Chapter 16

Two numbers seek which make eleven,Divide the larger by the less,The quotient is exactly seven,As all who solve it will confess.

Solution

36. If there are twenty sorts of things from which 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 different selections can be made, how many of each sort are there?

Solution

Three women, with no money, went to market. The first had thirty-three apples, the second twenty-nine, the third twenty-seven. Each woman sold the same number of apples for a penny. They all sold out, and yet each received an equal amount of money. How was this?

Solution

38.

Take five from five, oh, that is mean!Take five from seven, and this is seen.

Solution

If a bun and a half cost three halfpence, how many do you buy for sixpence?

Solution

How many times in a day would the hands of a watch meet each other, if the minute-hand moved backward and the hour-hand forward?

Solution

How can half-a-crown be equally divided between two fathers and two sons so that a penny is the coin of smallest value given to them?

Solution

42. If the number of the revolutions of the wheel of a bicycle in six seconds is equal to the number of miles an hour at which it is running, what is the circumference of the wheel?

Solution

Hearing a clock strike, and being uncertain of the hour, I asked a policeman. He had a turn for figures, and replied: “Take half, a third, and a fourth of the hour that has just struck, and the total will be one larger than that hour.” What o’clock was it?

Solution

44.

If cash is lent at five per cent.To those who choose to borrow,How soon shall I be worth a poundIf I lend a crown to-morrow?

Solution

45. If I jump off a table with a 20-lb. dumb-bell in my hand, what is the pressure upon me of its weight while I am in the air?

Solution

46. In charitable mood I went recently to a bazaar where there were four tents arranged to tempt a purchaser. At the door of each tent I paid a shilling, and in each tent I spent half of the money remaining in my purse, giving the door-keepers each another shilling as I came out.

It took my last shilling to pay the fourth door-keeper. How much money had I at first, and what did I spend in each tent?

Solution

47. Rain is falling vertically, at a speed of 5 miles an hour. I am walking through it at 4 miles an hour. At what angle to the vertical must I hold my umbrella, so that the raindrops strike its top at right angles?

Solution

48. The following interesting problem, translated from the original Sanscrit, is given by Longfellow in his“Kavanagh”:—

“A tree, 100 cubits high, is distant from a well 200 cubits. From the top of this tree one monkey descends, and goes to the well.Another monkey leaps straight upwards from the top, and then descends to the well by the hypotenuse. Both pass over an equal space. How high does the second monkey leap?”

Solution

A steamboat 105 miles east of Tynemouth Lighthouse springs a leak. She puts back at once, and in the first hour goes at the rate of 10 miles an hour.

More and more water-logged, she decreases her speed each succeeding hour at the rate of one-tenth of what it has been during the previous hour. When will she reach the lighthouse?

Solution

If a hen and a half lays an egg and a half in a day and a half, how many eggs will twenty-one hens lay in a week?

Solution

If the population of Bristol exceeds by two hundred and thirty-seven the number of hairs on the head of any one of its inhabitants, how many of them at least, if none are bald, must have the same number of hairs on their heads?

Solution

A benevolent uncle has in his pocket a sovereign, a half-sovereign, a crown, a half-crown, a florin, a shilling, and a threepenny piece. In how many different ways can he tip his nephew, using only these coins, and how is this most easily determined?

Solution

Here is a prime problem, in more senses than one, which will tax the ingenuity of our solvers:—I am a prime number of three figures. Increased by one-third, ignoring fractions, I become a square number. Transpose my firsttwo figures and increase me by one-third, and again I am a square number. Put my first figure last, and increase me by one-third, and I am another square number. Reverse my three figures, and increase as before by one-third, and for a fourth time I become a square number. What are my original figures?

Solution

In how many different ways can six different things be divided between two boys?

Solution

What is quite the highest number that can be scored at six card cribbage by the dealer, if he has the power to select all the cards, and to determine the order in which every card shall be played?

Solution

56. A fanciful collector, who bought pictures with more regard to quantity than quality, gave instructions that the area of each frame should exactly equal that of the picture it contained, and that the frames should be of the same width all round.

At an auction he picked up a so-called “old master,” unframed, which measured 18 inches by 12 inches. What width of frame will fulfil his conditions?

Solution

57. Our family consists of my mother, a brother, a sister, and myself. Our average age is thirty-nine. My mother was twenty when I was born; my sister is two years my junior, and my brother is four years younger than she is. What are our respective ages?

Solution

A spider in a dockyard unwittingly attached her web to a mechanical capstan 1 foot in diameter, at the moment when it began to revolve. To hold her ground she paid out 73 feet of thread, when the capstan stopped, and she found herself drained of silk.

To make the best of a bad job she determined to unwind her thread, walking round and round the capstan at the end of the stretched thread. When she had gone a mile in her spiral path she stopped, tired and in despair. How far was she then from the end of her task?

Solution

A mountebank at a fair had six dice, each marked only on one face 1, 2, 3, 4, 5, or 6, respectively. He offered to return a hundredfold any stake to a player who should turn up all the six marked faces once in twenty throws. How far was this from being a fair offer?

Solution

60.

If ninety groats for twenty catsWill furnish three weeks’ fare,How many hounds for forty pounds,Less one, may winter there?Just ninety days and one assumeThe winter’s space to be;And note that what five cats consumeWill serve for dogs but three.

If ninety groats for twenty catsWill furnish three weeks’ fare,How many hounds for forty pounds,Less one, may winter there?

Just ninety days and one assumeThe winter’s space to be;And note that what five cats consumeWill serve for dogs but three.

(A groat = 4d.)

Solution

Two wineglasses of equal size are respectively half and one-third full of wine. They are filled up with water, and their contents are then mixed. Half of this mixture is finally poured back into one of the wineglasses. What part of this will be wine and what part will be water?

Solution

A legend goes that on a stout ship on which St Peter was carried with twenty-nine others, of whom fourteen were Christians and fifteen Jews, he so arranged their places, that when a storm arose, and it was decided to throw half of the passengers overboard, all the Christians were saved. The order was that every ninth man should be cast into the sea. How did he place the Christians and the Jews?

Solution

63. Farmer Southdown was the proud possessor of a prize cow, which had a fine calf every year for sixteen years. Each of these calves when two years old, and their calves also in their turn, followed this excellent example. How many head did they thus muster in sixteen years?

Solution

A shepherd was asked how many sheep he had in his flock. He replied that he could not say, but he knew that if he counted them by twos, by threes, by fours, by fives, or by sixes, there was always one over, but if he counted them by sevens, there was no remainder. What is the smallest number that will answer these conditions?

Solution

65. If a number of round bullets of equal size are arranged in rows one above another evenly graduated till a single bullet crowns the flat pyramid, how can their number be readily reckoned, however long the base line may be?

Solution

66.

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?

Solution

A farmer sends five pieces of chain, of three links each, to be made into one continuous length. He agrees to pay a penny for each link cut, and a penny for each link joined. What was the blacksmith entitled to charge if he worked in the best interest of thefarmer?

Solution

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, and the whole is worth eighteen shillings. How many are there of each?

Solution

69. Take the natural numbers 1 to 11, inclusive, and arrange them in five groups, not using any of them more than once, so that these groups are equal. Any necessary signs or indices may be used.

Solution

70. John Bull passed one-sixth of his life in childhood and one-twelfth as a youth. When one-seventh of his life had elapsed he had a son who died at half his father’s age, and John himself lived on four years more. How old was he at the last?

Solution

71. There are two numbers under two thousand, such that if unity is added to each of them, or to the half of each, the result is in every case a square number. Can you find them?

Solution

A cheese in one scale of a balance with arms of unequal length seems to weigh 16 lbs. In the other scale it weighs but 9 lbs. What is its true weight?

Solution

73. Can you divide 100 into two such parts that if the larger is divided by the lesser the quotient is also 100?

Solution

I have marbles in my two side pockets. If I add one to those in the right-hand pocket, and multiply its increased contents by the number it held at first, and then deal in a similar way with those in the other pocket, the difference between the two results is 90. If, however, I multiply the sum of the two original quantities by the square of their difference, the result is 176. How many marbles had I at first in each pocket?

Solution

75. A friendly circle of twenty-one persons agreed to meet each week, five at a time, for an afternoon of bridge, so long as they could do so without forming exactly the same party on any two occasions.

As a central room had to be hired, it was important to have some idea as to the length of time for which they would require it. How long could they keep up their weekly meetings?

Solution

A herring and a half costs a penny and a half; what is the price of a dozen?

Solution

What sum of money is in any sense seen to be the double of itself?

Solution

78. At the close of his lecture upon unknown quantities, Dr Bulbous Roots, in playful mood, wrote this puzzle on hisblackboard:—

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. What am I?

Solution

79. If we can imagine the earth at a standstill for the purpose of our experiment, and if a perfectly straight tunnel could be bored through its centre from side to side, what would be the course of a cannon ball dropped into it from one end, under the action of gravity?

Solution

80.

A lady to her lover cried,“How many notes have you of mine?”“Six more I’ve sent,” the youth replied,“Than I have had of thine.”“But if from one pound ten you takeThe pennies we on stamps have spent,One eighth their cost you thus will make.”How many had they had and sent?

A lady to her lover cried,“How many notes have you of mine?”“Six more I’ve sent,” the youth replied,“Than I have had of thine.”

“But if from one pound ten you takeThe pennies we on stamps have spent,One eighth their cost you thus will make.”How many had they had and sent?

Solution

A man, on the day of his marriage, made his will, leaving his money thus:—If a son should be born, two-thirds of the estate to that son and one-third to the widow. If a daughter should be born, two-thirds to the widow and one-third to that daughter. In the course of time twins were born, a boy and a girl. The man fell sick and died without making a fresh will. How ought his estate to be divided in justice to the widow, son, and daughter?

Solution

My carpet is 22 feet across. My stride, either backwards or forwards, is always 2 feet, and I make a stride every second. If I take three strides forwards and two backwards continuously until I cross the carpet, how long does it take me to reach the end of it?

Solution

83. A merchant at Lisbon has an urgent business call to New York. Taking these places to be, as they appear on a map of the world, on the same parallel of latitude, and at a distance measured along the parallel, of some 3600 miles, if the captain of a vessel chartered to go there sails along this parallel, will he be doing the best that he can for the impatient merchant?

Solution

84. Two schoolboys, John and Harry, start from the right angle of a triangular field, and run along its sides. John’s speed is to Harry’s as 13 is to 11.

They meet first in the middle of the opposite side, and again 32 yards from their starting point. How far was it round the field?

Solution

85. The following question is given and spelt exactly as it was contributed to a puzzle column by “John Hill, Gent.,” in1760:—

“A vintner has 2 sorts of wine, viz. A and B, which if mixed in equal parts a flagon of mixed will cost 15 pence; but if they be mixed in asesqui-alterproportion, as you should take two flagons of A as often as you take three of B, a flagon will cost 14 pence. Required the price of each wine singly.”

Solution

86. A man met a beggar and gave him half the money he had in his pocket, and a shilling besides. Meeting another he gave him half of what was left and two shillings, and to a third, he gave half of the remainder and three shillings. This left a shilling in his pocket. How much had he at first?

Solution

A young clerk wishes to start work at an office in the City on January 1st. He has two promising offers, one fromAof £100 a year, with a yearly rise of £20, the other fromBof £100 a year with a half-yearly rise of £5. Which should he accept, and why?

Solution

88. I have an abundance of florins and half-crowns, but no other coins. In how many different ways can I pay my tailor £11, 10s. without receiving change?

Solution

A monkey climbing up a greased pole ascends 3 feet and slips down 2 feet in alternate seconds till he reaches the top. If the pole is 60 feet high, how long does it take him to arrive there?

Solution

90. Old Adze, the village carpenter, who kept his tools in an open chest, found that his neighbours sometimes borrowed and forgot to return them.

To guard against this, he secured the lid of the chest with a letter lock, which carried six revolving rings, each engraved with twelve different letters. What are the chances against any one discovering the secret word formed by a letter on each ring, which will open the lock, and be the only key to the puzzle?

Solution

Five merry married couples happened to meet at a Swiss hotel, and one of the husbands laughingly proposed that they should dine together 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. How long would their nights at the round table be continued under these conditions?

Solution

92. In calm water the tip of a stiff rush is 9 inches above the surface of a lake. As a steady wind rises it is gradually blown aslant, until at the distance of a yard it is submerged. Can you decide from these data the depth of the water in which the rush grows?

Solution

93.

If to Aminta’s age exactIts square you add, and eighteen more,And from her age its third subtract,And to the difference add three score,The latter to the former thenWill just the same proportion bearAs eighteen does to nine times ten.Can you Aminta’s age declare?

Solution

94. A bag of nuts was to be divided thus among four boys:—Dick took a quarter, and finding that there was one over when he made the division, gave it to the parrot. Tom dealt in exactly the same way with the remainder, as did Jack and Harry in their turns, each finding one nut from the reduced shares to spare for theparrot. The final remainder was equally divided among the boys, and again there was one for the bird. How many nuts, at the lowest estimate, did the bag contain?

Solution

Here is an easyone:—

If five times four are thirty-three,What will the fourth of twenty be?

Solution

What fraction of a pound, added to the same fraction of a shilling, and the same fraction of a penny, will make up exactly one pound?

Solution

97. “Now, boys!” said Dr Tripos, “I think of a number, add 3, divide by 2, add 8, multiply by 2, subtract 2, and thus arrive at twice the number I thought of.” What was it?

Solution

Two club friends,AandB, deposit similar stakes withC, and agree that whoever first wins three games at billiards shall take the whole of them.Awins two games andBwins one. Upon this they determine to divide the stakes in proper shares. How must this division be arranged?

Solution

Not so simple as it sounds is the following compact little problem:—If I run by motor from London to Brighton at 10 miles an hour, and return over the same course at 15 miles an hour, what is my average speed?

Solution

“I can divide my sheep,” says Farmer Hodge, who from his schooldays had a turn for figures, “into two unequal parts, so that the larger part added to the square of the smaller part shall be equal to the smaller part added to the square of the larger part.”

How many sheep had the farmer?

Solution

101. The following question was proposed in an old book of Mathematical Curiosities published more than a hundred yearsago:—

“It often happens that if we take two horses, in every respect alike, yet, if both are put to the draught, that horse which is most loaded shall be capable of performing most work; so that the horse which carries the heavier weight can draw the larger load. How is this?”

Solution

In the king’s treasury were six chests. Two held sovereigns, two shillings, and two pence, in equal numbers of these coins. “Pay my guard,” said the king, “giving an equal share to each man, and three shares to the captain; give change if necessary.” “It may not be possible,” replied the treasurer, “and the captain may claim four shares.” “Tut, tut,” said the king, “it can be done whatever the amount of the treasure, and whether the captain has three shares or four.”

Was the king right? If so, how many men were there in the guard?

Solution

103. I bought a parcel of nuts at forty-nine for twopence. I divided it into two equal parts, one of which I sold at twenty-four, the other at twenty-five for a penny. I spent and received an integral number of pence, but bought the least possible number of nuts. How many did I buy? What did they cost? What did I gain?

Solution

104. My purse contained sovereigns and shillings. After I had spent half of its contents there were as many pounds left as I had shillings at first. With what sum did I start?

Solution

105. A lady was asked her age in a letter, and she replied by postcardthus:—

If first my age is multiplied by three,And then of that two-sevenths tripled be,The square root of two-ninths of this is four,Now tell my age, or never see me more!

What was her age?

Solution

If cars run at uniform speed on the twopenny tube, from Shepherd’s Bush to the Bank at intervals of two minutes, how many shall I meet in half an hour if I am travelling from the Bank to Shepherd’s Bush?

Solution

107. What would it cost me to keep my word if I were to offer my greengrocer a farthing for every different group of ten apples he could select from a basket of a hundred apples?

Solution

If the minute-hand of a clock moves roundin the opposite directionto the hour-hand, what will be the real time between three and four, when the hands are exactly together?

Solution

Two monkeys have stolen some filberts and some walnuts. As they begin their feast they see the owner of the garden coming with a stick. It will take him two and a half minutes to reach them. There are twice as many filberts as walnuts, and one monkey finishes the walnuts at the rate of fifteen a minute in four-fifths of the time and bolts. The other manages to finish the filberts just in time.

If the walnut monkey had stopped to help him till all was finished, when would they have got away if they ate filberts at equal rates?

Solution

A cashier, in payment for a cheque, gives by mistake pounds for shillings and shillings for pounds. The receiver spends half-a-crown, and then finds that he has twice as much as the cheque was worth. What was its value?

Solution

What five uneven figures can be added together so as to make up 14?

Solution

Three posts which vary in value are vacant in an office. In how many ways can the manager fill these up from seven clerks who apply for the appointments?

Solution

“It is now5⁄11of the time to midnight,” said the fasting man, who began his task at noon. What time was it?

Solution

114. If a clock takes six seconds to strike six how long will it take to strike eleven?

Solution

115. How would you arrange twenty horses in three stalls so as to have an odd number of horses in each stall?

Solution

Here is a pretty little problem, which has at any rate an Algebraic form, and is exceedinglyingenious:—

Givena,b,c, to findq.

Solution

117. Tom Evergreen was asked his age by some men at his club on his birthday in 1875. “The number of months,” he said, “that I have lived are exactly half as many as the number which denotes the year in which I was born.” How old was he?

Solution

Draw three circles of any size, and in any position, so long as they do not intersect, or lie one within another. How many different circles can be drawn touching all the three?

Solution

We have seen that the nine digits can be so dealt with, using each once, as to add up to 100. How can 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 be arranged so that they form a sum which is equal to 1?

Solution

How is it possible, by quite a simple method, to find the sum of the first fifty numbers without actually adding them together?

Solution

Two tram-cars,AandB, start at the same time.Aruns into a “lie by” in four minutes, and waits there five minutes, whenBmeets and passes it. Both complete the whole course at the same moment. In what time canAcomplete it without a rest?

Solution

122. Take 10, double it, deduct 10, and tell me what remains.

Solution

The average weight of the Oxford crew is increased by 2 lbs., when one of them, who weighs 12 stone, is replaced by a fresh man. What is the fresh man’s weight?

Solution

124. A motor car is twice as old as its tyres were when it was as old as its tyres are. When these tyres are as old as the car is now, the unitedages of car and tyres will be two years and a quarter. What are their respective ages now?

Solution

AandBon the edge of a desert can each carry provisions for himself for twelve days. How far into the desert can an advance be made, so that neither of them misses a day’s food?

Solution

A bottle of medicine and its cork cost half-a-crown, but the bottle and the medicine cost two shillings and a penny more than the cork. What did the cork cost?

Solution

127. A boat’s crew are afloat far from land with no sail or oars. How can they, without making any use of wind or stream, and without any outside help, regain the shore by means of a coil of rope which happens to be at hand.

Solution

What is the largest sum in silver that I can have in my pockets without being able to give change for a half-sovereign.

Solution

I have apples in a basket. Without cutting an apple I give half of the number and half an apple to one person; half of what then remains and half an apple to another, and half of what are still left and half an apple to a third. One apple now remains in the basket. How many were there at first?

Solution

130.

Back to Index Next