Henry Ernest Dudeney/Puzzles and Curious Problems

"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?

Geometrical Problems

Dissection Puzzles

$177$ - Square of Squares

Cutting only along the lines, what is the smallest number of square pieces into which the diagram can be dissected?

The largest number possible is, of course, $169$, where all the pieces will be of the same size -- one cell -- but we want the smallest number.

We might cut away the border on two sides, leaving one square $12 \times 12$, and cutting the remainder into $25$ little squares, making $25$ in all.

This is better than $169$, but considerably more than the fewest possible.

$178$ - Stars and Crosses

Cut the square into four parts by going along the lines, so that each part shall be exactly the same size and shape,

and each part contains a star and a cross.

$179$ - Greek Cross Puzzle

Cut the square into four pieces as shown, and put them together to form a regular Greek cross.

$180$ - Square and Cross

Cut a regular Greek cross into five pieces,

so that one piece shall be a smaller regular Greek cross,

and so that the remaining four pieces will fit together and form a perfect square.

$181$ - Three Greek Crosses from One

How can you cut a regular Greek cross into as few pieces as possible

so as to reassemble them into $3$ identical smaller regular Greek crosses?

$182$ - Making a Square

Cut the figure into four pieces, each of the same size and shape,

that will fit together to form a perfect square.

$183$ - Table-Top and Stools

Some people may be familiar with the old puzzle of the circular table-top cut into pieces to form two oval stools, each with a hand-hole.

Those who remember the puzzle will be interested in a solution in as few as four pieces by the late Sam Loyd.

Can you cut the circle into four pieces that will fit together (two and two) and form two oval stool-tops, each with a hand-hole?

$184$ - Dissecting the Letter E

In Modern Puzzles readers were asked to cut this $\text E$ into five pieces

that would fit together and form a perfect square.

It was understood that no piece was to be turned over,

but we remarked that it can be done in four pieces if you are allowed to turn over pieces.

I give the puzzle again, with permission to make the reversals.

Can you do it in four pieces?

$185$ - The Dissected Chessboard

Here is an ancient and familiar fallacy.

If you cut a chessboard into four pieces in the manner indicated by the black lines in Figure $\text A$,

and then reassemble the pieces as in Figure $\text B$,

you appear to gain a square by the operation,

since this second figure would seem to contain $13 \times 5 = 65$ squares.

I have explained this fallacy over and over again, and the reader probably understands all about it.

The present puzzle is to place the same four pieces together in another way

so that it may appear to the novice that instead of gaining a square we have lost one,

the new figure apparently containing only $63$ cells.

$186$ - Triangle and Square

Can you cut each of two equilateral triangles into three pieces,

so that the six pieces will fit together and form a perfect square?

$187$ - Changing the Suit

You are asked to cut the Spade into three pieces that will fit together and form a Heart.

$188$ - Squaring the Circle

The problem of squaring the circle depends on finding the ratio of the diameter to the circumference.

This cannot be found in numbers with exactitude,

but we can get near enough for all practical purposes.

But it is equally impossible, by Euclidean geometry, to draw a straight line equal to the circumference of a given circle.

You can roll a penny carefully on its edge along a straight line on a sheet of paper and get a pretty exact result,

but such a thing as a circular garden-bed cannot be so rolled.

Now, the line below, when straightened out

(it is bent for convenience in presentation),

is very nearly the exact length of the circumference of the accompanying circle.

The horizontal part of the line is half the circumference.

Could you have found it by a simple method, using only pencil, compasses and ruler?

$189$ - Problem of the Extra Cell

In diagram $A$ the square representing a chessboard is cut into $4$ pieces along the dark lines,

and these four pieces are seen re-assembled in Diagram $B$.

But in $A$ we have $64$ of these little squares, whereas in $B$ we have $65$.

Where does the additional cell come from?

$190$ - A Horseshoe Puzzle

Given a paper horseshoe, similar to the one in the illustration,

can you cut it into seven pieces, with two straight clips of the scissors,

so that each part shall contain a nail hole?

There is no objection to your shifting the pieces and putting them together after the first cut,

only you must not bend or fold the paper in any way.

$191$ - Two Squares in One

Two squares of any relative size can be cut into $5$ pieces, in the manner shown below,

that will fit together and form a larger square.

But this involves cutting the smaller square.

Can you show an easy method of doing it without in any way cutting the smaller square?

$192$ - The Submarine Net

The illustration is supposed to represent a portion of a long submarine net,

and the puzzle is to make as few cuts as possible from top to bottom,

to divide the net into two parts,

and so to make an opening for a submarine to pass through.

Where would you make the cuts?

No cut can be made through the knots.

Only remember the cuts must be made from the top line to the bottom.

$193$ - Square Table-Top

The illustration represents a $7 \times 7$ piece of veneer which has been cut into a number of pieces,

of which the shaded pieces are unusable.

A cabinet maker had to fit together the remaining $8$ pieces of veneer to form a small square table-top, $6 \times 6$,

and he stupidly cut that piece No. $8$ into three parts.

How would you form the square without cutting any one of the pieces?

$194$ - Cutting the Veneer

A cabinetmaker had a perfect square of beautiful veneer

which he wished to cut into $6$ pieces to form three separate squares, all different sizes.

How might this have been done without any waste?

$195$ - Improvised Chessboard

Cut this piece of checkered linoleum into only two pieces,

that will fit together and form a perfect chessboard,

without disturbing the checkering of black and white.

Of course, it would be easy to cut off those two overhanging white squares and put them in the vacant places,

but that would be doing it in three pieces.

$196$ - The Four Stars

Can you cut the square into four pieces, all of exactly the same size and shape,

each piece to contain a star, and each piece to contain one of the four central squares?

$197$ - Economical Dissection

Take a block of wood $8$ units long by $4$ units wide by $3 \tfrac 3 4$ units deep.

How many pieces, each measuring $2 \tfrac 1 2$ by $1 \tfrac 1 2$ by $1 \tfrac 1 4$ can be cut out of it?

Patchwork Puzzles

$198$ - The Patchwork Cushion

A lady had $20$ pieces of silk, all of the same triangular shape and size.

She found that four of these would fit together and form a perfect square, as in the illustration.

How was she to fit together these $20$ pieces to form a perfectly square patchwork cushion?

There must be no waste, and no allowance need be made for turnings.

$199$ - The Hidden Star

The illustration represents a square tablecloth of choice silk patchwork.

This was put together by the members of a family as a little birthday present for one of its number.

One of the contributors supplied a portion in the form of a perfectly symmetrical star,

and this has been worked in exactly as it was received.

But the triangular pieces so confuse the eye that it is quite a puzzle to find the hidden star.

Can you discover it, so that, if you wished, by merely picking put the stitches,

you could extract it from the other portions of the patchwork?

Various Geometrical Puzzles

$200$ - Measuring the River

A traveller reaches a river at the point $A$,

and wishes to know the width across to $B$.

As he has no means of crossing the river, what is the easiest way of finding its width?

$201$ - Square and Triangle

Take a perfectly square piece of paper,

and fold it as to form the largest possible equilateral triangle.

A triangle in which the sides are the same length as those of the square, as shown in our diagram,

will not be the largest possible.

Of course, no markings or measurements may be made except by the creases themselves.

$202$ - A Garden Puzzle

The four sides of a garden are known to be $20$, $16$, $12$ and $10$ rods,

and it has the greatest possible area for these sides.

What is the area?

$203$ - A Triangle Puzzle

In the solution to our puzzle No. $162$, we said that:

"there is an infinite number of rational triangles composed of three consecutive numbers like $3$, $4$, $5$, and $13$, $14$, $15$."

We here show these two triangles.

In the first case the area ($6$) is half of $3 \times 4$,

and in the second case, the height being $12$, the area ($84$) is a half of $12 \times 14$.

It will be found interesting to discover such a triangle with the smallest possible three consecutive numbers for its sides,

that has an area that may be exactly divided by $20$ without remainder.

$204$ - The Donjon Keep Window

In The Canterbury Puzzles Sir Hugh de Fortibus calls his chief builder, and, pointing to a window, says:

"Methinks yon window is square, and measures, on the inside, one foot every way,

and is divided by the narrow bars into four lights, measuring half a foot on every side."

See our Figure $A$.

"I desire that another window be made higher up,

whose four sides shall also be each one foot,

but it shall be divided by bars into eight lights, whose sides shall be all equal."

This the craftsman was unable to do, so Sir Hugh showed him our Figure $B$, which is quite correct.

But he added, "I did not tell thee that the window must be square, as it is most certain it never could be."

Now, an ingenious correspondent, Mr. George Plant, found a flaw in Sir Hugh's conditions.

Something that was understood is not actually stated,

and the window may, as the conditions stand, be perfectly square.

How is it to be done?

$205$ - The Square Window

A man had a window a yard square, and it let in too much light.

He blocked up one half of it, and still had a square window a yard high and a yard wide.

How did he do it??

$206$ - The Triangular Plantation

A man had a plantation of twenty-one trees set out in the triangular form shown in the diagram.

If he wished to enclose a triangular piece of ground with a tree at each of the three angles,

how may different ways of doing it are there from which he might select?

The dotted lines show three ways of doing it.

How many are there altogether?

$207$ - Six Straight Fences

A man had a small plantation of $36$ trees, planted in the form of a square.

Some of these died, and had to be cut down in the positions indicated by crosses in the diagram.

How is it possible to put up $6$ straight fences across the field,

so that every one of the remaining $20$ trees shall be in a separate enclosure?

As a matter of fact, $22$ trees might be so enclosed by $6$ straight fences if their positions were a little more accommodating,

but we have to deal with the trees as they stand in regular formation, which makes all the difference.

$208$ - Dividing the Board

A man had a board measuring $10$ feet in length, $6$ inches wide at one end, and $12$ inches wide at the other,

as shown in the diagram.

How far from $B$ must the straight cut at $A$ be made in order to divide it into two equal pieces?

$209$ - A Running Puzzle

$ABCD$ is a square field of $40$ acres.

The line $BE$ is a straight path, and $E$ is $110$ yards from $D$.

In a race Adams runs direct from $A$ to $D$,

but Brown has to start from $B$, go from $B$ to $E$, and thence to $D$.

Each keeps to a uniform speed throughout, and when Brown reaches $E$, Adams is $30$ yards ahead of him.

Who wins the race, and by how much?

$210$ - Pat and his Pig

The diagram represents a field $100$ yards square.

Pat is at $A$ and his pig is at $B$.

The pig runs straight for the gateway at $C$.

As Pat can run twice as fast as the pig, you would expect that he would first make straight for the gate and close it.

But this is not Pat's way of doing things.

He goes directly for the pig all the time, thus taking a curved course.

Now, does the pig escape, or does Pat catch it?

And if he catches it, exactly how far does the pig run?

$211$ - The Twenty Matches

The diagram shows how $20$ matches, divided into two groups of $14$ and $6$,

may form two enclosures so that one space enclosed is exactly $3$ times as large as the other.

Now divide the $20$ matches into two groups of $13$ and $7$,

and with them again make two enclosures,

one exactly three times as large as the other.

$212$ - Transplanting the Trees

A man has a plantation of $22$ trees arranged in the manner shown.

How is he to transplant only six of the trees so that they shall then form $20$ rows with $4$ trees in every row?.

$213$ - A Swastikaland Map

Swastikaland is divided in the manner shown in our illustration.

The Lord High Keeper of the Maps was ordered so to colour this map of the country

that there should be a different colour on each side of every boundary line.

What was the smallest number of colours that he required?

$214$ - Colouring the Map

Colonel Crackham asked his young son one morning to colour all the $26$ districts in this map

in such a way that no two contiguous districts should be the same colour.

The lad looked at it for a moment, and replied,

"I haven't enough colours by one in my box."

This was found to be correct.

How many colours had he?

He was not allowed to use black or white -- only colours.

$215$ - The Damaged Rug

A lady had a valuable Persian rug, $12$ feet by $9$ feet, which was badly damaged by fire.

So she cut from the middle a strip $8$ feet by $1$ foot, as shown in the diagram,

and then cut the remainder into two pieces that fitted together

and made a perfectly square rug $10$ feet by $10$ feet.

How did she do it?

$216$ - The Four Householders

The diagram represents a square plot of land with four houses, four trees, a well (W) in the centre,

and hedges planted across with four gateways.

can you divide the ground so that each householder shall have an equal portion of land,

one tree, one gateway, an equal length of hedge, and free access to the well without trespass?

$217$ - The Three Fences

A man had a circular field, and he wished to divide it into four equal parts by three fences, each of the same length.

How might this be done?

$218$ - The Farmer's Sons

A farmer once had a square piece of ground on which stood $24$ trees, exactly as shown in the illustration.

He left instructions in his will that each of his eight sons should receive the same amount of ground and the same number of trees.

How was the land to be divided in the simplest possible manner?

$219$ - The Three Tablecloths

A person had $3$ tablecloths, each $4$ feet square.

What is the length of the side of the largest square table top that they will cover together?

$220$ - The Five Fences

A man owned a large, fenced-in field in which were $16$ oak trees, as depicted in the diagram.

He wished to put up five straight fences so that every tree should be in a separate enclosure.

How did he do it?

$221$ - The Fly's Journey

A fly, starting from point $A$, can crawl around the four sides of the base of this cubical block in $4$ minutes.

Can you say how long it will take to crawl from $A$ to the opposite corner $B$?

$222$ - Folding a Pentagon

Given a ribbon of paper, as in the diagram, of any length -- say more than $4$ times as long as broad --

it can all be folded into a perfect pentagon,

with every part lying within the boundaries of the figure.

The only condition is that the angle $ABC$ must be the correct angle of two contiguous sides of a regular pentagon.

How are you to fold it?

$223$ - The Tower of Pisa

Suppose you were on the top of the Tower of Pisa, at a point where it leans exactly $179$ feet above the ground.

Suppose you were to drop an elastic ball from there such that on each rebound it rose exactly one-tenth of the height from which it fell.

What distance would the ball travel before it came to rest?

$224$ - The Tank Puzzle

The area of the floor of a tank is $6$ square feet,

the water in it is $9$ inches deep.

$(1)$ How much does the water rise if a $1$ foot metal cube is put in it,

$(2)$ How much farther does it rise if another cube like it is put in by its side?

$225$ - An Artist's Puzzle

An artist wished to obtain a canvas for a painting which would allow for

the picture itself occupying $72$ square inches,

a margin of $4$ inches on top and on bottom,

and $2$ inches on each side.

What is the smallest dimensions possible for such a canvas?

$226$ - The Circulating Motor-Car

A car was running on a circular track such that the outside wheels were going twice as fast as the inside ones.

What was the length of the circumference described by the outer wheels?

The wheels were $5$ feet apart at the axle-tree.

$227$ - A Match-Boarding Order

A man gave an order for $297$ feet of match-boarding of usual width and thickness.

There were to be $16$ pieces, all measuring an exact number of feet -- no fractions of a foot.

He required $8$ pieces of the greatest length, the remaining pieces to be $1$ foot, $2$ feet, or $3$ feet shorter than the greatest length.

How was the order carried out?

$228$ - The Ladder

A ladder was fastened on end against a high wall of a building.

It was unfastened and pulled out $4$ yards at the bottom.

It was then found that the ladder had descended just one-fifth of the length of the ladder.

What was the length of the ladder?

$229$ - Geometrical Progression

Write out a series of whole numbers in geometrical progression, starting from $1$,

so that the numbers should add up to a square.

Thus, $1 + 2 + 4 + 8 + 16 + 32 = 63$.

But this is one short of being a square.

$230$ - In a Garden

Consider a rectangular flower-bed.

If it were $2$ feet broader and $3$ feet longer, it would have been $64$ square feet larger;

if it were $3$ feet broader and $2$ feet longer, it would have been $68$ square feet larger.

What is its length and breadth?

$231$ - The Rose Garden

A man has a rectangular garden, and wants to make exactly half of it into a large bed of roses,

with a gravel path of uniform width round it.

Can you find a general rule that will apply equally to any rectangular garden, whatever its proportions?

$232$ - A Pavement Puzzle

Two square floors had to be paved with stones each $1$ foot square.

The number of stones in both together was $2120$, but each side of one floor was $12$ feet more than each side of the other floor.

What were the dimensions of the two floors?

$233$ - The Nougat Puzzle

A block of nougat is $16$ inches long, $8$ inches wide, and $7 \tfrac 1 2$ inches deep.

What is the greatest number of pieces that I can cut from it measuring $5$ inches by $3$ inches by $2 \tfrac 1 2$ inches?

$234$ - Pile Driving

During some bridge-building operations a pile was being driven into the bed of the river.

A foreman remarked that at high water a quarter of the pile was embedded in the mud,

one-third was under water,

and $17$ feet $6$ inches above water.

What was the length of the pile?

$235$ - An Easter Egg Problem

I have an easter egg exactly $3$ inches in length, and $3$ other eggs all similar in shape,

having together the same contents as the large egg.

Can you tell me the exact measurements for the lengths of the three smaller ones?

$236$ - The Pedestal Puzzle

A man had a block of wood measuring $3$ feet by $1$ foot by $1$ foot,

which he gave to a wood-turner with instructions to turn from it a pedestal,

saying that he would pay him a certain sum for every cubic inch of wood taken from the block in the process of turning.

The ingenious turner weighed the block and found it to contain $30$ pounds.

After he had finished the pedestal it was again weighed, and found to contain $20$ pounds.

As the original block contained $3$ cubic feet, and it had lost just one-third of its weight,

the turner asked payment for $1$ cubic foot.

But the gentleman objected, saying that the heart of the wood might be heavier or lighter than the outside.

How did the ingenious turned contrive to convince his customer that he had taken not more and not less than $1$ cubic foot from the block?

$237$ - The Mudbury War Memorial

The inhabitants of Mudbury recently erected a war memorial,

and they proposed to enclose a piece of ground on which it stands with posts.

They found that if they set up the posts $1$ foot asunder they would have too few by $150$.

But if they placed them a yard asunder there would be too many by $70$.

How many posts had they in hand?

$238$ - A Maypole Puzzle

During a gale a maypole was broken in such a manner that it struck the level ground at a distance of $20$ feet from the base of the pole,

where it entered the earth.

It was repaired, and broken by the wind a second time at a point $5$ feet lower down,

and struck the ground at a distance of $30$ feet from the base.

What was the original height of the pole?

$239$ - The Bell Rope

A bell rope, passing through the ceiling above, just touches the belfry floor,

and when you pull the rope to the wall, keeping the rope taut, it touches a point just $3$ inches above the floor,

and the wall was $4$ feet from the rope, when it hung at rest.

How long was the rope from floor to ceiling?

$240$ - Counting the Triangles

Draw a pentagon, and connect each point with every other point with straight lines, as in the diagram.

How many different triangles are contained in this figure?

$241$ - A Hurdles Puzzle

The answers given in the old books to some of the best-known puzzles are often clearly wrong.

Yet nobody ever seems to detect their faults.

Here is an example.

A farmer had a pen made of fifty hurdles, capable of holding a hundred sheep only.

Supposing he wanted to make it sufficiently large to hold double that number, how many additional hurdles must he have?

$242$ - Correcting a Blunder

Mathematics is an exact science, but first-class mathematicians are apt, like the rest of humanity, to err badly on occasions.

On referring to Peter Barlow's Elementary Investigation of the Theory of Numbers, we hit on this problem:

"To find a triangle such that its three sides, perpendicular, and the line drawn from one of the angles bisecting the base

may all be expressed in rational numbers."

He gives as his answer the triangle $480$, $299$, $209$, which is wrong and entirely unintelligible.

Readers may like to find a correct solution when we say that all the five measurements may be in whole numbers,

and every one of them less than a hundred.

It is apparently intended that the triangle must not itself be right-angled.

$243$ - The Squirrel's Climb

A squirrel goes spirally up a cylindrical post, making the circuit in $4$ feet.

How many feet does it travel to the top if the post is $16$ feet high and $3$ feet in circumference?

$244$ - Sharing a Grindstone

Three men bought a grindstone $20$ inches in diameter.

How much must each grind off so as to share the stone equally,

making an allowance of $4$ inches off the diameter as waste for the aperture?

We are not concerned with the unequal value of the shares for practical use --

only with the actual equal quantity of stone each receives.

Moving Counter Problems

$245$ - Magic Fifteen Puzzle

This is Loyd's famous $14$-$15$ puzzle,

in which you were asked to get the $14$ and $15$ in their proper order by sliding them about in the box.

It was, of course, impossible of solution.

I now propose to slide them about until they shall form a perfect magic square

in which the four columns, four rows and two diagonals all add up to $30$.

It will be found convenient to use numbered counters in place of the blocks.

What are your fewest possible moves?

$246$ - Transferring the Counters

Place ten counters on the squares of a chessboard as here shown,

and transfer them to the other corner as indicated by the ten crosses.

A counter may jump over any counter to the next square beyond, if vacant,

either horizontally or vertically, but not diagonally,

and there are no captures and no simple moves -- only leaps.

Not to waste the reader's time it can be conclusively proved that this is impossible.

You are now asked to add two more counters so that it may be done.

If you place these, say, on $\text {AA}$, they must, in the end, be found in the corresponding positions $\text {BB}$.

Where will you place them?

$247$ - The Counter Cross

Arrange twenty counters in the form of a cross, in the manner shown in the diagram.

Now, in how many different ways can you point out four counters that will form a perfect square if considered alone?

Thus the four counters composing each arm of the cross, and also the four in the centre, form squares.

Squares are also formed by the four counters marked $\text A$, the four marked $\text B$, and so on.

And in how many ways can you remove six counters so that not a single square can be so indicated from those that remain?

$248$ - Four in Line

Here we have a board of $36$ squares, and four counters are so placed in a straight line

that every square of the board is in line horizontally, vertically, or diagonally with at least one counter.

In other words, if you regard them as chess queens, every square on the board is attacked by at least one queen.

The puzzle is to find in how many different ways the four counters may be placed in a straight line so that every square shall thus be in line with a counter.

Every arrangement in which the counters occupy a different set of four squares is a different arrangement.

Thus, in the case of the example given, they can be moved to the next column to the right with equal effect,

or they may be transferred to either of the two central rows of the board.

This arrangement, therefore, produces $4$ solutions by what we call reversals or reflections of the board.

Remember that the counters must always be disposed in a straight line.

$249$ - Odds and Evens

Place eight counters in a pile on the middle circle so that they shall be in proper numerical order, with $1$ on the top and $8$ on the bottom.

It is required to transfer $1$, $3$, $5$, $7$ to the circle marked "Odds", and $2$, $4$, $6$, $8$ to the circle marked "Evens".

You can only move one counter at a time from circle to circle, and you must never place a number on a smaller number,

nor an odd number and an even number together on the same circle.

What are the fewest possible moves?

$250$ - Adjusting the Counters

Place $25$ counters in a square in the order shown.

Then it is a good puzzle to put them all into regular order so that the first line reads $1 \ 2 \ 3 \ 4 \ 5$, and the second $6 \ 7 \ 8 \ 9 \ 10$,

and so on to the end, by taking up one counter in each hand and making them change places.

The puzzle is to determine the fewest possible exchanges in which this can be done.

$251$ - Nine Men in a Trench

Here are nine men in a trench.

No. $1$ is the sergeant, who wishes to place himself at the other end of the line -- at point $1$ --

all the other men returning to their proper places at present.

There is no room to pass in the trench, and for a man to climb over another would be a dangerous exposure.

But it is not difficult with these three recesses, each of which will hold a man.

How is it to be done with the fewest possible moves?

A man may go any distance that is possible in a move.

$252$ - Black and White

Place four light and four dark counters alternately in a row as here shown.

The puzzle is to transfer two contiguous counters to one end and then move two contiguous counters to the vacant space,

and so on until in four such moves they form a continuous line of four dark counters followed by four light ones.

Then try this variant.

The conditions are exactly the same, only in moving a contiguous pair you must make them change sides.

How many moves do you now require?

$253$ - The Angelica Puzzle

Draw a square with three lines in both direction and place on the intersecting points eight lettered counters as shown in our illustration.

The puzzle is to move the counters, one at a time, along the lines from point to vacant point until you get them in the order $\text {ANGELICA}$ thus:

$\begin{array} \\ A & N & G \\ E & L & I \\ C & A & .\end {array}$

Try to do this in the fewest possible moves.

$254$ - The Flanders Wheel

Place eight lettered counters on the wheel as shown.

Now move them one at a time along the line from circle to circle

until the word $\text {FLANDERS}$ can be correctly read round the rim of the wheel as at present,

only that the $\text F$ is in the upper circle now occupied by the $\text N$.

Of course two counters cannot be in a circle at the same time.

Find the fewest possible moves.

$255$ - A Peg Puzzle

The diagram represents a square board with $49$ holes in it.

There are $10$ pegs to be placed in the positions shown,

and the puzzle is to remove only $3$ of these pegs to different holes,

so that the ten shall form $5$ rows with $4$ pegs in every row.

Which three would you move, and where would you place them?

$256$ - Catching the Prisoners

Make a rough diagram on a sheet of paper, and use counters to indicate the two warders (marked as $W$) and the two prisoners (marked as $P$).

At the beginning the counters must be placed in the squares shown.

The first player moves each of the warders to an adjacent cell, in any direction.

Then the second player moves each prisoner to an adjoining cell;

and so on until each warder captures his prisoner.

If one warder makes a capture, both he and his captive are out of the game, and the other player continues alone.

You may come to the conclusion that it is a hopeless chase, but it can really be done if you use a little cunning.

$257$ - Five Lines of Four

The diagram shows how ten counters may be placed on the points of the grid where the lines intersect,

so that they form five straight lines with four counters in every line.

Can you find a second way of doing this?

$258$ - Deploying Battleships

Ten battleships were anchored in the form here shown.

The puzzle is for four ships to move to new positions (the others remaining where they are)

until the ten form five straight rows with four ships in each row.

How should the admiral do it?

$259$ - Flies on Window Panes

The diagram represents a window with $81$ panes.

The dots represent nine flies, on as many panes,

and no fly is in line with another one horizontally, vertically, or diagonally.

Six of those flies are very torpid and do not move,

but each of the remaining three goes to an adjoining pane.

And yet, after this change of station, no fly is in line with another.

Which are the three lively flies, and to which three panes (at present unoccupied), do they pass?

$260$ - Stepping Stones

The diagram represents eight stepping-stones across a stream.

The puzzle is to start from the lower bank and land twice on the upper bank (stopping there),

having returned once to the lower bank.

But you must be careful to use each stepping-stone the same number of times.

In how few steps can you make the crossing?

Unicursal and Route Problems

$261$ - The Twenty-Two Bridges

We have a rough map of a district with an elaborate system of irrigation,

as the various waterways and numerous bridges will show.

A man set out from one of the lettered departments to pay a visit to a friend living in a different department.

For the purpose of pedestrian exercise he crossed every one of the bridges once, and once only.

The puzzle is to show in which two departments their houses are situated.

$262$ - A Monmouth Tombstone

In the burial ground attached to St. Mary's Church, Monmouth, is this arrangement of letters on one of the tombstones.

$\qquad \begin{array} \\ E & I & N & E & R & N & H & O & J & S & J & O & H & N & R & E & N & I & E \\ I & N & E & R & N & H & O & J & S & E & S & J & O & H & N & R & E & N & I \\ N & E & R & N & H & O & J & S & E & I & E & S & J & O & H & N & R & E & N \\ E & R & N & H & O & J & S & E & I & L & I & E & S & J & O & H & N & R & E \\ R & N & H & O & J & S & E & I & L & E & L & I & E & S & J & O & H & N & R \\ N & H & O & J & S & E & I & L & E & R & E & L & I & E & S & J & O & H & N \\ H & O & J & S & E & I & L & E & R & E & R & E & L & I & E & S & J & O & H \\ O & J & S & E & I & L & E & R & E & \mathbf H & E & R & E & L & I & E & S & J & O \\ H & O & J & S & E & I & L & E & R & E & R & E & L & I & E & S & J & O & H \\ N & H & O & J & S & E & I & L & E & R & E & L & I & E & S & J & O & H & N \\ R & N & H & O & J & S & E & I & L & E & L & I & E & S & J & O & H & N & R \\ E & R & N & H & O & J & S & E & I & L & I & E & S & J & O & H & N & R & E \\ N & E & R & N & H & O & J & S & E & I & E & S & J & O & H & N & R & E & N \\ I & N & E & R & N & H & O & J & S & E & S & J & O & H & N & R & E & N & I \\ E & I & N & E & R & N & H & O & J & S & J & O & H & N & R & E & N & I & E \\ \end{array}$

In how many different ways can these words "$\text{HERE LIES JOHN RENIE}$" be read,

starting at the central $H$ and always passing from one letter to another that is contiguous?

$263$ - Footprints in the Snow

Four schoolboys, living respectively in the houses $A$, $B$, $C$, and $D$, attended different schools.

After a snowstorm one morning their footprints were examined, and it was found that no boy had ever crossed the track of another boy,

or gone outside the square boundary.

Take your pencil and continue their tracks, so that the boy $A$ goes to the school $A$, the boy $B$ to the school $B$, and so on,

without any line crossing another line.

$264$ - The Fly's Tour

A fly pitched on the square in the top left-hand corner of a chessboard,

and then proceeded to visit every white square.

He did this without ever entering a black square or ever passing through the same corner more than once.

Can you show his route?

It can be done in seventeen continuous straight courses.

$265$ - Inspecting the Roads

A man starting from the town $A$, has to inspect throughout all the roads shown from town to town.

Their respective lengths, $13$, $12$, and $5$ miles, are all shown.

What is the shortest route he can adopt, ending his journey wherever he likes?

$266$ - Railway Routes

The diagram below represents a simplified railway system,

and we want to know how many different ways there are of going from $A$ to $E$, if we never go twice along the same line in any journey.

$267$ - A Motor-Car Tour

A man started in a motor-car from town $A$, and wished to make a complete tour of these roads,

going along every one of them once, and once only.

How many different routes are there from which he can select?

Every route must end at the town $A$, from which you start,

and you must go straight from town to town -- never turning off at crossroads.

$268$ - Mrs. Simper's Holiday Tour

The diagram shows a plan, very much simplified, of a tour that Mrs. Simper proposes to take next autumn.

It will be seen that there are $20$ towns, all connected by railway lines.

Mrs. Simper lives at $H$, and wants to visit every other town once and once only, ending her tour at home.

There are in fact $60$ possible routes she can select from, counting the reverse of a route as different.

There is a tunnel between $N$ and $O$, and one between $R$ and $S$, but Mrs. Simper does not want to go through these.

She also wants to delay her visit to $D$ as long as possible so as to meet a friend who lives there.

The puzzle is to show Mrs. Simper the best route under these circumstances.

$269$ - Sixteen Straight Runs

A commercial traveller started in his car from the point $A$ shown,

and wished to go $76$ miles in $16$ straight runs, never going along the same road twice.

The dots represent the towns and villages, and these are one mile apart.

The lines show the route he selected.

It will be seen that he carried out his plan correctly, but $6$ towns or villages were unvisited.

Can you show a better route by which he could have gone $76$ miles in $16$ straight runs, and left only $3$ towns unvisited?

$270$ - Planning Tours

The diagram represents a map (considerably simplified for our purposes) of a certain district.

The circles and dots are towns and villages, and the lines roads.

Can you show how $5$ motor-car drivers can go from $A$ to $A$, from $B$ to $B$, from $C$ to $C$, from $D$ to $D$, from $E$ to $E$, respectively,

without ever crossing the track or going along the same road as another car?

$271$ - Avoiding the Mines

Here we have a portion of the North Sea thickly sown with mines by the enemy.

A cruiser made a safe passage through them from south to north in two straight courses, without striking a single mine.

Take your pencil and try to discover how it is done.

Go from the bottom of the chart to any point you like on the chart in a straight line,

and then from that point to the top in another straight line without touching a mine.

$272$ - A Madam Problem

In how many different ways is it possible to read the word $\text {MADAM}$ in the diagram?

You may go as you please, upwards and downwards, forwards and backwards,

any way possible along the open paths.

But the letters in every case must be contiguous, and you may never pass a letter without using it.

Combination and Group Problems

$273$ - City Luncheons

The clerks attached to the firm of Pilkins and Popinjay arranged that three of them would lunch together every day at a particular table

so long as they could avoid the same three men sitting down twice together.

The same number of clerks of Messrs. Radson, Robson, and Ross decided to do precisely the same, only with four men at a time instead of three.

On working it out they found that Radson's staff could keep it up exactly three times as many days as their neighbours.

What is the least number of men there could have been in each staff?

$274$ - Halfpennies and Tray

What is the greatest number of halfpennies that can be laid flat on a circular tray

(with a small brim to prevent overlapping the edge)

of exactly $9$ inches in diameter, inside measurements?

No halfpenny may rest, however slightly, on another.

Of course, everybody should know that a halfpenny is exactly one inch in diameter.

$275$ - The Necklace Problem

How many different necklaces can be made with $8$ beads, where each bead may be either black or white,

the beads being indistinguishable except by colour?

$276$ - An Effervescent Puzzle

In how many ways can the letters in the word $\text {EFFERVESCES}$ be arranged in a line without two $\text E$s ever appearing together?

Of course, two occurrences of the same letter, such as $\text {F F}$, have no separate identity,

so that to interchange them will make no difference.

When the reader has done that, he should try the case where the letters have to be arranged differently in a circle, with no two $\text E$s together.

We are here, of course, only concerned with their positions on the circumference, and you must always read in a clockwise direction.

$277$ - Tessellated Tiles

Here we have $20$ tiles, all coloured with the same four colours.

The puzzle is to select any $16$ of these tiles that you choose and arrange them in the form of a square,

always placing same colours together -- white against white, red against red, and so on.

$278$ - The Thirty-Six Letter Puzzle

If you try to fill up this square by repeating the letters $A$, $B$, $C$, $D$, $E$, $F$,

so that no $A$ shall be in a line across, downwards, or diagonally, with another $A$,

no $B$ with another $B$, no $C$ with another $C$, and so on,

you will find that it is impossible to get in all the $36$ letters under these conditions.

The puzzle is to place as many letters as possible.

$279$ - Roses, Shamrocks, and Thistles

Place the numbers $1$ to $12$ (one number in every design) so that they shall add up to the same sum in the following $7$ different ways --

viz., each of the two centre columns, each of the two central rows,

the four roses together, the four shamrocks together, and the four thistles together.

$280$ - The Ten Barrels

A merchant had ten barrels of sugar, which he placed in the form of a pyramid, as shown.

Every barrel bore a different number, except one, which was not marked.

It will be seen that he had accidentally arranged them so that the numbers in the three sides added up alike --

that is, to $16$.

Can you arrange them so that the three sides shall sum to the smallest number possible?

Of course the central barrel (which happens to be $7$ in the diagram) does not come into the count.

$281$ - A Match Puzzle

The $16$ squares of a chessboard are enclosed by $16$ matches.

It is required to place an odd number of matches inside the square so as to enclose $4$ groups of $4$ squares each.

There are $4$ distinct ways to do this, up to reflection and rotation.

$282$ - The Magic Hexagon

In the diagram it will be seen how the numbers from $1$ to $19$ are arranged so that all $12$ lines add up to $23$.

Six of the lines are the six sides, and the other six lines radiate from the centre.

Can you find a different arrangement that will still add up to $23$ in all the $12$ directions?

$283$ - Pat in Africa

Many years ago, when the world was different, a team of explorers consisting of $5$ men from Western Civilization and $5$ natives

fell into the hands of a hostile local chief, who, after receiving a number of gifts, consented to let them go,

but only after half of them had been flogged by the head of the security services.

The Westerners cruelly hatched a plot to make the flogging fall upon the $5$ natives.

They were all to be arranged in a circle, and Pat, in position no. $1$, was given a number to count round and round in the clockwise direction.

In the diagram, $W$ represents a Westerner, and $N$ represents a native.

When that number fell on a man, he was to be taken out for flogging,

while the counting went on from where it left off until another man fell out,

and so on until the five men had been selected for punishment.

If Pat had remembered the number correctly, and had begun at the right man,

the flogging would all have fallen upon the $5$ natives.

But Pat was humane at heart, and did not hold with the casual cruelty of his fellows,

and so deliberately used the wrong number and started at the wrong man,

with the result that the Westerners all got the flogging instead.

Can you find:

$(1)$ the number Pat selected, and the man he started the count at,

$(2)$ the number he had been expected to use, and the man he was supposed to have begun at?

The smallest possible number is required in each case.

$284$ - Lamp Signalling

Two spies on the opposite sides of a river devised a method for signalling by night.

They each put up a stand, like the diagram, and each had three lamps which could show either white, red or green.

They constructed a code in which every different signal meant a sentence.

Note that a single lamp on any one of the hooks could only mean the same thing,

that two lamps hung on the upper hooks $1$ and $2$ could not be distinguished from two on, for example, $4$ and $5$.

However, two red lamps on $1$ and $5$ could be distinguished from two on $1$ and $6$,

and two on $1$ and $2$ from two on $1$ and $3$.

Remembering the variations of colour as well as of position, what is the greatest number of signals that could be sent?

$285$ - The Teashop Check

We give an example of the check supposed to be used at certain popular teashops.

The waitress punches holes in the tickets to indicate the amount of the purchase.

$\boxed {\begin{array} {rcl} \\

\tfrac 1 2 \oldpence & --- & \bullet \\ 1 \oldpence & --- \\ 1 \tfrac 1 2 \oldpence & --- \\ 2 \oldpence & --- \\ 2 \tfrac 1 2 \oldpence & --- \\ 3 \oldpence & --- & \bullet \\ 4 \oldpence & --- \\ 6 \oldpence & --- \\ 7 \oldpence & --- \\ 8 \oldpence & --- \\ 1 \shillings & --- \\

& & \\

\end{array} }$

Thus, in the example, the two holes indicate that the customer has to pay $3 \tfrac 1 2 \oldpence$

But the girl might, if she had chosen, have punched in any one of three other ways --

$2 \tfrac 1 2 \oldpence$ and $1 \oldpence$, or $2 \oldpence$ and $1 \tfrac 1 2 \oldpence$, or $2 \oldpence$, $1 \oldpence$ and $\tfrac 1 2 \oldpence$

On one occasion a waitress said, "I can punch this ticket in any one of $10$ different ways, and no more."

Her coworker, whose customer owed a different amount, said, "Same here."

What were the amounts of the purchases of each of their customers?

Only one hole is allowed to be punched against any given amount.

$286$ - Unlucky Breakdowns

On a day of great festivities, a large crowd gathered for a day's outing and pleasure.

They all agreed to pile into a bunch of wagons, each of which was to carry the same number of people.

But ten of the wagons broke down half way, so each of the other wagons then had to carry one more person than had been planned.

As they were about to start back, it was discovered that $15$ more of these wagons had become unserviceable,

and so there were three more people in each working wagon on the way back than started out.

How many people were there in the party?

$287$ - The Handcuffed Prisoners

Nine dangerous convicts needed to be guarded.

Every day except Sunday they were taken out for exercise, handcuffed together in groups of three, as in the diagram:

On no day in any one week were the same two men to be handcuffed together.

If will be seen how they were sent out on Monday.

Can you arrange the nine men in triplets for the remaining $5$ days?

It will be seen that No. $1$ cannot be handcuffed to No. $2$ again, but $1$ and $3$ can subsequently be so.

$288$ - Seating the Party

On a family outing, Dora asked in how many ways they could all be seated.

There were $6$ of them, three of each gender, and $6$ seats:

one beside the driver, two with their backs to the driver, and two behind them, facing the driver,

No two of the same gender were allowed to sit side by side.

The only people who were able to drive were the men.

So, how many ways could they all be seated?

Magic Square, Measuring, Weighing, and Packing Problems

$289$ - Magic Square Trick

Place in the empty squares such figures (different in every case, and no two squares containing the same figure)

so that they shall add up to $15$ in as many straight directions as possible.

$\qquad \begin{array} {|c|c|c|} \hline \ \ & \ \ & \ \ \\ \hline \ \ & 5 & \ \ \\ \hline \ \ & \ \ & \ \ \\ \hline \end{array}$

$290$ - A Four-Figure Magic Square

In this square, as every cell contains the same number -- $1234$ -- the three columns, three rows and two long diagonals naturally add up alike.

$\qquad \begin{array} {|c|c|c|} \hline 1234 & 1234 & 1234 \\ \hline 1234 & 1234 & 1234 \\ \hline 1234 & 1234 & 1234 \\ \hline \end{array}$

The puzzle is to form and place nine different $4$-figure numbers (using the same figures) so that they shall form a perfect magic square.

That is, the numbers all together must contain nine of each of $1$, $2$, $3$ and $4$, and they must be proper numbers without using fractions or any other trick like that.

$291$ - Progressive Squares

This is a magic square, adding up to $287$ in every row, every column, and each of the two diagonals.

If we remove the outer margin of numbers we have another square giving sums of $205$.

If we again remove the margin there is left a magic square adding up to $123$.

Now fill up the vacant spaces in the diagram with such numbers from $1$ to $81$ inclusive as have not already been given,

so that there shall be formed a magic square adding up to $369$ in each of twenty directions.

$292$ - Conditional Magic Square

Can you form a magic square with all the columns rows, and two long diagonals, adding up alike,

with numbers $1$ to $25$ inclusive, placing only the odd numbers on the shaded squares in the diagram,

and the even numbers on the other squares?

$293$ - The Twenty Pennies

If sixteen pennies are arranged in the form of a square

there will be the same number of pennies in every row, column and each of the long diagonals.

Can you do the same with twenty pennies?

$294$ - The Keg of Wine

A man had a $10$-gallon keg of wine and a jug.

One day he drew off a jugful of wine and filled up the keg with water.

Later on, when the wine and water had got thoroughly mixed, he drew off another jugful, and again filled up the keg with water.

The keg then contained equal quantities of wine and water.

What was the capacity of the jug?

$295$ - Blending the Teas

A grocer buys two kinds of tea --

one at $2 \shillings 8 \oldpence$ per pound,

and the other, a better quality, at $3 \shillings 4 \oldpence$ per pound.

He mixes together some of each, which he proposes to sell at $3 \shillings 7 \oldpence$ a pound,

and so make a profit of $25$ per cent on the cost.

How many pounds of each kind must he use to make a mixture of $100$ pounds weight?

$296$ - Water Measurement

A maid was sent to the brook with two vessels that exactly measured $7$ pints and $11$ pints exactly.

She had to bring back exactly $2$ pints of water.

What is the smallest possible number of transactions necessary?

$297$ - Mixing the Wine

A glass is one-third full of wine,

and another glass, with equal capacity, is one-fourth full of wine.

Each is filled with water and their contents mixed in a jug.

Half of the mixture is poured into one of the glasses.

What proportion of this is wine and what part water?

$298$ - The Stolen Balsam

Three men robbed a gentleman of a vase containing $24$ ounces of balsam.

While running away, they met in the forest a glass seller, of whom, in a great hurry, they purchased three vessels.

On reaching a place of safety they wished to divide the booty,

but they found that their vessels contained $5$, $11$, and $13$ ounces respectively.

How could they divide the balsam into equal portions?

$299$ - The Weight of the Fish

A man caught a fish.

The tail weighed $9$ ounces.

The head weighed as much as the tail and half the body,

and the body weighed as much as the head and tail together.

What is the weight of the fish?

$300$ - Fresh Fruits

Some fresh fruit was being weighed for some domestic purpose.

It was found that the apples, pears and plums exactly balanced each other as follows:

One pear and three apples weigh the same as $10$ plums;

and one apple and six plums weigh the same as one pear.

How many plums alone would weigh the same as one pear?

$301$ - Weighing the Tea

A grocer proposed to put up $20$ pounds of China tea into $2$-pound packets,

but the weights had been misplaced by somebody, and he could only find the $5$-pound and the $9$-pound weights.

What is the quickest way for him to do the business?

We will say at once that only nine weighings are really necessary.

$302$ - Delivering the Milk

A milkman one morning was driving to his dairy with two $10$-gallon cans full of milk,

when he was stopped by two countrywomen, who implored him to sell them a quart of milk each.

Mrs. Green had a jug holding exactly $5$ pints, and Mrs. Brown a jug holding exactly $4$ pints,

but the milkman had no measure whatsoever.

How did he manage to put an exact quart into each of the jugs?

It was the second quart that gave all the difficulty.

But he contrived to do it in as few as nine transactions --

and by a "transaction" we mean the pouring from a can into a jug, or from one jug to another, or from a jug back to the can.

How did he do it?

Crossing River Problem, and Problems Concerning Games and Puzzle Games

$303$ - Crossing the River

During the Turkish stampede in Thrace, a small detachment found itself confronted by a wide and deep river.

However, they discovered a boat in which two children were rowing about.

It was so small that it would only carry the two children, or one grown up person.

How did the officer get himself and his $357$ soldiers across the river and leave the two children finally in joint possession of their boat?

And how many times need the boat pass from shore to shore?

$304$ - Grasshoppers' Quadrille

It is required to make the white men change places with the black men in the fewest possible moves.

There is no diagonal play, nor are there captures.

The white men can only move to the right or downwards, and the black men to the left or upwards,

but they may leap over one of the opposite colour, as in draughts.

$305$ - Domino Frames

Take an ordinary set of $28$ dominoes and return double $3$, double $4$, double $5$, and double $6$ to the box as not wanted.

Now, with the remainder form three square frames, in the manner shown, so that the pips in every side shall add up alike.

In the example given the sides sum to $15$.

If this were to stand, the sides of the other two frames must also sum to $15$.

But you can take any number you like, and it will be seen that it is not required to place $6$ against $6$, $5$ against $5$, and so on, as in play.

$306$ - A Puzzle in Billiards

Alfred Addlestone can give Benjamin Bounce $20$ points in $100$, and beat him;

Bounce can give Charlie Cruikshank $25$ points in $100$, and beat him.

Now, how many points can Addlestone give Cruikshank in order to beat him in a game of $200$ up?

Of course we assume that the players play constantly with the same relative skill.

$307$ - Scoring at Billiards

What is the highest score that you can make in two consecutive shots at billiards?

$308$ - Domino Hollow Squares

It is required with the $28$ dominoes to form $7$ hollow squares, like the example given,

so that the pips in the four sides of every square shall add up alike.

All these seven squares need not have the same sum, and, of course, the example given need not be one of your set.

$309$ - Domino Sequences

A boy who had a complete set of dominoes, up to double $9$, was trying to arrange them all in sequence, in the usual way --

$6$ against $6$, $3$ against $3$, blank against blank, and so on.

His father said to him, "You are attempting an impossibility, but if you let me pick out $4$ dominoes it can them be done.

And those I take shall contain the smallest total number of pips possible in the circumstances.

Now, which dominoes might the father have selected?

$310$ - Two Domino Squares

Arrange the $28$ dominoes as shown in the diagram to form two squares

so that the pips in every one of the eight sides shall add up alike.

The constant addition must of course be within limits to make the puzzle possible,

and it will be interesting to find those limits.

Of course, the dominoes need not be laid according to the rule, $6$ against $6$, blank against blank, and so on.

$311$ - Domino Multiplication

Four dominoes may be so placed as to form a simple multiplication sum if we regard the pips as figures.

The example here shown will make everything perfectly clear.

Now, the puzzle is, using all the $28$ dominoes to arrange them so as to form $7$ such little sums in multiplication.

No blank may be placed at the left end of the multiplicand or product.

$312$ - Domino Rectangle

Arrange the $28$ dominoes exactly as shown in the diagram, where the pips are omitted,

so that the pips in every one of the seven columns shall sum to $24$, and the pips in every one of the eight rows to $21$.

The dominoes need not be $6$ against $6$, $4$ against $4$, and so on.

$313$ - The Domino Column

Arrange the $28$ dominoes in a column so that the three sets of pips, taken anywhere,

shall add up alike on the left side and on the right.

Such a column has been started in the diagram.

This is merely an example, so you can start afresh if you like.

$314$ - Card Shuffling

The rudimentary method of shuffling a pack of cards is to take the pack face downwards in the left hand and then transfer them one by one to the right hand,

putting the second on top of the third, the third under, the fourth above, and so on until all are transferred.

If you do this with any even number of cards and keep on repeating the shuffle in the same way,

the cards will in due time return to their original order.

Try with $4$ cards, and you will find the order is restored in $3$ shuffles.

In fact, where the number of cards is $2$, $4$, $8$, $16$, $32$, $64$,

the number of shuffles to get them back to the original arrangement is $2$, $3$, $4$, $5$, $6$, $7$ respectively.

Now, how many shuffles are necessary in the case of $14$ cards?

$315$ - Arranging the Dominoes

The number of ways the set of $28$ dominoes may be arranged in a straight line, in accordance with the original rule of the game,

left to right and right to left, in any arrangement counting as different ways,

is $7 \, 959 \, 229 \, 931 \, 520$.

After discarding all dominoes bearing a $5$ or a $6$, how many ways may the remaining $15$ dominoes be so arranged in a line?

$316$ - Queer Golf

A certain links had nine holes, $300$, $250$, $200$, $325$, $275$, $350$, $225$, $375$, and $400$ yards apart.

If a man could always strike the ball in a perfectly straight line and send it exactly one of two distances,

so that it would either go towards the hole, pass over it, or drop into it,

what would those two distances be that would carry him in the least number of strokes round the whole course?

Two very good distances are $125$ and $75$, which carry you round in $28$ strokes,

but this is not the correct answer.

$317$ - The Archery Match

On a target on which the scoring was $40$ for the bull's-eye, and $39$, $24$, $23$, $17$ and $16$ respectively for the rings from the centre outwards, as shown in the diagram,

three players had a match with six arrows each.

The result was:

Miss Dora Talbot: $120$ points;

Reggie Watson, $110$ points;

Mrs. Finch, $100$ points.

Every arrow scored, and the bull's-eye was only once hit.

Can you, from these facts, determine the exact six hits made by each competitor?

$318$ - Target Practice

Three people in an archery competition had each had six shots at a target, and the result is shown in the diagram,

where they all hit the target every time.

The bull's-eye scores $50$, then the scores are $25$, $20$, $10$, $5$, $3$, $2$, $1$ for the rings from the centre outwards respectively.

It is seen that the hits on target are $1$ bull's-eye, two $25$s, three $20$s, three $10$s, three $1$s and two hits in each of the other rings.

The three men tied with an equal score.

Can you work out who hit what?

$319$ - The Ten Cards

Place any ten playing cards in a row face up.

There are two players.

The first player may turn face down any single card he chooses.

Then the second player can turn face down any single card or any $2$ adjacent cards.

And so on.

Thus the first player must turn face down a single, but afterwards either player may turn down either a single or two adjacent cards.

The player who turns down the last card wins.

Should the first or second player win?

Unclassified Problems

$320$ - An Awkward Time

When I told someone the other morning that I had to catch the $12:50$ train, he told me it was a very awkward time for a train to start.

I asked him to explain why.

Can you guess his answer?

$321$ - Cryptic Addition

Can you prove that the following addition sum is correct?

$322$ - The New Gun

An inventor undertook that a new gun which he had manufactured, which when once loaded,

would fire fifteen shots at the rate of a shot a minute.

A series of tests were made, and the gun certainly fired fifteen shots in a quarter of an hour.

However, the Government refused to buy the gun, on the grounds that it did not do the job as advertised.

Why?

$323$ - Cats and Mice

A number of cats (more than one) killed between them $999 \, 919$ mice, and every cat killed an equal number of mice (more than one).

Each cat killed more mice than there were cats.

How many cats were there?

$324$ - The Two Snakes

Suppose two snakes started swallowing one another simultaneously,

each getting the tail of the other in its mouth,

so that the circle formed by the snakes becomes smaller and smaller.

What will eventually happen?

$325$ - The Price of a Garden

A neighbour told you he was offered a triangular piece of ground for a garden.

Its sides were $55$ yards, $62$ yards and $117$ yards.

The price was $10$ shillings per square yard.

What will its cost be?

$326$ - Strange Though True

There is a district in Sussex where any healthy horse can travel, quite regularly, $30$ miles per day,

yet while its legs on one side travel $30$ miles, the legs on its other side travel $31$ miles.

The horse, apparently, does not seem to mind this.

How can this be?

$327$ - Two Paradoxes

$(1): \quad$ Imagine a man going to the North Pole.

The points of the compass are, as everyone knows:

$\qquad \qquad \qquad \begin{array} {ccc} & \text N & \\ \text W & & \text E \\ & \text S & \\ \end{array}$

He reaches the pole and, having passed over it, must turn about to look North.

East is now on his left-hand side, West on his right-hand side, and the points of the compass therefore:

$\qquad \qquad \qquad \begin{array} {ccc} & \text N & \\ \text E & & \text W \\ & \text S & \\ \end{array}$

which is absurd.

What is the explanation?

$(2): \quad$ When you look in the mirror, you are turned right round, so that right is left and left is right,

and yet top is not bottom and bottom is not top.

If it reverses sideways, why does it not reverse lengthways?

Why are you not shown standing on your head?

$328$ - Choosing a Site

A man bought an estate enclosed by three straight roads forming an equilateral triangle.

He wished to build a house somewhere on the estate so that if he should have a straight drive from the front to each of the three roads,

he might be put to least expense.

Where should be build the house?

$329$ - The Four Pennies

Take four pennies and arrange them on the table without the assistance of another coin or any means of measurement,

so that when a fifth penny is produced it may be placed in exact contact with each of the four (without moving them)

in the manner shown in the diagram.

The shaded circle represents the fifth penny.

How should you proceed?

$330$ - The Encircled Triangles

Draw the design of circle and triangles in as few continuous strokes as possible.

You may go over a line twice if you wish to do so, and begin and end wherever you like.

How should you proceed?

$331$ - The Siamese Serpent

Draw as much of the serpent as possible using one continuous line,

without taking the pencil off the paper or going over the same line twice.

$332$ - A Bunch of Grapes

Here is a rough conventionalized sketch of a bunch of grapes.

The puzzle is to make a copy of it with one continuous stroke of the pencil,

never lifting the pencil from the paper,

nor going over a line twice throughout.

$333$ - A Hopscotch Puzzle

We saw some boys playing hopscotch, and wondered whether the figure marked on the ground could be drawn in one continuous stroke.

Can the reader draw it without taking the pencil off the paper or going over the same line twice?

$334$ - A Little Match Trick

We pulled open a box of matches the other day, and showed some friends that there were only about $12$ matches in it.

When opened at that end no heads were visible.

The heads were all at the other end of the box.

We told them after they had closed the box in front of them that we would give it a shake, and on reopening,

they would find a match turned round with its head visible.

They afterwards examined it to see that the matches were all sound.

$335$ - Three Times the Size

Lay out $20$ matches in the way shown in the diagram.

You will see that the two groups of $6$ and $14$ matches form two enclosures, so that one space is exactly $3$ times as large as the other.

Now transfer one match from the larger to the smaller group, and with the $7$ and $13$ enclose two spaces again, one exactly $3$ times as large as the other.

Twelve of the matches must remain unmoved from their present positions --

and there must be no duplicated matches or loose ends.

The dotted lines are just there to indicate the respective areas.

$336$ - A Six-Sided Figure

Here are $6$ matches arranged to form a regular hexagon.

Can you take $3$ more matches and so arrange the $9$ as to show another regular $6$-sided figure?

$337$ - Twenty-Six Matches

Make a rough square diagram, like the one shown, where the side of each square is the length of a match,

and put the stars and crosses in their given positions.

It is required to put $26$ matches along the lines so as to enclose $2$ parts of exactly the same size and shape,

one part enclosing two stars, and the other part enclosing two crosses.

In the example given, each part is correctly the same size and shape, and each part contains either two stars or two crosses,

but unfortunately only $20$ matches have been used.

$338$ - The Three Matches

Can you place $3$ matches on the table, and support the matchbox on them,

without allowing the heads of the matches to touch the table, to touch one another, or to touch the box?

$339$ - Equilateral Triangles

Place $16$ matches, as shown, to form $8$ equilateral triangles.

Now take away $4$ matches so as to leave $4$ equal triangles.

No superfluous matches or loose ends to be left.

$340$ - Squares with Matches

Arrange $12$ matches on the table, as shown in the diagram.

Now it is required to remove $6$ of these matches and replace them so as to form $5$ squares.

Of course $6$ matches must remain unmoved, and there must be no duplicated matches or loose ends.

$341$ - Hexagon to Diamonds

Arrange $6$ matches form a hexagon, as here shown.

Now, by moving only $2$ matches and adding $1$ more, can you form two diamonds?

$342$ - A Wily Puzzle

A life prisoner appealed to the king for pardon.

Not being ready to favour the appeal, the king proposed a pardon on condition that the prisoner should start at cell $A$

and go in and out of each cell of the prison, coming back to the cell $A$ without going into any cell twice.

$343$ - Tom Tiddler's Ground

I am on Tom Tiddler's ground picking up gold and silver.

Here we have a piece of land marked off with $36$ circular plots,

on each of which is deposited a bag containing as many sovereigns as the figures indicate in the diagram.

I am allowed to pick up as many bags of gold as I like,

provided I do not take two lying on the same line.

What is the greatest amount of money I can secure?

$344$ - Coin and Hole

We have before us a specimen of every coin which was current in Britain in $1930$.

And we have a sheet of paper with a circular hole cut in it $\tfrac 3 4$ of an inch in diameter.

What is the largest coin I can pass through that hole without tearing the paper?

$345$ - The Egg Cabinet

A man has a cabinet for holding birds' eggs.

There are $12$ drawers, and all -- except the first drawer, which holds the catalogue -- are divided into cells by intersecting wooden strips,

running the entire length or width of a drawer.

The number of cells in any drawer is greater than that of the drawer above.

The bottom drawer, No. $12$, has $12$ times as many cells as strips,

No. $11$ has $11$ times as many cells as strips, and so on.

Can you show how the drawers were divided -- how many cells and strips in each drawer?

Give the smallest possible number in each case.

$346$ - A Leap Year Puzzle

The month of February in $1928$ contained five Wednesdays.

There is, of course, nothing remarkable in this fact, but it will be found interesting to discover

when was the last year and when will be the next year that had, and that will have, $5$ Wednesdays in February.

$347$ - The Iron Chain

Two pieces of iron chain were picked up on the battlefield.

What purpose they had originally served is not certain, and does not immediately concern us.

They were formed of circular iron links (all of the same size) out of metal half an inch thick.

One piece of chain was exactly $3$ feet long, and the other $22$ inches in length.

Now, as one piece contained exactly six links more than the other, how many links were there in each piece of chain?

$348$ - Blowing Out the Candle

Candles were lighted on Colonel Crackham's breakfast-table one foggy morning.

When the fog lifted, the Colonel rolled a sheet of paper into the form of a hollow cone, like a megaphone.

He then challenged his young friends to use it in blowing out the candles.

They failed, until he showed them the trick.

Of course, you blow through the small end.

$349$ - Releasing the Stick

It is simply a loop of string passed through one end of a stick as here shown, but not long enough to pass round the other end.

The puzzle is to suspend it in the manner shown from the top hole of a man's coat, and then get it free again.

$350$ - The Keys and Ring

Colonel Crackham the other day produced a ring and two keys, as here shown,

cut out of a solid piece of cardboard, without a break or join anywhere.

$351$ - The Entangled Scissors

If you start on the loop at the bottom, the string can readily be got into position.

The puzzle is, of course, to let someone hold the two ends of the string until you disengage the scissors.

A good length of string should be used to give you free play.

$352$ - Locating the Coins

Said Dora to her brother:

"Put a shilling in one of your pockets and a penny in the pocket in the opposite side.

Now the shilling represents $12$ and the penny $1$.

Triple the coin in your right pocket, and double that in your left pocket.

Add these products together and tell me whether the result is odd or even."

He said the result was even, and she immediately told him that the shilling was in the right pocket and the penny in the left one.

Every time he tried it she told him correctly how the coins were located.

How did she do it?

$353$ - The Three Sugar Basins

Three basins each contain the same number of lumps of sugar,

and nine cups are empty.

If we transfer to each cup one-eighteenth of the number of lumps that each basin contains,

we then find that each basin holds $12$ more lumps than each of the cups.

How many lumps are there in each basin before they are removed?

$354$ - The Wheels of the Car

"You see, sir," said the motor-car salesman, "at present the fore-wheel of the car I am selling you makes four revolutions more than the hind-wheel in going $120$ yards;

but if you have the circumference of each wheel reduced by $3$ feet, it would make as many as six revolutions more than the hind-wheel in the same distance."

Why the buyer wished that the difference in the number of revolutions between the two wheels should not be increased does not concern us.

The puzzle is to discover the circumference of each wheel in the first case.

$355$ - The Seven Children

Four boys and three girls are seated in a row at random.

What are the chances that the two children at the end of each row will be girls?

$356$ - A Rail Problem

There is a garden railing similar to our design.

In each division between two uprights there is an equal number of ornamental rails,

and a rail is divided in halves and a portion stuck on each side of every upright,

except that the uprights at the end have not been given half rails.

Idly counting the rails from one end to another, we found that there were $1223$ rails, counting two halves as one rail.

We also noticed that the number of those divisions was five more than twice the number of whole rails in a division.

How many rails were there in each division?

$357$ - The Wheel Puzzle

Place the numbers $1$ to $19$ in the $19$ circles,

so that wherever there are three in a straight line they shall add up to $30$.

$358$ - Simple Addition

Can you show that four added to six will make eleven?

$359$ - Queer Arithmetic

Can you take away seven-tenths from five so that exactly four remains?

$360$ - Fort Garrisons

Here we have a system of fortifications.

It will be seen that there are ten forts, connected by lines of outworks,

and the numbers represent the strength of the small garrisons.

The General wants to dispose these garrisons afresh so that there shall be $100$ men in every one of the five lines of four forts.

The garrison must be moved bodily -- that is to say, you are not allowed to break them up into other numbers.

$361$ - Constellation Puzzle

The arrangement of stars in the diagram is known as "The British Constellation".

It is not given in any star map or books, and it is very difficult to find on the clearest night for the simple reason that it is not visible.

The $21$ stars form seven lines with $5$ stars in every line.

Can you rearrange these $21$ stars so that they form $11$ straight lines with $5$ stars in every line?

$362$ - Intelligence Tests

An English officer fell asleep in church during a sermon.

He was dreaming that the executioner was approaching him to cut off his head,

and just as the sword was descending on the officer's unhappy neck

his wife lightly touched her husband on the back of his neck with her fan to awaken him.

The shock was too great, and the officer fell forward dead.

Now, there is something wrong with this.

What is it?

Another such question goes along these lines:

If we sell apples by the cubic inch,

how can we really find the exact number of cubic inches in, say, a dozen dozen apples?

$363$ - At the Mountain Top

"When I was in Italy I was taken to the top of a mountain

and shown that a mug would hold less liquor at the top of the mountain than in the valley beneath.

Can you tell me," asked Professor Rackbrane, "what mountain this might be that has so strange a property?"

$364$ - Cupid's Arithmetic

Dora Crackham one morning produced a slip of paper bearing the jumble of figures shown in our diagram.

She said that a young mathematician had this poser presented to him by his betrothed when she was in a playful mood.

"What am I to do with it?" asked George.

"Just interpret its meaning," she replied. "If it is properly regarded it should not be difficult to decipher."

$365$ - Tangrams

In the diagram the square is shown cut into the $7$ pieces.

If you mark the point $B$, midway between $A$ and $C$, on one side of a square of any size,

and $D$, midway between $C$ and $E$, on an adjoining side, the direction of the cuts is obvious.

Henry Ernest Dudeney/Puzzles and Curious Problems

Henry Ernest Dudeney: Puzzles and Curious Problems