Curious Calculations

1.

There is a sum of money of such sort that its pounds, shillings, and pence, written down as one continuous number, represent exactly the number of farthings which it contains. What is it?

Solution

The Slip Carriage

2. If on a level track a train, running all the time at 30 miles an hour, slips a carriage, which is uniformly retarded by the brakes, and which comes to rest in 200 yards, how far has the train itself then travelled?

Solution

3.

A traveller said to the landlord of an inn, “Give me as much money as I have in my hand, and I will spend sixpence with you.” This was done, and the process was twice repeated, when the traveller had no money left. How much had he at first?

Solution

4.

How can we obtain eleven by adding one-third of twelve to four-fifths of seven?

Solution

Fill in the Gaps

5. Can you replace the missing figures in this mutilated long division sum?

2 1 5 ) * 7 * 9 * ( 1 * *

* 5 * 9

* 5 * 5

* 4 *

Solution

6.

I buy as many heads of asparagus in the market as can be contained by a string 1 foot long. Next day I take a string 2 feet long, and buy as many as it will gird, offering double the price that I have given before. Was this a reasonable offer?

Solution

Alike from Either End

7. As “one good turn deserves another,” first reverse me, then reverse my square, then my cube, then my fourth power. When all this is done no change has been made. What am I?

Solution

The Sealed Bags

8. How can a thousand pounds be so conveniently stored in ten sealed bags that any sum in pounds from £1 to £1000 can be paid without breaking any of the seals?

Solution

9.

This is at once a problem and a puzzle:—

Though you twist and turn me over,
Yet no change can you discover.
Take me thrice, and cut in twain,
You will find but one remain.

Solution

10.

Three gamblers, when they sit down to play, agree that the loser shall always double the sum of money that the other two have before them. After each of them has lost once, it is found that each has eight sovereigns on the table. How much money had each at starting?

Solution

Find the Multiplier

11. When Tom’s back was turned, the boy sitting next to him rubbed out almost all his sum. Tom could not remember the multiplier, and only this remained on his slate—

3 4 5

. . 7 6 .

Can you reconstruct the sum?

Solution

Juggling with the Digits

12. The sum of the nine digits is 45. Can you hit upon other arrangements of 1, 2, 3, 4, 5, 6, 7, 8, 9, writing each of them once only, which will produce the same total. Of course fractions may be used.

Solution

A Brain Twister

13. The combined ages of Mary and Ann are forty-four years. Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann. How old, then, is Mary?

Solution

14.

Mr Oldboy was playing backgammon with his wife on the eve of his golden wedding, and could not make up his mind whether he should leave a blot where it could be taken up by an ace, or one which a tré would hit.

His grandson, at home from Cambridge for the Christmas vacation, solved the question for him easily. What was his decision?

Solution

“ASK WHERE’S THE NORTH?”—Pope.

15. I am aboard a steamer, anchored in a bay where the needle points due north, and exactly 1200 miles from the North Pole. If the course is perfectly clear, and I steam continuously at the rate of 20 miles an hour, always steering north by the compass needle, how long will it take me to reach the North Pole?

Solution

16.

Three persons, A, B, and C, share twenty-one wine casks of equal capacity, of which seven are full, seven are half full, and seven are empty. How can these be so apportioned that each person shall have an equal number of casks, and an equal quantity of wine, without transferring any of it from cask to cask?

Solution

17.

A hungry mouse, in search of provender, came upon a box containing ears of corn. He could carry three ears home at a time, and only had opportunity to make fourteen journeys to and fro. How many ears could he add to his store?

Solution

18.

Take exactly equal quantities of lard and butter; mix a small piece of the butter intimately with all the lard. From this blend take a piece just as large as the fragment removed from the butter, and mix this thoroughly with the butter. Is there now more lard in the butter or more butter in the lard?

Solution

Where Is the Fallacy?

19. Here is an apparent proof that 2 = 3:—

4 - 10 = 9 - 15

4 - 10 + ⁄4 = 9 - 15 + ⁄4

and the square roots of these:—

2 - ⁄2 = 3 - ⁄2

therefore 2 = 3.

Solution

Parcel Post Limitations

20. Our Parcel Post regulations limit the length of a parcel to 3 feet 6 inches, and the length and girth combined to 6 feet. What is the largest parcel of any size that can be sent through the post under these conditions?

Solution

21.

How can we show, or seem to show, that either four, five, or six nines amount to one hundred?

Solution

Test Your Skill

22. Can you arrange nine numbers in the nine cells of a square, the largest number 100, and the least 1, so that the product by multiplication of each row, column and diagonal is 1000?

Solution

Catching Crabs

23. Seven London boys were at the seaside for a week’s holiday, and during the six week-days they caught four fine crabs in pools under the rocks, when the tide was out at Beachy Head.

Hearing of this, the twenty-one boys of a school in the neighbourhood determined to explore the pools; but with the same rate of success they only caught one large crab. For how long were they busy searching under the seaweed?

Solution

Setting a Watch

24. On how many nights could a watch be set of a different trio from a company of fifteen soldiers, and how often on these terms could one of them, John Pipeclay, be included?

Solution

“Imperial Cæsar”

25. If Augustus Cæsar was born on September 23rd, B.C. 63, on what day and in what year did he celebrate his sixty-third birthday, and by what five-letter symbol can we express the date?

Solution

26.

A was born in 1847, B in 1874. In what years have the same two digits served to express the ages of both, if they are still living?

Solution

This Should “Amuse”

27.

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.

Solution

Over the Ferry

28. A man started one Monday morning with money in his purse to buy goods in a neighbouring town. He paid a penny to cross the ferry, spent half of the money he then had, and paid another penny at the ferry on his return home.

He did exactly the same for the next five days, and on Saturday evening reached home with just one penny in his pocket. How much had he in his pocket on Monday before he reached the ferry?

Solution

The Men, the Monkey, and the Mangoes

29. Three men gathered mangoes, and agreed that next day they would give one to their monkey and divide the rest equally. The first who arrived gave one to the monkey, and then took his proper share; the second came later and did likewise, and the third later still, neither knowing that any one had preceded him. Finally they met, and, as there were still mangoes, gave one to the monkey, and shared the rest equally. How many mangoes at least must there have been if all the divisions were accurate?

Solution

30.

I look at my watch between four and five, and again between seven and eight. The hands have, I find, exactly changed places, so that the hour-hand is where the minute-hand was, and the minute-hand takes the place of the hour-hand. At what time did I first look at my watch?

Solution

A Large Order

31. There is a number consisting of twenty-two figures, of which the last is 7. If this is moved to the first place, the number is increased exactly sevenfold. Can you discover this lengthy number?

Solution

32.

A farmer borrowed from a miller a sack of wheat, 4 feet long and 6 feet in circumference. He sent in repayment two sacks, each 4 feet long and 3 feet in circumference. Was the miller satisfied?

Solution

The Five Gamblers

33. Five gamblers, whom we will call A, B, C, D, and E, play together, on the condition that after each hazard he who loses shall give to all the others as much as they then have in hand.

Each loses in turn, beginning with A, and when they leave the table each has the same sum in hand, thirty-two pounds. How much had each at first?

Solution

34.

Knowing that the square of 87 is 7569, how can we rapidly, without multiplication, determine in succession the squares of 88, 89, and 90?

Solution

Not Whole Numbers

35.

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

An Ample Choice

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

37.

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

Simple Subtraction

38.

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

Solution

39.

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

Solution

40.

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

41.

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

Size and Speed

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

43.

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

After Long Years

44.

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

Solution

On the Jump

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

At a Bazaar

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

Out in the Rain

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

A Monkey Puzzler

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

49.

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

50.

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

51.

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

52.

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

53.

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 first two 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

54.

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

Solution

55.

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

Covering the Walls

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

A Family Register

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

58.

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

59.

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

Cats’-Meat for Dogs

60.

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

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

(A groat = 4d.)

Solution

61.

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

62.

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

A Prolific Cow

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

64.

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

Ready Reckoning

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

The Tithe of War

66.

Old General Host
A battle lost,
And reckoned on a hissing,
When he saw plain
What men were slain,
And prisoners, and missing.

To his dismay
He learned next day
What havoc war had wrought;
He had, at most,
But half his host
Plus ten times three, six, ought.

One-eighth were lain
On 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 night
Before the fight
The deadly cannon slumbered?

Solution

67.

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 the farmer?

Solution

68.

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

A Feat with Figures

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

How Old Was John?

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

Figure It Out

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

72.

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

“Division Is as Bad!”

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

Solution

74.

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

A Surfeit of Bridge

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

76.

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

Solution

77.

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

Solution

Comic Arithmetic

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

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

Dropped Through the Globe

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

Love Letters

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 take
The pennies we on stamps have spent,
One eighth their cost you thus will make.”
How many had they had and sent?

Solution

81.

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

82.

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

No Time to Be Lost

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

Round the Angles

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

An Echo from the Past

85. The following question is given and spelt exactly as it was contributed to a puzzle column by “John Hill, Gent.,” in 1760:—

“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 a sesqui-alter proportion, 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

Promiscuous Charity

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

87.

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

Solution

How Can I Pay My Bill?

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

89.

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

“Safe Bind, Safe Find”

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

91.

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

Sounding the Depth

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

Aminta’s Age

93.

If to Aminta’s age exact
Its 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 then
Will just the same proportion bear
As eighteen does to nine times ten.
Can you Aminta’s age declare?

Solution

One for the Parrot

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 the parrot. 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

95.

Here is an easy one:—

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

Solution

96.

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

Mental Arithmetic

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

98.

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

Solution

99.

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

100.

“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

A Helpful Burden

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

“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

102.

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

Nuts to Crack

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

Money Matters

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

A Delicate Question

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

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

106.

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

Payment by Results

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

108.

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

Solution

109.

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

110.

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

111.

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

Solution

112.

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

113.

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

Solution

A Striking Time

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

Solution

What Are the Odds?

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

Solution

116.

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

Given a, b, c, to find q.

Solution

When Was He Born?

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

118.

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

119.

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

120.

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

Solution

121.

Two tram-cars, A and B, start at the same time. A runs into a “lie by” in four minutes, and waits there five minutes, when B meets and passes it. Both complete the whole course at the same moment. In what time can A complete it without a rest?

Solution

A Curious Deduction

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

Solution

123.

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

Ask Any Motorist

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 united ages of car and tyres will be two years and a quarter. What are their respective ages now?

Solution

125.

A and B on 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

126.

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

In a Fix

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

128.

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

129.

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

A Queer Division

130.

A third of twelve divide
By just a fifth of seven;
And you will soon decide
That this must give eleven.

Solution

131.

A motor goes 9 miles an hour uphill, 18 miles an hour downhill, and 12 miles an hour on the level. How long will it take to run 50 miles and return at once over the same course?

Solution

132.

In firing at a mark A hits it in two out of three shots, B in three out of four, and C in four out of five. The mark was hit 931 times. If each fired the same number of shots, how many hits did each make, and how many shots were fired?

Solution

133.

If a cat and a dog, evenly matched in speed, run a race out and back over a course 75 yards in all, and the dog always takes 5 feet at a bound and the cat 3 feet, which will win?

Solution

In a Fog

134. In a fog a man caught up a wagon going in the same direction at 3 miles an hour. If the wagon was just visible to him at a distance of 55 yards, and he could see it for five minutes, at how many miles an hour was he walking?

Solution

135.

Three horses start in a race. In how many different ways can they be placed by the judge?

Solution

New Zealand Football

136. The New Zealanders, winning a match against Oxbridge, scored 34 points, from tries and tries converted into goals.

If every try had been converted they would have made four-fifths of the maximum which a score of 34 points from tries and goals can yield.

What is this maximum, and what was their actual score?

Solution

137.

What is the smallest number, of which the alternate figures are cyphers, that is divisible by 9 and by 11?

Solution

138.

Here is an interesting little problem:—A, with 8d. in his hand, meets B and C, who have five and three loaves respectively. In hungry mood they all agree to share the loaves equally, and to divide the money fairly between B and C. How much does each receive?

Solution

The Money-Boxes

139. When four money-boxes, containing pennies only, were opened and counted, it was found that the number in the first with half those in all the others, in the second with a third of all the others, in the third with a fourth of all the others, and in the fourth with a fifth of all the others, amounted in each case to 740. How much money did the boxes contain, and how was it divided?

Solution

140.

Two steamers start together to make a trip to a far-off buoy and back. Steamer A runs all the time at 10 miles an hour. Steamer B does the passage out at 8 miles an hour, and the return at 12 miles. Will they regain port together?

Solution

141.

A golf player has two clubs mended in London. One has a new head, the other a new leather face. The head costs four times as much as the face. At St Andrews it costs five times as much, and the leather face at St Andrews is half the London price. Including a shilling for a ball he pays twice the St Andrews charges for these repairs. What is the London charge for each?

Solution

From Two Wrongs to Make a Right

142. Two children were asked to give the total number of animals in a pasture, where sheep and cattle were grazing. They were told the numbers of sheep and of cattle, but one subtracted, and gave 100 as the answer, and the other arrived at 11,900 by multiplication. What was the correct total?

Solution

The Stone Carrier

143. Fifty-two stones are placed at intervals along a straight road. The distance between the first and the second is 1 yard, between the second and the third it is 3 yards, between the next two 5 yards, and so on, the intervals increasing each time by 2 yards.

How far would a tramp have to travel to earn five shillings promised to him when he had brought them one by one to a basket placed at the first stone?

Solution

144.

On a division in the House of Commons, if the Ayes had been increased by fifty from the Noes, the motion would have been carried by five to three. If the Noes had received sixty votes from the Ayes, it would have been lost by four to three. Did the motion succeed? How many voted?

Solution

A Watch Puzzle

145. How many positions are there on the face of a watch in which the places of the hour and minute-hands can be interchanged so as still to indicate a possible time?

Solution

Cricket Scores

146. In a cricket match the scores in each successive innings are a quarter less than in the preceding innings. The match was played out, and the side that went in first won by fifty runs. What was the complete score?

Solution

How High Can You Throw?

147. A boy throws a cricket ball vertically upwards, and catches it as it falls just five seconds later. How high from his hands does the ball go?

Solution

Measure the Carpet

148. It may be said of a section of the gigantic carpet at Olympia that had it been 5 feet broader and 4 feet longer it would have contained 116 more feet; but if 4 feet broader and 5 longer the increase would have been but 113 feet. What were its actual breadth and length?

Solution

149.

In estimating the cost of a hundred similar articles, the mistake was made of reading pounds for shillings and shillings for pence in each case, and under these conditions the estimated cost was £212 18s. 4d. in excess of the real cost. What was the true cost of the articles?

Solution

Squares in Streets

150. The square of the number of my house is equal to the difference of the squares of the numbers of my next door neighbour’s houses on either side.

My brother, who lives in the next street, can say the same of the number of his house, though his number is not the same as mine. How are our houses numbered?

Solution

151.

Two men of unequal strength are set to move a block of marble which weighs 270 lbs., using for the purpose a light plank 6 feet long. The stronger man can carry 180 lbs. How must the block be placed so as to allow him just that share of the weight?

Solution

152.

A man has twenty coins, of which some are shillings and the rest half-crowns. If he were to change the half-crowns for sixpences and the shillings for pence he would have 156 coins. How many shillings has he?

Solution

Count the Coins

153. Some coins are placed at equal distances apart on a table, so that they form the sides of an equilateral triangle.

From the middle of each side as many are then taken as equal the square root of the number on that side, and placed on the opposite corner coin. The number of coins on each side is then to the original number as 5 is to 4. How many coins are there in all?

Solution

Putting in the Posts

154. A gardener, wishing to fence round a piece of ground with some light posts, found that if he set them a foot apart there would be 150 too few, but if placed a yard apart there would be 70 to spare. How many posts had he?

Solution

155.

A gives B £100 to buy 100 animals, which must be cows at £5 each, sheep at £1, and geese at 1s. How many of each sort can he buy?

Solution

156.

John is twice as old as Mary was when he was as old as Mary is. John is now twenty-one. How old is Mary?

Solution

157.

In a cricket match A made 35 runs; C and D made respectively half and one-third as many as B, and B’s score was as much below A’s as C’s was above D’s. What did B, C, and D each score?

Solution

Settled by Remainders

158. What is the least number which, divided by 2, 3, 4, 5, 6, 7, 8, 9, or 10, leaves respectively as remainders 1, 2, 3, 4, 5, 6, 7, 8, and 9?

Solution

Table Turning

159. A square table stands on four legs, which are set at the middle point of its sides. What is the greatest weight that this table can uphold upon one of its corners?

Solution

Bribing the Boys

160. Well pleased with the inspector’s report, the rector of a country parish came into his school with 99 new pennies in his pocket, and said that he would give them to the five boys in Standard VII. if they could, within an hour, show him how to divide them so that the first share should exceed the second by 3, be less than the third by 10, be greater than the fourth by 9, and less than the fifth by 16. What was the answer which would satisfy these conditions?

Solution

Sheep-Stealing

161. Some Indian raiders carried off a third of a flock of sheep, and a third of a sheep. Another party took a fourth of what remained, and a fourth of a sheep. Others took a fifth of the rest and three-fifths of a sheep. What was the number of the full flock, if there were then 409 left?

Solution

162.

Three boys begin to fill a cistern. A brings a pint at the end of every three minutes, B a quart every five minutes, and C a gallon every seven minutes. If the cistern holds fifty-three gallons, in what time will it be filled, and who will pour in the last contribution?

Solution

Estimate of Age

163. A man late in the last century said that his age was the square root of the year in which he was born. In what year did he say this?

Solution

A Deal in China

164. A dealer in Eastern curios sold a Satzuma vase for £119, and on calculation found that the number which expressed his profit per cent. expressed also the cost price in pounds of the vase. What was this number?

Solution

165.

What is the chance of throwing at least one ace in a single throw with a pair of dice?

Solution

166.

A thief starts running from a country house as fast as he can. Four minutes later a policeman starts in pursuit. If both run straight along the road, and the policeman gets over the ground one-third faster than the thief, how soon will he catch him?

Solution

167.

Twenty-seven articles are exposed for sale on one of the stalls of a bazaar. What choice has a purchaser?

Solution

Very Personal

168. “How old are you, dad?” said Nellie on her birthday, as her father gave her as many shillings as she was years old. His answer was quite a puzzle for a time, but with the help of her schoolfellows Nellie worked it out.

This is what he said:—

“I was twice as old as you are
The day that you were born.
You will be just what I was then
When fourteen years are gone.”

How old was Nellie, and how old was her dad?

Solution