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

Henry Ernest Dudeney: Modern Puzzles: Arithmetical and Algebraical Problems

$78$ - The Miller's Toll

A miller was accustomed to take as toll one-tenth of the flour that he ground for his customers.

How much did he grind for a man who had just one bushel after the toll had been taken?

$79$ - Egg Laying

If a hen and a half lays an egg and a half in a day and a half,

how many and a half who lay better by half will lay half a score and a half in a week and a half?

$80$ - The Flocks of Sheep

Four brothers were comparing the number of sheep that they owned.

It was found that Claude had ten more sheep than Dan.

If Claude gave a quarter of his sheep to Ben,

then Claude and Adam would together have the same number as Ben and Dan together.

If, then, Adam gave one-third to Ben,

and Ben gave a quarter of what he then held to Claude,

who then passed on a fifth of his holding to Dan,

and then Ben divided one-quarter of the number he then possessed equally among Adam, Claude and Dan,

they would all have an equal number of sheep.

How many sheep did each possess?

$81$ - Pussy and the Mouse

"There's a mouse in one of these barrels," said the dog.

"Which barrel?" asked the cat.

"Why, the five hundredth barrel."

"What do you mean, the five hundredth? There are only five barrels in all."

"It's the five hundredth if you count backwards and forwards this way."

And the dog explained that if you count like this:

 1   2   3   4   5
 9   8   7   6
    10  11  12  13

so that the seventh barrel would be the one marked $3$ and the twelfth barrel the one numbered $4$.

The story goes on laboriously to its inevitable conclusion that the mouse escapes before the cat has finished counting, until:

Now, which was the five hundredth barrel?

Can you find a quick way of arriving at the answer without making the actual count?

$82$ - Army Figures

A certain division in an army was composed of a little over twenty thousand men, made up of five brigades.

It was know that one third of the first brigade,

two-sevenths of the second brigade,

seven-twelfths of the third,

nine-thirteenths of the fourth,

and fifteen-twenty-seconds of the fifth brigades happened in every case to be the same number of men.

Can you discover how many men there were in every brigade?

$83$ - A Critical Vote

A meeting of a charitable society was held to decide whether the members should expand their operations.

It was arranged that during the count those in favour of the motion should remain standing,

and those who voted against should sit down.

"Ladies and gentlemen," said the chairman in due course, "I have the pleasure to announce that the motion is carried by a majority exactly equal to exactly a quarter of the opposition."

"Excuse me, sir," called somebody from the back, "but some of us over here could not sit down, because there are not enough chairs."

"Then those who wanted to sit down but couldn't are to hold up their hands ... I find there are a dozen of you, so the motion is lost by a majority of one."

Now, how many people voted at that meeting?

$84$ - The Three Brothers

The discussion arose before one of the tribunals as to which of a tradesman's three sons could best be spared for service in the Army.

"All I know as to their capabilities," said the father, "is this:

Arthur and Benjamin can do a certain quantity of work in eight days,

which Arthur and Charles will do in nine days,

and which Benjamin and Charles will take ten days over."

Of course, it was at once seen that as longer time was taken over the job whenever Charles was one of the pair,

he must be the slowest worker.

This was all they wanted to know, but it is an interesting puzzle to ascertain just how long each son would be required to do that job alone.

Can you discover?

$85$ - The House Number

A man said the house of his friend was in a long street,

numbered on his side one, two, three, and so on,

and that all the numbers on one side of him added up exactly the same as all the numbers on the other side of him.

He said he knew there were more than fifty houses on that side of the street,

but not as many as five hundred.

Can you discover the number of that house?

$86$ - A New Street Puzzle

Brown lived in a street which contained more than twenty houses, but fewer than five hundred,

all numbered one, two, three, four, etc., throughout.

Brown discovered that all the numbers from one upwards to his own number inclusive summed to exactly half the sum of all the numbers from one up to, and including, the last house.

Now what was the number of his house?

$87$ - Another Street Puzzle

A long street in Brussels has all the odd numbers of the houses on one side and all the even numbers on the other.

$(1)$ If a man lives in an odd-numbered house and all the numbers on one side of him, added together, equal the numbers on the other side,

how many houses are there, and what is the number of his house?

$(2)$ If a man lives on the even side and all the numbers on one side of him equal those on the other side,

how many houses are there, and what is his number?

We will assume that there are more than fifty houses on each side of the street and fewer than five hundred.

$88$ - Correcting an Error

Hermione was given a certain number to multiply by $409$,

but she made a blunder that is very common with mudbloods when learning the elements of simple arithmetic:

she placed the first figure of her product by $4$ below the second figure from the right instead of below the third.

We have all done that as youngsters (speak for yourself, Harry, old boy) when there has happened to be a $0$ in the multiplier.

The result of Hermione's mistake was that her answer was wrong by $328,320$, entirely in consequence of that little slip.

Now, what was the multiplicand?

$89$ - The Seventeen Horses

"I suppose you all know this old puzzle," said Jeffries.

"A farmer left seventeen horses to be divided among his three sons in the following proportions:

To the eldest, one-half;

to the second, one-third;

and to the youngest, one-ninth.

How should they be divided?

"Yes; I think we all know that," said Robinson, "but it can't be done.

The answer always given is a fallacy."

(Considerable pointless argument ensues.)

... The terms of the will can be exactly carried out, without any mutilation of a horse.

... How should the horses be divided in strict accordance with the directions?

$90$ - Equal Perimeters

Rational right-angled triangles have been a fascinating subject for study since the time of Pythagoras, before the Christian era.

Every schoolboy knows that the sides of these, generally expressed in whole numbers,

are such that the square of the hypotenuse is exactly equal to the sum of the squares of the other two sides.

Now, can you find $6$ rational right-angled triangles each with a common perimeter, and the smallest possible?

$91$ - Counting the Wounded

When visiting with a friend one of our hospitals for wounded soldiers, I was informed that

exactly two-thirds of the men had lost an eye,

three-fourths had lost an arm,

and four-fifths had lost a leg.

"Then," I remarked to my friend, "it follows that at least twenty-six of the men must have lost all three -- an eye, an arm, and a leg."

That being so, can you say exactly how many men were in the hospital?

$92$ - A Cow's Progeny

"Supposing," said my friend Farmer Hodge, "that cow of mine to have a she-calf at the age of two years,

and supposing she goes on having the like every year,

and supposing every one of her young to have a she-calf at the age of two years,

and afterwards every year likewise, and so on.

Now, how many do you suppose would spring from that cow and all her descendants in the space of twenty-five years?"

$93$ - Sum Equals Product

There are two numbers whose sum equals their product, that is, $2$ and $2$.

What other numbers have that property?

$94$ - Adding their Cubes

The numbers $407$ and $370$ have this peculiarity, that they exactly equal the sum of the cubes of their digits.

Thus the cube of $4$ is $64$, the cube of $0$ is $0$, and the cube of $7$ is $343$.

Add together $64$, $0$ and $343$, and you get $407$.

Again, the cube of $3$ ($27$), added to the cube of $7$ ($343$), is $370$.

Can you find a number not containing a nought that will work in the same way?

Of course, we bar the absurd case of $1$.

$95$ - Squares and Cubes

Can you find two whole numbers, such that the difference of their squares is a cube and the difference of their cubes is a square?

What is the answer in the smallest possible numbers?

$96$ - Concerning a Cube

What is the length in feet of the side of a cube when

$(1)$ the superficial area equals the cubical contents;

$(2)$ the superficial area equals the square of the cubical contents;

$(3)$ the square of the superficial area equals the cubical contents?

$97$ - A Common Divisor

Find a common divisor for the three numbers $480 \, 608$, $508 \, 811$, and $723 \, 217$, so that the remainder shall be the same in every case.

$98$ - Curious Multiplication

If a person can add correctly but is incapable of multiplying or dividing by a number higher than $2$,

it is possible to obtain the product of any two numbers in this curious way.

Multiply $97$ by $23$.

In the first column we divide by $2$, rejecting the remainders, until $1$ is reached.

In the second column we multiply $23$ by $2$ the same number of times.

If we now strike out those products that are opposite ton the even numbers in the first column

(we have enclosed these in brackets for convenience in printing)

and add up the remaining numbers we get $2231$, which is the correct answer.

Why is this?

$99$ - The Rejected Gun

An inventor offered a new large gun to the committee appointed by our Government for the consideration of such things.

He declared that when once loaded it would fire sixty shots at the rate of a shot a minute.

The War Office put it to the test and found that it fired sixty shots an hour,

but declined it, "as it did not fulfil the promised condition."

"Absurd, said the inventor, "for you have shown that it clearly does all that we undertook it should do."

"Nothing of the sort," said the experts. "It has failed."

Now, can you explain this extraordinary mystery?

Was the inventor, or were the experts, right?

$100$ - Odds and Evens

Ask a friend to take an even number of coins in one hand and an odd number in the other.

You then undertake to tell him which hand holds the odd and which the even.

Tell him to multiply the number in the right hand by $7$ and the number in the left by $6$,

add the two products together, and tell you the result.

You can then immediately give him the required answer.

How are you to do it?

$101$ - Twenty Questions

I think of a number containing six figures.

Can you discover what it is by putting to me twenty questions,

each of which can only be answered by "yes" or "no"?

After the twentieth question you must give the number.

$102$ - The Nine Barrels

In how many different ways may these nine barrels be arranged in three tiers of three

so that no barrel shall have a smaller number than its own below it or to the right of it?

The first correct arrangement that will occur to you is $1 \ 2 \ 3$ at the top then $4 \ 5 \ 6$ in the second row, and $7 \ 8 \ 9$ at the bottom,

and my sketch gives a second arrangement.

How many are there altogether?

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

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