other the quotient is again exactly 100 are 991⁄101and100⁄101.
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If, with marbles in two pockets, I add one to those in that on the right, and then multiply its contents by the number it held at first, and after dealing in a similar way with those on the left, find the difference between the two results to be 90; while if I multiply the sum of the two original quantities by the square of their difference the result is 176, I started with twenty-three in the right-hand pocket and twenty-one in the other.
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The circle of twenty-one friends who arranged to meet each week five at a time for Bridge so long as exactly the same party did not meet more than once, and who wished to hire a central room for this purpose, would need it for no less than 20,349 weeks, or more than 390 years, to carry out their plan.
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If a herring and a half costs (not cost) a penny and a half, the price of a dozen such quantities is eighteenpence.
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The sum of money which in a sense appears to be the double of itself is 1s. 10d., for we may write itoneandtenpence ortwoandtwentypence.
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The “comic arithmetic” question set by Dr BulbousRoots—
Divide my fifth by my first, and you have my fourth; subtract my first from my fifth, and you have my second; multiply my first by my fourth followed by my second, and you have my third; place my second after my first, and you have my third multiplied by my fourth—is solved by COMIC.
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If the earth could stand still, and a straight tunnel could be bored through it, a cannon ball dropped into it, if there is no air or other source of friction, would oscillate continually from end to end.
Taking air into account, the ball would fall short of the opposite end at its first lap, and in succeeding laps its path would become shorter and shorter, until its initial energy was exhausted, when it would come to rest at the centre.
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He sent 163. She sent 157.
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When twins were born the estate was properly dividedthus:—
So the son takes four-sevenths, the widow two-sevenths, and the daughter one-seventh of the estate.
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If each of my strides forwards or backwards across a 22 feet carpet is 2 feet, and I make a stride every second; and if I take three strides forwards and two backwards until I cross the carpet, I reach the end of it in forty-three seconds. In three steps I advance 6 feet. Then in two steps I retrace 4 feet, thus gaining only 2 feet in five steps,i.e., in five seconds. I therefore advance 16 feet in forty seconds, and three more strides cover the remaining 6 feet.
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If the captain of a vessel chartered to sail from Lisbon to New York, which appear on a map of the world to be on the same parallel of latitude, and which are, along the parallel,about 3600 miles apart, takes his ship along this parallel, he will not be doing his best for the impatient merchant who has had an urgent business call to New York.
The shortest course between the two points is traced by a segment of a “great circle,” having its centre at the centre of the earth, and touching the two points. This segment lies wholly north of the parallel, and is the shortest possible course.
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When John and Harry, starting from the right angle of a triangular field, run along its sides, and meet first in the middle of the opposite side, and again 32 yards from their starting point, if John’s speed is to Harry’s as 13 to 11, the sides of the field measure 384 yards.
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If two sorts of wine when mixed in a flagon in equal parts cost 15d., but when mixed so that there are two parts ofAto three ofBcost 14d., a flagon ofAwould cost 20d., and a flagon ofB10d.
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If, when a man met a beggar, he gave him half of his loose cash and a shilling, and meeting another gave him half what was left and two shillings, and to a third half the remainder and three shillings, he had two guineas at first.
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The clerk who has two offers of work from January 1, one fromAof £100 a year, with an annual rise of £20, and the other fromBof £100 a year, with a half-yearly rise of £5, should acceptB’s offer.
The half-yearly payments fromA(allowing for the rise), would be 50, 50, 60, 60, 70, 70, etc., etc.; and fromBthey would be 50, 55, 60, 65, 70, 75, etc., etc., so thatB’s offer is worth £5 a year more thanA’s always.
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If I have a number of florins and half-crowns, but no other coins, I can pay my tailor £11, 10s. in 224 different ways.
This can be found thus by rule of thumb: Start with 0 half-crowns and 115 florins. Then 4 half-crowns and 110 florins. Add 4 half-crowns and deduct 5 florins each time till 92 half-crowns and 0 florins is reached.
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The monkey climbing a greased pole, 60 feet high, who ascended 3 feet, and slipped back 2 feet in alternate seconds, reached the top in 1 minute, 55 seconds, for he did not slip back from the top.
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When Adze, the carpenter, secured his tool-chest with a puzzle lock of six revolving rings, each engraved with twelve different letters, the chances against any one discovering the secret word formed by a letter on each ring was 2,985,983 to 1; for the seventy-two letters may be placed in 2,985,984 different arrangements, only one of which is the key.
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The five married couples who arranged to dine together in Switzerland at a round table, with the ladies always in the same places, so long as the men could seat themselves each between two ladies, but never next to his own wife, were able under these conditions to enjoy thirteen of these nights at the round table.
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If in a calm the tip of a rush is 9 inches above the surface of a lake, and as the wind rises it is gradually blown aslant, until at the distance of a yard it is submerged, it is growing in water that is 5 feet 71⁄2inches deep.
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Aminta was eighteen.
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When Dick took a quarter of the bag of nuts, and gave the one over to the parrot, and Tom and Jack and Harry dealt in the same way with the remainders in their turns, each finding a nut over from the reduced shares for the bird, and one was again over when they divided the final remainder equally, there were, at the lowest estimate, 1021 nuts in the bag.
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Eight and a quarter is the answer to the nonsensequestion—
If five times four are thirty-three,What will the fourth of twenty be?
If five times four are thirty-three,What will the fourth of twenty be?
If five times four are thirty-three,What will the fourth of twenty be?
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The similar fraction of a pound, a shilling, and a penny which make up exactly a pound are asfollows:—
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When Dr Tripos thought of a number, added 3, divided by 2, added 8, multiplied by 2, subtracted 2, and thus arrived at double the number, he started with 17.
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WhenAandBdeposited equal stakes withC, and agreed that the one who should first win three games of billiards should take all, but consented to a division in proper shares whenAhad won two games andBone, it was evident that ifAwon the next game all would go to him,while if he lost he would be entitled to one half. One case was as probable as the other, therefore he was entitled tohalf of these sums taken together; that is, to three quarters of the stakes, andBto a quarter only.
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The average speed of a motor which runs over any course at 10 miles an hour, and returns over the same course at 15 miles an hour, is 12 miles an hour, and not 121⁄2, as might be imagined. Thus a run of 60 miles out takes, under the conditions, six hours, and the return takes four hours; so that the double journey of 120 miles is done in ten hours, at an average speed of 12 miles an hour.
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Farmer Hodge, who proposed to divide his sheep into two unequal parts, so that the larger part added to the square of the smaller part should equal the smaller part added to the square of the larger part, had but one sheep.
Faithful to his word, he divided this sheep into two unequal parts,2⁄3and1⁄3, and was able to show that2⁄3+1⁄9=7⁄9, and that1⁄3+4⁄9=7⁄9. He was heard to declare further, and he was absolutely right, thatno number larger than1 can be so divided as to satisfy the conditions which he had laid down.
The fact thatsheepis both singular and plural, adds much to the perplexing points of this attractive problem.
Here is a very simple proof that the numbermust be1:—
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A horse that carries a load can draw a greater weightup the shaft of a minethan a horse that bears no burden. The load holds him more firmly to the ground, and thus gives him greater power over the weight he is raising from below.
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In the six chests, of which two contained pence, two shillings, and two pounds, there must have been at least the value of 506 pence. This can be divided into 22 (or 19 + 3) shares of 23d. each, or 23 (19 + 4) shares of 22d. each. Evidently then the treasure can be divided so that 19 men have equal shares, while their captain has either 3 shares or 4 shares.
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If I bought a parcel of nuts at 49 for 2d., and divided it into two equal parts, one of which I sold at 24, the other at 25 a penny; and if I spent and received an integral number of pence, but bought the least possible number of nuts, I bought 58,800 nuts, at a cost of £10, and I gained a penny.
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When, with a purse containing sovereigns and shillings, after spending half of its contents, I found as many pounds left as I had shillings at first, I started with £13, 6s.
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When the lady replied to a question as to herage—