Henry Ernest Dudeney/Puzzles and Curious Problems/Arithmetical and Algebraical Problems

Henry Ernest Dudeney: Puzzles and Curious Problems: Arithmetical and Algebraical Problems

Money Puzzles

$1$ - The Money Bag

"A bag," said Rackbrane, when helping himself to the marmalade, "contained fifty-five coins consisting entirely of crowns and shillings,

and their total value was $\pounds 7, \ 3 \shillings 0 \oldpence$

How many coins were there of each kind?"

$2$ - A Legacy Puzzle

A man left legacies to his three sons and to a hospital, amounting in all to $\pounds 1,320$.

If he had left the hospital legacy also to his first son, that son would have received as much as the other two sons together.

If he had left it to his second son, that son would have received twice as much as the other two sons together.

If he had left the hospital legacy to his third son, he would have received then thrice as much as the first son and second son together.

Find the amount of each legacy.

$3$ - Buying Toys

George and William were sent out to buy toys for the family Christmas tree,

and, unknown to each other, both went at different times to the same little shop,

where they had sold all their stock of small toys

except engines at $4 \oldpence$, balls at $3 \oldpence$ each, dolls at $2 \oldpence$ each, and trumpets at $\tfrac 1 2 \oldpence$ each.

They both bought some of all, and obtained $21$ articles, spending $2 \shillings$ each.

But William bought more trumpets than George.

What were their purchases?

$4$ - Puzzling Legacies

A man bequeathed a sum of money, a little less than $\pounds 1500$, to be divided as follows:

The five children and the lawyer received such sums that

the square root of the eldest son's share,

the second son's share divided by two,

the third son's share minus $\pounds 2$,

the fourth son's share plus $\pounds 2$,

the daughter's share multiplied by two,

and the square of the lawyer's fee

all worked out at exactly the same sum of money.

No pounds were divided, and no money was left over after the division.

What was the total amount bequeathed?

$5$ - Dividing the Legacy

A man left $\pounds 100$ to be divided between his two sons Alfred and Benjamin.

If one-third of Alfred's legacy be taken from one-fourth of Benjamin's, the remainder would be $\pounds 11$.

What was the amount of each legacy?

$6$ - A New Partner

Two partners named Smugg and Williamson have decided to take a Mr. Rogers into partnership.

Smugg has one and a half times as much capital invested in the business as Williamson

and Rogers has to pay down $\pounds 2500$, which sum shall be divided between Smugg and Williamson,

so that the three partners shall have an equal interest in the business.

How shall that sum be divided?

$7$ - Squaring Pocket-Money

A man has four different English coins in his pocket,

and their sum in pence was a square number.

He spent one of the coins, and the sum of the remainder in shillings was a square number.

He then spent one of the three, and the sum of the other two in pence was a square number.

And when he deducted the number of farthings in one of them from the number of halfpennies in the other, the remainder was a square number.

What were the coins?

$8$ - Equal Values

A lady and her daughter set out on a walk the other day,

and happened to notice that they both had money of the same value in their purses,

consisting of three coins each, and all six coins were different.

During the afternoon they made slight purchases,

and on returning home found that they again had similar value in their purses made up of three coins each, and all six different.

How much money did they set out with, and what was the value of their purchases?

$9$ - Pocket-Money

"When I got to the station this morning," said Harold Tompkins, at his club, "I found I was short of cash.

I spend just one-half of what I had on my railway ticket, and then bought a penny newspaper.

When I got to the terminus I spent half of what I had left and twopence more on a telegram.

Then I spent half of the remainder on a bus, and gave threepence to that old match-seller outside the club.

Consequently I arrive here with this single penny.

Now, how much did I start out with?"

$10$ - Mental Arithmetic

If a tobacconist offers a cigar at $7 \tfrac 3 4 \oldpence$,

but says we can have the box of $100$ for $65 \shillings$,

shall we save much by buying the box?

In other words, what would $100$ at $7 \tfrac 3 4 \oldpence$ cost?

By a little rule that we shall give the calculation takes only a few moments.

$11$ - Distribution

Nine persons in a party, $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, $K$, did as follows:

First $A$ gave each of the others as much money as he (the receiver) already held;

then $B$ did the same; then $C$; and so on to the last,

$K$ giving to each of the other eight persons the amount the receiver then held.

Then it was found that each of the nine persons held the same amount.

Can you find the smallest amount in pence that each person could have originally held?

$12$ - Reductions in Price

"I have often been mystified," said Colonel Crackham, "at the startling reductions some people make in their prices,

and wondered on what principles they went to work.

For example, a man offered me a motor-car two years ago for $\pounds 512$;

a year later his price was $\pounds 320$;

a little while after he asked a level $\pounds 200$;

and last week he was willing to sell for $\pounds 125$.

The next time he reduces I shall buy.

At what price shall I purchase if he makes a consistent reduction?"

$13$ - The Three Hospitals

Colonel Crackham said that a hospital collection brought in the following contributions:

A cheque for $\pounds 2, 10 \shillings$,

two cheques for $\pounds 1, 5 \shillings$ each,

three $\pounds 1$ Treasury notes,

three $10 \shillings$ Treasury notes,

two crowns,

two postal orders for $3 \shillings$ each,

two florins,

and three shillings.

As this money had to be divided amongst three hospitals, just as it stood,

since nobody happened to have any change in his pocket,

how was it to be done?

$14$ - Horses and Bullocks

A dealer bought a number of horses at $\pounds 17, 4 \shillings$ each,

and a number of bullocks at $\pounds 13, 5 \shillings$ each.

He then discovered that the horses had cost him in all $33 \shillings$ more than the bullocks.

Now, what is the smallest number of each that he must have bought?

$15$ - Buying Turkeys

A man bought a number of turkeys at a cost of $\pounds 60$,

and after reserving fifteen of the birds he sold the remainder for $\pounds 54$,

thus gaining $2 \shillings$ a head by these.

How may turkeys did he buy?

$16$ - The Thrifty Grocer

A grocer in a small way of business had managed to put aside (apart from his legitimate profits) a little sum in $\pounds 1$ notes, $10 \shillings$ notes, and crowns,

which he kept in eight bags,

there being the same number of crowns and of each kind of note in each bag.

One night he decided to put the money into only seven bags, again with the same number of each kind of currency in every bag.

And the following night he further reduced the number of bags to six, again putting the same number of each kind of note and of crowns in every bag.

The next night the poor demented miser tried to do the same with five bags, but after hours of trial he utterly failed, had a fit, and died, greatly respected by his neighbours.

What is the smallest possible amount of money he had put aside?

$17$ - The Missing Penny

Two market women were selling their apples, one at three a penny and the other at two a penny.

One day they were both called away when each had thirty apples unsold:

these they handed to a friend to sell at five for twopence.

Now it will be seen that if they had sold their apples separately they would have fetched $2 \shillings 1 \oldpence$,

but when they were sold together they fetched only $2 \shillings$

Can you explain this little mystery?

$18$ - The Red Death League

In a story too tedious to relate, we are given to find the number of members and cost of membership when the total subscription is $\pounds 323, 5 \shillings 4 \tfrac 1 4 \oldpence$

We are also given that the number of members is under $500$.

$19$ - A Poultry Poser

Three chickens and one duck sold for as much as two geese;

one chicken, two ducks, and three geese were sold together for $25 \shillings$

What was the price of each bird in an exact number of shillings?

$20$ - Boys and Girls

Nine boys and three girls agreed to share equally their pocket-money.

every boy gave an equal sum to every girl,

and every girl gave another equal sum to every boy.

Every child then possessed exactly the same amount.

What was the smallest possible amount that each then possessed?

$21$ - The Cost of a Suit

Melville bought a suit.

The jacket cost as much as the trousers and waistcoat.

The jacket and two pairs of trousers would cost $\pounds 7, 17 \shillings 6 \oldpence$

The trousers and two waistcoats would cost $\pounds 4, 10 \shillings$

Can you tell me the cost of the suit?

$22$ - The War Horse

Farmer Wurzel bought a old war horse for $\pounds 13$ and sold it later for $\pounds 30$.

After having paid for its keep, it turned out he lost half the price he paid and one-quarter the cost of his keep.

How much did Farmer Wurzel lose on the transaction?

$23$ - A Deal in Cucumbers

"How much to you pay for these cucumbers?" someone asked.

The reply: "I pay as many shillings for six dozen cucumbers of that size as I get cucumbers for $32 \shillings$"

What was the price per cucumber?

$24$ - The Two Turkeys

"I sold those two turkeys," said Tozer.

"They weighed $20$ pounds together.

Mrs. Burkett paid $24 \shillings 8 \oldpence$ for the large one, and Mrs. Suggs paid $6 \shillings 10 \oldpence$ for the small one.

I made $2 \oldpence$ a pound more on the little one than on the other."

What did the big one weigh?

$25$ - Flooring Figures

A correspondent accidentally discovered the following when making out an invoice with the items:

$148 \ \mathrm {ft.}$ flooring boards at $2 \oldpence$ $\pounds 1, 4 \shillings 8 \oldpence$

$150 \ \mathrm {ft.}$ flooring boards at $2 \oldpence$ $\pounds 1, 5 \shillings 0 \oldpence$

where it will be seen that in each case the three digits are repeated in the same order.

He thought this coincidence so extraordinary that he tried to find another similar case.

This seems to have floored him. But it is possible.

$26$ - Cross and Coins

Take any $11$ of the $12$ current coins of the realm,

and using one duplicate coin, can you place the $12$ coins, one in each division of the cross,

so that they add up to the same value in the upright and in the horizontal?

$27$ - Buying Tobacco

A box of $50$ cigarettes cost the same in shillings and pence as some tobacco bought in pence and shillings.

The change out of a $10 \shillings$ note was the same as the cost of the cigarettes.

What did the cigarettes cost?

$28$ - A Farthings Puzzle

Find a sum of money expressed in pounds, shillings and pence

which, when you take the currency indicators and punctuation away, reads the number that you get when you reduce the sum to farthings.

$29$ - The Shopkeeper's Puzzle

A shopkeeper uses a code word where each letter stands for the digits from $0$ to $9$.

What is the code used to encode this addition sum?

  GAUNT
+ OILER
 ------
 RGUOEI

$30$ - Subscriptions

Seven men agreed to subscribe towards a certain fund,

and the first six gave $\pounds 10$ each.

The other man gave $\pounds 3$ more than the average of the seven.

What amount did the seventh man subscribe?

$31$ - A Queer Settling Up

Person 1: "Here is my purse, give me just as much money as you find in it."

Person 2, having done that: "If you give me as much as I have left of my own, we shall be square."

After Person 2 has done that, Person 1 find his purse contains three shillings and sixpence,

while Person 2 has three shillings.

How much did each possess at the start?

$32$ - Apple Transactions

A man was asked what price per $100$ he paid for some apples, and his reply was as follows:

"If they had been $4 \oldpence$ more per $100$ I should have got $5$ less for $10 \shillings$"

Can you say what was the price per $100$?

$33$ - Prosperous Business

A man started business with a capital of $\pounds 2000$, and increased his wealth by $50$ per cent every three years.

How much did he possess at the expiration of eighteen years?

$34$ - The Banker and the Note

A banker in a country town was walking down the street when he saw a $\pounds 5$ note on the kerb-stone.

He picked it up, noted the number, and went to his private house for luncheon.

His wife said that the butcher had sent in his bill for $\pounds 5$,

and, as the only money he had was the note he had found, he gave it to her and she paid the butcher.

The butcher paid it to a farmer in buying a calf,

the farmer paid it to a merchant

who in turn paid it to a laundry-woman,

and she, remembering that she owed the bank $\pounds 5$, went there and paid the note.

The banker recognised the note as the one he had found,

and by that time it had paid $\pounds 25$ worth of debts.

On careful examination he discovered that the note was counterfeit.

Now, what was lost in the whole transaction, and by whom?

$35$ - The Reapers' Puzzle

Three men were to receive $90 \shillings$ for harvesting a field, conditionally upon the work being done in $5$ days.

Jake could do it alone in $9$ days, but as Ben was not as good a workman they were compelled to engage Bill for $2$ days,

in consequence of which Ben got $3 \shillings 9 \oldpence$ less than he would otherwise have received.

How long would it have taken Ben and Bill together to complete the work?

$36$ - The Flagons of Wine

A quart of Burgundy costs $4 \shillings 9 \oldpence$, but $3 \oldpence$ is returnable on the empty flagon,

so that the Burgundy seems to be worth $4 \shillings 6 \oldpence$

For $12$ of the capsules with which each of the quart flagons is sealed, a free flagon of the same value is obtained.

What is the value of a single capsule?

Obviously a twelfth of $4 \shillings 6 \oldpence$ which is $4 \tfrac 1 2 \oldpence$

But the free flagon also has a capsule worth $4 \tfrac 1 2 \oldpence$, so that this full flagon appears to be worth $4 \shillings 10 \tfrac 1 2 \oldpence$,

which makes the capsule worth a twelfth of $4 \shillings 10 \tfrac 1 2 \oldpence$, or $4 \tfrac 7 8 \oldpence$,

and so on ad infinitum, with an ever-increasing value.

Where is the fallacy, and what is the real worth of a capsule?

$37$ - A Wages Paradox

"I want a rise, sir," said the office-boy.

"That's nonsense," said the employer.

"If I give you a rise you will really be getting less wages per week than you are getting now."

The boy pondered over this, but was unable to see how such a thing could happen.

Can you explain it?

Age and Kinship Puzzles

$38$ - The Picnic

Four married couples had a picnic together, and their refreshments included $32$ bottles of lemonade.

Mary only disposed of one bottle,

Anne had two,

Jane swallowed the contents of three,

and Elizabeth emptied four bottles.

The husbands were more thirsty,

except John MacGregor, who drank the same quantity as his better half.

Lloyd Jones drank twice as much as his wife,

William Smith three times as much as his wife,

and Patrick Dolan four times as much as his wife demanded.

The puzzle is to find the surnames of the ladies.

Which man was married to which woman?

$39$ - Surprising Relationship

Angelina: "You say that Mr. Tomkins is your uncle?"

Edwin: "Yes, and I am his uncle!"

Angelina: Then -- let me see -- you must be nephew to each other, of course! Funny, isn't it?"

Can you say quite simply how this might be, without any breach of the marriage law or disregard of the Table of Affinity?

$40$ - An Epitaph (A.D. $1538$)

Two grandmothers, with their two granddaughters;

Two husbands, with their two wives;

Two fathers, with their two daughters;

Two mothers, with their two sons;

Two maidens, with their two mothers;

Two sisters, with their two brothers;

Yet only six in all lie buried here;

All born legitimate, from incest clear.

How might this happen?

$41$ - Ancient Problem

Demochares has lived one-fourth of his life as a boy;

one-fifth as a youth;

one-third as a man;

and has spent thirteen years in his dotage.

How old is this gentleman?

$42$ - Family Ages

A man and his wife had three children, John, Ben, and Mary,

and the difference between their parents' ages was the same as between John and Ben and between Ben and Mary.

The ages of John and Ben, multiplied together, equalled the age of the father,

and the ages of Ben and Mary multiplied together equalled the age of the mother.

The combined ages of the family amounted to ninety years.

What was the age of each person?

$43$ - Mike's Age

Pat O'Connor is $1 \tfrac 1 3$ times as old as when he built the pigsty.

Little Mike, who was $40$ months old when Pat built the sty, is now two years more than half as old as Pat's wife, Biddy, was when Pat built the sty,

so that when Little Mike is as old as Pat was when he built the sty,

their three ages combined will amount to just one hundred years.

How old is Little Mike?

$44$ - Their Ages

A man, on being asked the ages of his two sons, stated that

eighteen more than the sum of their ages is double the age of the elder,

and six less than the difference of their ages is the age of the younger.

What are their ages?

$45$ - Brother and Sister

A boy on being asked the age of himself and his sister replied:

"Three years ago I was seven times as old as my sister;

two years ago I was four times as old;

last year I was three times as old;

and this year I am two and one-half times as old."

What are their ages?

$46$ - A Square Family

A man had nine children, all born at regular intervals,

and the sum of the squares of their ages was equal to the square of his own.

What were the ages of each?

Every age was an exact number of years.

$47$ - The Quarrelsome Children

A man married a widow, and they each already had children.

Ten years later there was a pitched battle engaging the present family of $12$ children.

The mother ran to the father and cried,

"Come at once! Your children and my children are fighting our children!"

As the parents now had each nine children of their own, how many were born during the ten years?

$48$ - Robinson's Age

Robinson said.

My brother is two years older than I,

my sister is four years older than he,

my mother was $20$ when I was born,

and I was told yesterday that the average age of the four of is is $39$ years.

What was Robinson's age?

$49$ - The Engine-Driver's Name

Three business men -- Smith, Robinson and Jones -- all live in the Leeds-Sheffield district.

Three railwaymen of similar names live in the same district.

The business man Robinson and the guard live at Sheffield,

the business man Jones and the stoker live at Leeds,

while the business man Smith and the engine-driver live half-way between Leeds and Sheffield.

The guard's namesake earns $\pounds 1000, 10 \shillings 2 \oldpence$ per annum,

and the engine-driver earns exactly one-third of the business man living nearest to him.

Finally, the railwayman Smith beats the stoker at billiards.

What is the engine driver's name?

$50$ - Buying Ribbon

Four mothers, each with one daughter, went into a shop to buy ribbon.

Each mother bought twice as many yards as her daughter,

and each person bought as many yards of ribbon as the number of farthings she paid for each yard.

Mrs. Jones spent $1 \shillings 7 \oldpence$ more than Mrs. White;

Nora bought three yards less than Mrs. Brown;

Gladys bought two yards more than Hilda,

who spent $1 \shillings$ less than Mrs. Smith.

What is the name of Mary's mother?

$51$ - Sharing the Apples

Eight children had a basket containing $32$ apples.

They divided them amongst themselves as follows:

Anne got one apple,

Mary two,

Jane three,

and Kate four.

Ned Smith took as many as his sister,

Tom Brown twice as many as his sister,

Bill Jones three times as many as his sister,

and Jack Robinson four times as many as his sister.

What are the full names of the girls?

$52$ - In the Year $1900$

A man's age at death was one-twenty-ninth of the year of his birth.

How old was he in the year $1900$?

$53$ - Finding a Birthday

A correspondent informs us incidentally that on Armistice Day (Nov. 11, 1928)

he would have lived as long in the $20$th century as he lived in the $19$th.

This tempted us to work out the day of his birth.

Perhaps the reader may like to do the same.

$54$ - The Birth of Boadicea

Boadicea died $129$ years after Cleopatra was born.

Their united ages (that is, the combined years of their complete lives) were $100$ years.

Cleopatra died $\text {30}$ $\text {BCE}$.

When was Boadicea born?

$55$ - Eliza's Surname

Smith, Brown and Robinson have provided themselves with a penny pencil each,

and took their wives to a stockbroker's office to buy shares.

Mary bought $50$ more shares than Brown,

and Robinson $120$ more than Jane.

Each man paid as many shillings per share as he bought shares,

and each wife as many pence per share as she bought shares,

and every man spent one guinea more than his wife.

What was Eliza's surname?

Clock Puzzles

$56$ - The Ambiguous Clock

A man had a clock with an hour hand and minute hand of the same length and indistinguishable.

If it was set going at noon, what would be the first time that it would be impossible, by reason of the similarity of the hands, to be sure of the correct time?

$57$ - The Broken Clock Face

How may a clock dial with Roman numerals be broken into four parts

so that the numerals on each part add up in every case to $20$?

$58$ - When did the Dancing Begin?

"The guests at that ball the other night," said Dora at the breakfast-table,

"thought that the clock had stopped,

because the hands appeared in exactly the same position as when the dancing began.

But it was found that they had really only changed places.

As you know, the dancing commenced between ten and eleven o'clock.

What was the exact time of the start?"

$59$ - Mistaking the Hands

"Between two and three o'clock yesterday," said Colonel Crackham,

"I looked at the clock and mistook the minute hand for the hour hand,

and consequently the time appeared to be fifty-five minutes earlier than it actually was.

What was the correct time?"

$60$ - Equal Distances

At what time between three and four o'clock is the minute hand the same distance from $\text {VIII}$ as the hour hand is from $\text {XII}$?

$61$ - Right and Left

At what time between three and four o'clock will the minute hand be as far from $12$ on the left side of the dial plate as the hour hand is from $12$ on the right side of the dial plate?

$62$ - At Right Angles

How soon between the hours of five and six will the hour and minute hands of a clock be exactly at right angles?

$63$ - Westminster Clock

A man crossed over Westminster Bridge one morning between eight and nine o'clock by the tower clock

(often mistakenly called Big Ben, which is the name of the large bell only. But this is by the way).

On his return between four and five o'clock he noticed that the hands were exactly reversed.

What were the exact times he made the crossings?

Locomotion and Speed Puzzles

$64$ - The Bath Chair

A correspondent informs us that a friend's house at $A$, where he was invited to lunch at $1$ p.m., is a mile from his own house at $B$.

He is an invalid, and at $12$ noon started in his Bath chair from $B$ towards $C$.

His friend, who had arranged to join him and help push back, left $A$ at $12.15$ p.m., walking at $5$ miles per hour towards $C$.

He joined him, and with his help they went back at $4$ miles per hour, and arrived at $A$ at exactly $1$ p.m.

How far did our correspondent go towards $C$?

$65$ - The Pedestrian Passenger

A train is travelling at the rate of $60$ miles per hour.

A passenger at the back of the train wishes to walk to the front along the corridor,

and in doing so walks at the rate of three miles per hour.

At what rate is the man travelling over the permanent way?

$66$ - Meeting Trains

At Wurzeltown Junction an old lady put her her head out of the window and shouted:

"Guard! how long will the journey be from here to Mudville?"

"All the trains take five hours ma'am, either way," replied the official.

"And how many trains shall I meet on the way?"

This absurd question tickled the guard, but he was ready with his reply:

"A train leaves Wurzletown for Mudville, and also one from Mudville to Wurzletown, at five minutes past every hour. Right away!"

The old lady induced one of her fellow passengers to work out the answer for her.

What is the correct number of trains?

$67$ - Carrying Bags

A gentleman had to walk to his railway station, four miles from his house,

and was encumbered by two heavy bags of equal weight, but too heavy for him to carry alone.

His gardener and the boy both insisted on carrying the luggage;

but the gardener is an old man, and the boy not sufficiently strong,

while the gentleman believes in a fair division of labour, and wished to take his own share.

They started off with the gardener carrying one bag and the boy the other,

while the gentleman worked out the best way of arranging that the three should share the burden equally among them.

Now, how would you have managed it?

$68$ - The Moving Staircase

"I counted $50$ steps that I made in going down the moving staircase," said Walker.

"I counted $75$ steps," said Trotman; "but I was walking down three times as quickly as you."

If the staircase were stopped, how many steps would be visible?

$69$ - The Four Cyclists

The four circles represent cinder paths.

The four cyclists started at noon.

Each person rode round a different circle,

one at the rate of $6$ miles an hour,

another at the rate of $9$ miles an hour,

another at the rate of $12$ miles an hour,

and the fourth at the rate of $15$ miles an hour.

They agreed to ride until all met at the centre, from which they started, for the fourth time.

The distances around each circle was exactly one-third of a mile.

When did they finish their ride?

$70$ - The Donkey Cart

Atkins, Brown and Cranby had to go an journey of $40$ miles.

Atkins could walk $1$ mile an hour,

Brown could walk $2$ miles an hour,

and Cranby could go in his donkey-cart at $8$ miles an hour.

Cranby drove Atkins a certain distance, and, dropping him to walk the remainder,

drove back to meet Brown on the way and carried him to their destination,

where they all arrived at the same time.

How long did the journey take?

$71$ - The Three Motor-Cars

Three motor-cars travelling along a road in the same direction are, at a certain moment, in the following positions in relation to one another.

Andrews is a certain distance behind Brooks,

and Carter is twice that distance in front of Brooks.

Each car travels at its own uniform rate of speed,

with the result that Andrews passes Brooks in seven minutes,

and passes Carter five minutes later.

Now, in how many minutes after Andrews would Brooks pass Carter?

$72$ - The Fly and the Motor-Cars

A road is $300$ miles long.

A motor-car, $A$, starts at noon from one end and goes throughout at $50$ miles an hour,

and at the same time another car, $B$, going uniformly at $100$ miles an hour, starts from the other end,

together with a fly travelling $150$ miles an hour.

When the fly meets the car $A$, it immediately turns and flies towards $B$.

$(1)$ When does the fly meet $B$?

The fly then turns towards $A$ and continues flying backwards and forwards between $A$ and $B$.

$(2)$ When will the fly be crushed between the two cars if they collide and it does not get out of the way?

$73$ - The Tube Stairs

We ran up against Percy Longman, a young athlete, the other day when leaving Curley Street tube station.

He stopped at the lift, saying, "I always go up by the stairs.

A bit of exercise, you know.

But this is the longest stairway on the line -- nearly $1000$ steps.

I will tell you a queer thing about it that only applies to one other smaller stairway on the line.

If I go up two steps at a time, there is one step left for the last bound;

if I go up three at a time, there are two steps left;

if I go up four at a time, there are three steps left;

five at a time, four are left;

six at a time, five are left;

and if I went up seven at a time there would be six risers left over for the last bound.

Now, why is that?"

As he went flying up the stairs, three steps at a time, we laughed and said,

He little suspects that if he went up twenty steps at a time there would be nineteen risers for his last bound!"

How many risers are there in the Curley Street tube stairway?

The platform does not count as a riser, and the top landing does.

$74$ - The Omnibus Ride

George treated his best girl to a ride on a motor omnibus,

but on account of his limited resources it was necessary that they should walk back.

Now, if the bus goes at the rate of nine miles an hour and they walk at the rate of three miles an hour,

how far can they ride so they may be back in eight hours?

$75$ - A Question of Transport

Twelve soldiers had to get to a place twenty miles distant with the quickest possible dispatch,

and all had to arrive at the same time.

They requisitioned the services of a man with a small motor-car.

"I can do twenty miles an hour," he said, "but I cannot carry more than four men at a time.

At what rate do you walk?"

"All of us can do a steady four miles an hour," they replied.

"Very well," exclaimed the driver, "then I will go ahead with four men,

drop them somewhere on the road to walk,

then return and pick up four more (who will then be somewhere on the road),

drop them off also, and return for the last four.

So all you have to do is to keep walking while you are on your feet, and I will do the rest."

As they started at noon, what was the exact time that they all arrived together?

$76$ - How Far Was It?

"The steamer," remarked one of our officers home from the East, "was able to go twenty miles an hour down-stream,

but could only do fifteen miles an hour upstream.

So, of course, she took five hours longer in coming up than in going down."

One could not resist working out mentally the distance from point to point.

What was it?

$77$ - Out and Home

Mr Wilkinson walks from his country house into the neighbouring town at the rate of five miles per hour,

and, as he is a little tired, he makes the return journey at the rate of three miles per hour.

As the double journey takes him exactly seven hours, can you tell me the distance from his house to the town?

$78$ - The Meeting Cars

The Crackhams made their first stop at Bugleminster, where they were to spend the night at a friend's house.

This friend was to leave home at the same time and ride to London to put up at the Crackhams' house.

They took the same route, and each car went at its own uniform speed.

They kept a look-out for one another, and met forty miles from Bugleminster.

George that evening worked out the following little puzzle:

"I find that if, on our respective arrivals, we had each at once proceeded on the return journey at the same speeds

we should meet $48$ miles from London."

If this were so, what is the distance from London to Bugleminster?

$79$ - A Cycle Race

Two cyclists race on a circular track.

Brown can ride once round the track in six minutes,

and Robinson in four minutes.

In how many minutes will Robinson overtake Brown?

$80$ - A Little Train Puzzle

A non-stop express going sixty miles an hour starts from Bustletown for Ironchester,

and another non-stop express going forty miles an hour starts at the same time from Ironchester for Bustletown.

How far apart are they exactly an hour before they meet?

As I have failed to find these cities on any map or in any gazetteer, I cannot state the distance between them,

so we will just assume that it is somewhere over $250$ miles.

$81$ - An Irish Jaunt

Colonel Crackham was going from Boghooley to Ballyfoyne, using Pat Doyle's horse and cart,

which moved at a steady rate, but more slowly than would normally be expected.

After they had been on the road for $20$ minutes, they had travelled half as far from Boghooley than it was to Pigtown.

They stopped for refreshment at Pigtown when they arrived there.

Five miles further on, it was half as far to Ballyfoyne as it was from Pigtown.

After another hour they had arrived in Ballyfoyne.

What is the distance from Boghooley to Ballyfoyne?

$82$ - A Walking Problem

A man taking a walk in the country on turning round saw a friend of his walking $400$ yards behind in his direction.

They each walked $200$ yards in a direct line, with their faces towards each other,

and you would suppose that they must have met.

Yet they found that after their $200$ yards walk that they were still $400$ yards apart.

Can you explain?

Digital Puzzles

$83$ - Three Different Digits

Find all $3$-digit numbers with distinct digits that are divisible by the square of the sum of those digits.

$84$ - Find the Cube

A number increased by its cube is $592 \, 788$.

What is that number?

$85$ - Squares and Triangulars

What is the third lowest number that is both a triangular and a square?

$1$ and $36$ are the two lowest which fulfil the conditions.

What is the next number?

$86$ - Digits and Cubes

Find all $5$-digit squares such that:

the number formed from the first $2$ digits added to that formed by the last $2$ digits form a cube.

$87$ - Reversing the Digits

What $9$-digit number, when multiplied by $123\, 456 \, 789$, gives a product ending in $987 \, 654 \, 321$?

$88$ - Digital Progression

If you arrange the nine digits in three numbers thus, $147$, $258$, $369$,

they have a common difference of $111$ and are therefore in arithmetic progression.

Can you find $4$ ways of rearranging the $9$ digits so that in each case the number shall have a common difference,

and the middle number be in every case the same?

$89$ - Forming Whole Numbers

Can the reader give the sum of all the whole numbers that can be formed with the four figures $1$, $2$, $3$, $4$?

That is, the addition of all such numbers as $1234$, $1423$, $4312$, etc.

$90$ - Summing the Digits

What is the sum of all the numbers that can be formed with all $9$ digits ($0$ excluded),

using each digit once and once only, in every number?

$91$ - Squaring the Digits

Take $9$ counters numbered $1$ to $9$, and place them in a row: $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$.

It is required in as few exchanges of pairs as possible to convert this into a square number.

$92$ - Digits and Squares

$(1) \quad$ What is the smallest square number, and

$(2) \quad$ what is the largest square number

that contains all the ten digits ($1$ to $9$ and $0$) once, and once only?

$93$ - Figures for Letters

Solve this cryptarithm:

           A B C D
 x       E F G H I
 -----------------
 A C G E F H I B D

Each of the letters $A$ to $I$ represents one of the digits from $1$ to $9$ inclusive.

$94$ - Simple Multiplication

 * * * * * * * * * *
x                  2
 -------------------
 * * * * * * * * * *

Substitute for the $*$ symbol each of the $10$ digits in each row,

so arranged as to form a correct multiplication operation.

$0$ is not to appear at the beginning or end of either answer.

$95$ - Beeswax

The word BEESWAX represents a number in a criminal's secret code,

but the police had no clue until they discovered among his papers the following sum:

 E A S E B S B S X
 B P W W K S E T Q
------------------
 K P E P W E K K Q

The detectives assumed that it was an addition sum, and utterly failed to solve it.

Then one man hit on the brilliant idea that perhaps it was a case of subtraction.

This proved to be correct, and by substituting a different figure for each letter, so that it worked out correctly,

they obtained the secret code.

What number does BEESWAX represent?

$96$ - Wrong to Right

Solve this cryptarithm:

  W R O N G
+ W R O N G
-----------
  R I G H T

Each letter represents a different digit, and no $0$ is allowed.

There are several different ways of doing this.

$97$ - Letter Multiplication

In this little multiplication sum the five letters represent $5$ different digits.

What are the actual figures?

There is no $0$.

    S E A M
x         T
-----------
  M E A T S

$98$ - Digital Money

Every letter in the following multiplication represents one of the digits, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, all different.

What is the value obtained if $K = 8$?

  A   B   C
x         K
-----------
 DE  FG   H

$99$ - The Conspirators' Code

Two conspirators had a secret code.

Their letters sometimes contained little arithmetical sums related to some quite plausible discussion,

and having an entirely innocent appearance.

But in their code each of the ten digits represented a different letter of the alphabet.

Thus, on one occasion, there was a little sum in simple addition which, when the letters were substituted for the figures, read as follows:

    F L Y
    F O R
+ Y O U R
-----------
  L I F E

It will be found an interesting puzzle to reconstruct the addition sum with the help of the clue that $I$ and $O$ stand for the figures $1$ and $0$ respectively.

$100$ - Digital Squares

Find a number which, together with its square, shall contain all the $9$ digits once, and once only, the $0$ disallowed.

Thus, if the square of $378$ happened to be $152 \, 694$, it would be a perfect solution.

But unfortunately the actual square is $142 \, 884$, which gives us repeated $4$s and $8$s, and omits the $6$, $5$, and $9$.

$101$ - Finding a Square

Here are six numbers:

$4 \, 784 \, 887$, $2 \, 494 \, 651$, $8 \, 595 \, 087$, $1 \, 385 \, 287$, $9 \, 042 \, 451$, $9 \, 406 \, 087$

It is known that three of these numbers added together will form a square.

Which are they?

$102$ - Juggling with Digits

Arrange the ten digits in three arithmetical sums,

employing three of the four operations of addition, subtraction, multiplication and division,

and using no signs except the ordinary ones implying those operations.

Here is an example to make it quite clear:

$3 + 4 = 7$; $9 - 8 = 1$; $30 \div 6 = 5$.

But this is not correct, because $2$ is omitted, and $3$ is repeated.

$103$ - Expressing Twenty-Four

In a book published in America was the following:

"Write $24$ with three equal digits, none of which is $8$.

(There are two solutions to this problem.)"

Of course, the answers given are $22 + 2 = 24$, and $3^3 - 3 = 24$.

Readers who are familiar with the old "Four Fours" puzzle, and others of the same class,

will ask why there are supposed to be only these solutions.

With which of the remaining digits is a solution equally possible?

$104$ - Letter-Figure Puzzle

$\text {(0)}: \quad$

$\ds A \times B$

$=$

$\ds B$

$\text {(1)}: \quad$

$\ds B \times C$

$=$

$\ds A C$

$\text {(2)}: \quad$

$\ds C \times D$

$=$

$\ds B C$

$\text {(3)}: \quad$

$\ds D \times E$

$=$

$\ds C H$

$\text {(4)}: \quad$

$\ds E \times F$

$=$

$\ds D K$

$\text {(5)}: \quad$

$\ds F \times H$

$=$

$\ds C J$

$\text {(6)}: \quad$

$\ds H \times J$

$=$

$\ds K J$

$\text {(7)}: \quad$

$\ds J \times K$

$=$

$\ds E$

$\text {(8)}: \quad$

$\ds K \times L$

$=$

$\ds L$

$\text {(9)}: \quad$

$\ds A \times L$

$=$

$\ds L$

Every letter represents a different digit, and, of course, $A C$, $B C$ etc., are two-figure numbers.

Can you find the values in figures of all the letters?

$105$ - Equal Fractions

Can you construct three ordinary vulgar fractions

(say, $\tfrac 1 2$, $\tfrac 1 3$, or $\tfrac 1 4$, or anything up to $\tfrac 1 9$ inclusive)

all of the same value, using in every group all the nine digits once, and once only?

The fractions may be formed in one of the following ways:

$\dfrac a b = \dfrac c d = \dfrac {e f} {g h j}$, or $\dfrac a b = \dfrac c {d e} = \dfrac {f g} {h j}$.

We have only found five cases, but the fifth contains a simple little trick that may escape the reader.

$106$ - Digits and Primes

Using the $9$ digits once, and once only,

can you find prime numbers that will add up to the smallest total possible?

$\qquad \begin{array}{|c|c|c|} \hline 2 & 1 & 8 \\ \hline 4 & 3 & 9 \\ \hline 6 & 5 & 7 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|} \hline 2 & 7 & 3 \\ \hline 5 & 4 & 6 \\ \hline 8 & 1 & 9 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|} \hline 3 & 2 & 7 \\ \hline 6 & 5 & 4 \\ \hline 9 & 8 & 1 \\ \hline \end{array}$

The $9$ digits may be arranged in a square in many ways,

so that the numbers formed in the first row and second row will sum to the third row.

We give $3$ examples, and it will be found that the difference between the first total, $657$, and the second total, $819$,

is the same as the difference between the second, $819$, and the third, $981$ --

that is, $162$.

Now, can you form $8$ such squares, every one containing the $9$ digits,

so that the common difference between the $8$ totals is throughout the same?

$108$ - The Nine Digits

It will be found that $32 \, 547 \, 891$ multiplied by $6$ (thus using all the $9$ digits once, and once only)

gives the product $195 \, 287 \, 346$ (also containing all the $9$ digits once, and once only).

Can you find another number to be multiplied by $6$ under the same conditions?

$109$ - Perfect Squares

Find $4$ numbers such that the sum of every two and the sum of all four may be perfect squares.

$110$ - An Absolute Skeleton

Here is a good skeleton puzzle.

The only conditions are:

$(1)$ No digit appears twice in any row of figures except the dividend.

$(2)$ If $2$ be added to the last figure in the quotient it equals the last but one,

and if $2$ be added to the third figure from the end it gives the last figure but $3$ in the quotient.

That is to say, the quotient might end in, say, $9742$, or in $3186$.

        ********
    ------------
 ***)***********
      ***
       ---
       ***
       ***
       ----
       ****
       ****
       -----
         ***
         ***
         ----
         ****
         ****
         -----
          ****
          ****
          -----
           ****
           ****
           -----
            ****
            ****
            ----

$111$ - Odds and Evens

Every asterisk and letter represents a figure,

and "$\mathtt O$" stands for an odd figure ($1$, $3$, $5$, $7$, or $9$)

while "$\mathtt E$" represents an even figure ($2$, $4$, $6$, $8$, or $0$).

Can you construct an arrangement complying with these conditions?

There are $6$ solutions.

Can you find one, or all of them?

       *******
    ----------
 ***)*********
     OE*
     -----
      ****
      OO**
      -----
        ***
        EE*
        ----
         ***
         EO*
         ----
         ****
         EE**
         -----
           ***
           OO*
           ---

$112$ - Simple Division

There is a simple division sum.

Can you restore it by substituting a figure for every asterisk, without altering or removing the sevens?

If you start out with the assumption that all the sevens are given, and that you must not use another,

you will attempt an impossibility, though the proof is difficult;

but when you are told that though no additional sevens may be used in divisor, dividend or quotient,

any number of extra sevens may be used in the working.

It is comparatively easy.

             **7**
       -----------
 ****7*)**7*******
        ******
        -------
        *****7*
        *******
        -------
          *7****
          *7****
          -------
          *******
           ***7**
          --------
            ******
            ******
            ------

$113$ - A Complete Skeleton

It will be remembered that a skeleton puzzle, where the figures are represented by stars,

has not been constructed without at least one figure, or some added condition, being used.

Perhaps the following comes a little nearer to the ideal,

though there are two division sums and not one,

and they are related by the fact that the six-figure quotient of the first happens to be the dividend of the second.

There appears to be only one solution.

        ******
    ----------
 ***)*********
      ***
     ----
        ****
         ***
        -----
          ***
          ***
          ----
          ****
          ****
          ----

     *****
   -------
 **)******
    **
    ----
     ***
      **
     ----
      ***
      ***
      ----
       ***
       ***
       ---

Various Arithmetical and Algebraical Problems

$114$ - Elementary Arithmetic

If a quarter of twenty is four, what would a third of ten be?

$115$ - The Eight Cards

Rearrange these cards, moving as few as possible, so that the two columns add up alike.

Can it be done?

$\begin{array} {} \boxed 1 & \boxed 3 \\ \boxed 2 & \boxed 4 \\ \boxed 7 & \boxed 5 \\ \boxed 9 & \boxed 8 \\ \end{array}$

$116$ - Transferring the Figures

If we wish to multiply $571 \, 428$ by $5$ and divide by $4$,

we need only transfer the $5$ from the beginning to the end for the correct answer $714 \, 285$.

Can you find a number that we can multiply by $4$ and then divide the product by $5$ in the same simple manner,

by moving the first figure to the end?

Of course $714 \, 285$, just given, would do if we were allowed to transfer from the end to the beginning.

But it must be from the beginning to the end!

$117$ - A Queer Addition

Write down $5$ odd figures so that they will add up to make $14$.

$118$ - Six Simple Questions

$(1)$ Deduct four thousand eleven hundred and a half from twelve thousand twelve hundred and twelve.

$(2)$ Add $3$ to $182$, and make the total less than $20$.

$(3)$ What two numbers multiplied together will produce seven?

$(4)$ What three figures multiplied by five will make six?

$(5)$ If five times four are $33$, what is the fourth of $20$?

$(6)$ Find a fraction whose numerator is less than its denominator, but which, when reversed, shall remain of the same value.

$119$ - The Three Drovers

Three drovers with varied flocks met on the highway.

Said Jack to Jim: "If I give you six pigs for a horse then you will have twice as many animals in your drove as I will have in mine."

Said Dan to Jack: "If I give you fourteen sheep for a horse, then you'll have three times as many animals as I have got."

Said Jim to Dan: "But if I give you four cows for a horse, then you'll have six times as many animals as I."

There were no deals; but can you tell me how many animals there were in the three droves?

$120$ - Proportional Representation

In a local election, there were ten names of candidates on a proportional representation ballot paper.

Voters should place No. $1$ against the candidate of their first choice.

They might also place No. $2$ against the candidate of their second choice,

and so on until all the ten candidates have numbers placed against their names.

The voters must mark their first choice, and any others may be marked or not as they wish.

How many different ways might the ballot paper be marked by the voter?

$121$ - Find the Numbers

Can you find $2$ numbers composed only of ones which give the same result by addition and multiplication?

Of course $1$ and $11$ are very near, but they will not quite do,

because added they make $12$, and multiplied they make only $11$.

$122$ - A Question of Cubes

From Sum of Sequence of Cubes, the cubes of successive numbers, starting from $1$, sum to a square number.

Thus the cubes of $1$, $2$, $3$ (that is, $1$, $8$, $27$), add to $36$, which is the square of $6$.

If you are forbidden to use the $1$, the lowest answer is the cubes of $23$, $24$ and $25$, which together equal $204^2$.

What is the next lowest number, using more than three consecutive cubes and as many more as you like, but excluding $1$?

$123$ - Two Cubes

Can you find two cube numbers in integers whose difference shall be a square number?

Thus the cube of $3$ is $27$, and the cube of $2$ is $8$,

but the difference, $19$, is not here a square number.

What is the smallest possible case?

$124$ - Cube Differences

If we wanted to find a way of making the number $1 \, 234 \, 567$ the difference between two squares,

we could of course write down $517 \, 284$ and $617 \, 283$ --

a half of the number plus $\tfrac 1 2$ and minus $\tfrac 1 2$ respectively to be squared.

But it will be found a little more difficult to discover two cubes the difference of which is $1 \, 234 \, 567$.

$125$ - Accommodating Squares

Can you find two three-digit square numbers (no noughts) that, when put together, will form a six-digit square number?

Thus, $324$ and $900$ (the squares of $18$ and $30$) make $324 \, 900$, the square of $570$, only there it happens there are two noughts.

There is only one answer.

$126$ - Making Squares

Find three whole numbers in arithmetic progression,

the sum of every two of which shall be a square.

$127$ - Find the Squares

What is the number which, when added separately to $100$ and $164$, make them both perfect square numbers?

$128$ - Forming Squares

An officer arranged his men in a solid square, and had $39$ men left over.

He then started increasing the number of men on a side by one, but found that $50$ new men would be needed to complete the new square.

Can you tell me how many men the officer had?

$129$ - Squares and Cubes

Find two different numbers such that the sum of their squares shall equal a cube, and the sum of their cubes equals a square.

$130$ - Milk and Cream

A dairyman found that the milk supplied by his cows was $5$ per cent cream and $95$ per cent skimmed milk.

He wanted to know how much skimmed milk he must add to a quart of whole milk to reduce the percentage of cream to $4$ per cent.

$131$ - Feeding the Monkeys

A man went to the zoo with a bag of nuts to feed the monkeys.

He found that if he divided them equally amongst the $11$ monkeys in the first cage he would have $1$ nut over;

if he divided them equally amongst the $13$ monkeys in the second cage there would be $8$ left;

if he divided them amongst the $17$ monkeys in the last cage $3$ nuts would remain.

He also found that if he divided them equally amongst the $41$ monkeys in all $3$ cages,

or amongst the monkeys in any $2$ cages,

there would always be some left over.

What is the smallest number of nuts that the man could have in his bag?

$132$ - Sharing the Apples

If $3$ boys had $169$ apples which they shared in the ratio of one-half, one-third and one-fourth, how many apples did each receive?

$133$ - Sawing and Splitting

Two men can saw $5$ cords of wood per day,

or they can split $8$ cords of wood when sawed.

How many cords must they saw in order that they may be occupied for the rest of the day in splitting it?

$134$ - The Bag of Nuts

There are $100$ nuts distributed between $5$ bags.

In the first and second there are altogether $52$ nuts;

in the second and third there are $43$;

in the third and fourth there are $34$;

in the fourth and fifth, $30$.

How many nuts are there in each bag?

$135$ - Distributing Nuts

Aunt Martha bought some nuts.

She gave Tommy one nut and a quarter of the remainder;

Bessie then received one nut and a quarter of what were left;

Bob, one nut and a quarter of the remainder;

and, finally, Jessie received one nut and a quarter of the remainder.

It was then noticed that the boys had received exactly $100$ nuts more than the girls.

How many nuts had Aunt Martha retained for her own use?

$136$ - Juvenile Highwaymen

Three juvenile highwaymen called upon an apple-woman to "stand and deliver."

Tom seized half of the apples, but returned $10$ to the basket;

Ben took one-third of what were left, but returned $2$ that he did not fancy;

Jim took half of the remainder, but threw back one that was worm-eaten.

The woman was then left with only $12$ in her basket.

How many had she before the raid was made?

$137$ - Buying Dog Biscuits

A salesman packs his dog biscuits (all of one quality) in boxes containing $16$, $17$, $23$, $24$, $39$ and $40 \ \mathrm{lbs.}$ (that is, pounds weight),

and he will not sell them in any other way, or break into a box.

A customer asked to be supplied with $100 \ \mathrm{lbs.}$ of the biscuits.

Could you have carried out the order?

If not, now near could you have got to making up the $100 \ \mathrm{lbs.}$ supply?

$138$ - The Three Workmen

"Me and Bill," said Casey, "can do the job for you in ten days,

but give me Alec instead of Bill, and we can get it done in nine days."

"I can do better than that," said Alec. "Let me take Bill as a partner, and we will do the job for you in eight days."

Then how long would each man take over the job alone?

$139$ - Working Alone

Alfred and Bill together can do a job of work in $24$ days.

If Alfred can do two-thirds as much as Bill, how long will it take each of them to do the work alone?

$140$ - A Curious Progression

A correspondent sent this:

"An arithmetical progression is $10, 20, 30, 40, 50$, the five terms of which sum is $150$.

Find another progression of five terms, without fractions, which sum to $153$."

$141$ - The First "Boomerang" Puzzle

You ask someone to think of any whole number between $1$ and $100$,

and then divide it successively by $3$, $5$ and $7$,

telling you the remainder in each case.

You then immediately announce the number that was thought of.

Can the reader discover a simple method of mentally performing this feat?

$142$ - Longfellow's Bees

If one-fifth of a hive of bees flew to the ladambra flower,

one-third flew to the slandbara,

three times the difference of these two numbers flew to an arbour,

and one bee continued to fly about, attracted on each side by the fragrant ketaki and the malati,

what was the number of bees?

$143$ - "Lilivati", A.D. $1150$

Beautiful maiden, with beaming eyes, tell me which is the number that, multiplied by $3$,

then increased by three-fourths of the product,

divided by $7$,

diminished by one-third of the quotient,

multiplied by itself,

diminished by $52$,

the square root found,

addition of $8$,

division by $10$,

gives the number $2$?

$144$ - Biblical Arithmetic

If you multiply the number of Jacob's sons by the number of times which the Israelites compassed Jericho on the seventh day,

and add to the product the number of measures of barley which Boaz gave Ruth,

divide this by the number of Haman's sons,

subtract the number of each kind of clean beasts that went into the Ark,

multiply by the number of men that went to seek Elijah after he was taken to Heaven,

subtract from this Joseph's age at the time he stood before Pharaoh,

add the number of stones in David's bag when he killed Goliath,

subtract the number of furlongs that Bethany was distant from Jerusalem,

divide by the number of anchors cast out when Paul was shipwrecked,

subtract the number of persons saved in the Ark,

and the answer will be the number of pupils in a certain Sunday school class.

How many people in the class?

$145$ - The Printer's Problem

A printer had an order for $10 \, 000$ bill forms per month,

but each month the name of the particular month had to be altered:

that is, he printed $10 \, 000$ "JANUARY", $10 \, 000$ "FEBRUARY", $10 \, 000$ "MARCH", etc.;

but as the particular types with which these words were to be printed had to be specially obtained, and were expensive,

he only purchased just enough movable types to enable him, by interchanging them,

to print in turn the whole of the months of the year.

How many separate types did he purchase?

Of course, the words were printed throughout in capital letters, as shown.

$146$ - The Swarm of Bees

The square root of half the number of bees in a swarm has flown out upon a jessamine bush;

eight-ninths of the whole swarm as remained behind;

one female bee flies about a male that is buzzing within the lotus flower into which he was allured in the night by its sweet odour,

but is now imprisoned in it.

Tell me the number of bees.

$147$ - Blindness in Bats

A naturalist was investigating (in a tediously long story) whether bats are in fact actually blind.

He discovered that blindness varied.

Two of his bats could see out of the right eye,

just three of them could see out of the left eye,

four could not see out of the left eye,

and five could not see out of the right eye.

He wanted to know the smallest number of bats that he could have examined in order to get these results.

$148$ - A Menagerie

A travelling menagerie contained two freaks of nature -- a four-footed bird and a six-footed calf.

An attendant was asked how many birds and beasts there were in the show, and he said:

"Well, there are $36$ heads and $100$ feet altogether.

You can work it out for yourself."

$149$ - Sheep Stealing

Some sheep stealers made a raid and carried off one-third of the flock of sheep, and one-third of a sheep.

Another party stole one-fourth of what remained, and one-fourth of a sheep.

Then a third party of raiders carried off one-fifth of the remainder and three-fifths of a sheep,

leaving $409$ behind.

What was the number of sheep in the flock?

$150$ - Sheep Sharing

An Australian farmer dies and leaves his sheep to his three sons.

Alfred is to get $20$ per cent more than John,

and $25$ per cent more than Charles.

John's share is $3600$ sheep.

How many sheep does Charles get?

$151$ - The Arithmetical Cabby

The driver of the taxi-cab was wanting in civility, so Mr. Wilkins asked him for his number.

"You want my number, do you?" said the driver.

"Well, work it out for yourself.

If you divide by number by $2$, $3$, $4$, $5$, or $6$ you will find there is always $1$ over;

but if you divide it by $11$ there ain't no remainder.

What's more, there's no other driver with a lower number who can say the same."

What was the fellow's number?

$152$ - The Length of a Lease

A friend's property had a $99$ years' lease.

When I asked him how much of this had expired, the reply was as follows:

Two-thirds of the time past was equal to four-fifths of the time to come,

so I had to work it out for myself.

$153$ - A Military Puzzle

An officer wished to form his men into $12$ rows, with $11$ men in every row,

so that he could place himself at a point that would be equidistant from every row.

"But there are only one hundred and twenty of us, sir," said one of the men.

Was it possible to carry out the order?

$154$ - Marching an Army

A body of soldiers was marching in regular column, with $5$ men more in depth than in front.

When the enemy came in sight the front was increased by $845$ men,

and the whole was drawn up in $5$ lines.

How many men were there in all?

$155$ - The Orchard Problem

A market gardener was planting a new orchard.

The young trees were arranged in rows so as to form a square,

and it was found that there were $146$ trees unplanted.

To enlarge the square by an extra row each way he had to buy $31$ additional trees.

How many trees were there in the orchard when it was finished?

$156$ - Multiplying the Nine Digits

They were discussing mental problems at the Crackham's breakfast-table,

when George suddenly asked his sister Dora to multiply as quickly as possible:

$1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 0$

How long would it have taken the reader?

$157$ - Counting the Matches

A friend bought a box of midget matches, each one inch in length.

He found he could arrange them all in the form of a triangle whose area was just as many square inches as there were matches.

He then used up $6$ of the matches,

and found that with the remainder he could again construct a triangle whose area was just as many square inches as there were matches.

And using another $6$ matches he could again do precisely the same.

How many matches were there in the box originally?

The number is less than $40$.

$158$ - Newsboys

A contest took place amongst some newspaper boys.

Tom Smith sold one paper more than a quarter of the whole lot they had secured;

Billy Jones disposed of one paper more than a quarter of the remainder;

Ned Smith sold one paper more than a quarter of what was left;

and Charlie Jones disposed of just one paper more than a quarter of the remainder.

At this stage it was found that the Smiths were exactly $100$ papers ahead,

but little Jimmy Jones, the youngest kid of the bunch, sold all that were left,

so that in this friendly encounter the Joneses won by how many papers do you think?

$159$ - The Year $1927$

Can you find values for $p$ and $q$ such that $p^q - q^p = 1927$?

To make it perfectly clear, we will give an example for the year $1844$, where $p = 3$ and $q = 7$:

$3^7 - 7^3 = 1844$

Can you express $1927$ in the same curious way?

$160$ - Boxes of Cordite

Cordite charges for $6$-inch howitzers were served out from ammunition dumps in boxes of $15$, $18$ and $20$.

"Why the three different sizes of boxes?" I asked the officer on the dump.

He answered: "So that we can give any battery the number of charges it needs without breaking a box.

This was an excellent system for the delivery of a large number of boxes,

but failed in small cases, like $5$, $10$, $25$ and $61$.

Now, what is the biggest number of charges that cannot be served out in whole boxes of $15$, $18$ and $20$?

It is not a very large number.

$161$ - Blocks and Squares

Three children each possess a box containing similar cubic blocks, the same number of blocks in every box.

The first girl was able, using all her blocks, to make a hollow square, as indicated by $A$.

The second girl made a still larger square, as $B$.

The third girl made a still larger square, as $C$ but had four blocks left over for the corners, as shown.

What is the smallest number of blocks that each box could have contained?

$162$ - Find the Triangle

The sides and height of a triangle are four consecutive whole numbers.

What is the area of the triangle?

$163$ - Domino Fractions

Taking an ordinary box, discard all doubles and blanks.

Then, substituting figures for the pips, regard the remaining $15$ dominoes as fractions.

Arrange these $15$ dominoes in $3$ rows of $5$ dominoes so that each row adds up to $10$.

$164$ - Cow, Goat and Goose

A farmer found

that his cow and goat would eat all the grass in a certain field in $45$ days,

that the cow and the goose would eat it in $60$ days,

but that it would take the goat and the goose $90$ days to eat it down.

Now, if he had turned cow, goat and goose into the field together, how long would it have taken them to eat all the grass?

$165$ - The Postage-Stamps Puzzle

A youth who collects postage stamps was asked how many he had in his collection, and he replied:

"The number, if divided by $2$, will give a remainder $1$;

divided by $3$, a remainder $2$;

divided by $4$, a remainder $3$;

divided by $5$, a remainder $4$;

divided by $6$, a remainder $5$;

divided by $7$, a remainder $6$;

divided by $8$, a remainder $7$;

divided by $9$, a remainder $8$;

divided by $10$, a remainder $9$.

But there are fewer than $3000$."

Can you tell how many stamps there were in the album?

$166$ - Hens and Tens

If ten hen-pens cost ten and tenpence (that is, $10 \shillings 10 \oldpence$),

and ten hens and one hen-pen cost ten and tenpence,

what will ten hens without any hen-pens cost?

$167$ - The Cancelled Cheque

Bankers at a certain bank would cancel their paid cheques by punching star-shaped holes in them.

There was a case in which they happened to punch out the $6$ figures that form the number of the cheque.

The puzzle is to find out what those figures were.

It was a square number multiplied by $113$, and when divided into three $2$-figure numbers,

each of these three numbers was a square number.

Can you find the number of the cheque?

$168$ - Mental Arithmetic

Find two whole numbers (each less than $10$)

such that the sum of their squares, added to their product, will make a square.

$169$ - Shooting Blackbirds

Twice four and twenty blackbirds

Were sitting in the rain;

I shot and killed a seventh part,

How many did remain?

$170$ - The Six Noughts

    A     B     C
  111   111   100
  333   333   000
  555   500   005
  777   077   007
  999   090   999
 ----  ----  ----
 2775  1111  1111
 ----  ----  ----

Write down the little addition sum $A$, which adds up to $2775$.

Now substitute $6$ noughts for $6$ of the figures, so that the total sum shall be $1111$.

It will be seen that in the case $B$ five noughts have been susbtituted, and in the case $C$ nine noughts.

But the puzzle is to do it with six noughts.

$171$ - Multiplication Dates

In the year $1928$ there were $4$ dates which, when written in the form dd/mm/yy,

the day multiplied by the month equal the year.

These are 28/1/28, 14/2/28, 7/4/28 and 4/7/28.

How many times in the $20$th century -- $\text {1901}$ – $\text {2000}$ inclusive -- does this so happen?

Or, you can try to find out which year in the century gives the largest number of dates that comply with the conditions.

There is one year that beats all the others.

$172$ - Curious Multiplicand

What number is it that can be multiplied by $1$, $2$, $3$, $4$, $5$, or $6$ and no new figures appear in the result?

$173$ - Short Cuts

Can you multiply $993$ by $879$ mentally?

It is remarkable that any two numbers of two figures each,

where the tens are the same, and the sum of the units make ten, can always be multiplied thus:

$97 \times 93 = 9021$

Multiply the $7$ by $3$ and set it down,

then add the $1$ to the $9$ and multiply by the other $9$, $9 \times 10 = 90$.

This is very useful for squaring any number ending in $5$, as $85^2 = 7225$.

With two fractions, when we have the whole numbers the same and the sum of the fractions equal unity,

we get an easy rule for multiplying them.

Take $7 \tfrac 1 4 \times 7 \tfrac 3 4 = 56 \tfrac 3 {16}$.

Multiply the fractions $= \tfrac 3 {16}$, add $1$ to one of the $7$'s, and multiply by the other, $7 \times 8 = 56$.

$174$ - More Curious Multiplication

What number is it that, when multiplied by $18$, $27$, $36$, $45$, $54$, $63$, $72$, $81$ or $99$,

gives a product in which the first and last figures are the same as those in the multiplier,

but which when multiplied by $90$ gives a product in which the last two figures are the same as those in the multiplier?

$175$ - Cross-Number Puzzle

Across:

1. a square number

4. a square number

5. a square number

8. the digits sum to $35$

11. square root of $39$ across

13. a square number

14. a square number

15. square of $36$ across

17. square of half $11$ across

18. three similar figures

19. product of $4$ across and $33$ across

21. a square number

22. five times $5$ across

23. all digits alike, except the central one

25. square of $2$ down

27. see $20$ down

28. a fourth power

29. sum of $18$ across and $31$ across

31. a triangular number

33. one more than $4$ times $36$ across

34. digits sum to $18$, and the three middle numbers are $3$

36. an odd number

37. all digits even, except one, and their sum is $29$

39. a fourth power

40. a cube number

41. twice a square

Down:

1. reads both ways alike

2. square root of $28$ across

3. sum of $17$ across and $21$ across

4. digits sum to $19$

5. digits sum to $26$

6. sum of $14$ across and $33$ across

7. a cube number

9. a cube number

10. a square number

12. digits sum to $30$

14. all similar figures

16. sum of digits is $2$ down

18. all similar digits except the first, which is $1$

20. sum of $17$ across and $27$ across

21. a multiple of $19$

22. a square number

24. a square number

26. square of $18$ across

28. a fourth power of $4$ across

30. a triangular number

32. digits sum to $20$ and end with $8$

34. six times $21$ across

35. a cube number

37. a square number

38. a cube number

$176$ - Counting the Loss

An officer explained that the force to which he belonged originally consisted of $1000$ men, but that it lost heavily in an engagement,

and the survivors surrendered and were marched down to a concentration camp.

On the first day's march one-sixth of the survivors escaped;

on the second day one-eighth of the remainder escaped, and one man died;

on the third day's march one-fourth of the remainder escaped.

Arrived in camp, the rest were set to work in four equal gangs.

How many had been killed in the engagement?