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

Henry Ernest Dudeney: Puzzles and Curious Problems: 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?