Henry Ernest Dudeney/Modern Puzzles/Geometrical Problems

Henry Ernest Dudeney: Modern Puzzles: Geometrical Problems

Dissection Puzzles

$103$ - A New Cutting-out Puzzle

Cut the figure into four pieces that will fit together and form a square.

$104$ - The Square Table-Top

A man had three pieces of beautiful wood, measuring $12$ units, $15$ units and $16$ units square respectively.

He wanted to cut those into the fewest pieces possible that would fit together and form a small square table-tip $25$ units by $25$ units.

How was he to do it?

$105$ - The Squares of Veneer

A man has two squares of valuable veneer, each measuring $25$ units by $25$ units.

One piece he cut, in the manner shown in our illustration, in four parts that will form two squares,

one $20$ units by $20$ units, and the other $15$ units by $15$ units.

Simply join $C$ to $A$ and $D$ to $B$.

How is he to cut the other square into four pieces that will form again two other squares, with sides in exact units,

but not $20$ and $15$ as before?

$106$ - Dissecting the Moon

In how large a number of pieces can this crescent moon be cut with five straight cuts of the knife?

The pieces may not be piled or shifted after a cut.

$107$ - Dissecting the Letter E

Can you cut this letter $\text E$ into only five pieces so that they will fit together to form a perfect square?

All the measurements have been given so that there should be no doubt as to the correct proportions of the letter.

In this case you are not allowed to turn over any piece.

$108$ - Hexagon to Square

Can you cut a regular hexagon into $5$ pieces that will fit together to form a square?

$109$ - Squaring a Star

This six-pointed star can be cut into as few as five pieces that will fit together to form a perfect square.

To perform the feat in $7$ pieces is quite easy,

but to do it in $5$ is more difficult.

The dotted lines are there to show the true shape of the star, which is made of $12$ equilateral triangles.

$110$ - The Mutilated Cross

Here is a regular Greek cross from which has been cut a square piece exactly equal to one of the arms of the cross.

The puzzle is to cut what remains into four pieces that will fit together and form a square.

$111$ - The Victoria Cross

Cut the cross shown into seven pieces that will fit together and form a perfect square.

Of course, there must be no trickery or waste of material.

$112$ - Squaring the Swastika

Cut out the swastika and then cut it up into four pieces that will fit together and form a square.

$113$ - The Maltese Cross

Can you cut the star into four pieces and place them inside the frame so as to show a perfect Maltese cross?

$114$ - The Pirates' Flag

Here is a flag taken from a band of pirates on the high seas.

The twelve stripes represented the number of men in the band,

and when a new man was admitted or dropped out a new stripe was added or one removed, as the case might be.

Can you discover how the flag should be cut into as few pieces as possible so that they may be put together again and show only ten stripes?

No part of the material may be wasted, and the flag must retain its oblong shape.

$115$ - The Carpenter's Puzzle

A ship's carpenter had to stop a hole $12$ inches square,

and the only piece of wood that was available measured $9 \ \mathrm{in.}$ in breadth by $16 \ \mathrm{in.}$ length.

How did he cut it into only two pieces that would exactly fit the hole?

The answer is based on the "step principle", as shown in the diagram.

If you move the piece marked $B$ up one step to the left,

it will exactly fit on $A$ and form a perfect square measuring $12$ inches on every side.

This is very simple and obvious.

But nobody has ever attempted to explain the general law of the thing.

As a consequence, the notion seems to have got abroad that the method will apply to any rectangle where the proportion of length to breadth is within reasonable limits.

This is not so, and I have had to expose some bad blunders in the case of published puzzles that were supposed to be solved by an application of this step principle,

but were really impossible of solution.$^*$

Let the reader take different measurements, instead of $9 \ \mathrm{in.}$ by $16 \ \mathrm{in.}$,

and see if he [or she] can find other cases in which this trick will work in two pieces and form a perfect square.

$116$ - The Crescent and the Star

Here is a little puzzle on the Crescent and the Star.

Look at the illustration, and see if you can determine which is the larger, the Crescent or the Star.

If both were cut out of a sheet of solid gold, which would be more valuable?

As it is very difficult to guess by the eye,

I will state that the outer arc is a semicircle;

the radius of the inner arc is equal to the straight line $BC$;

the distance in a straight line from $A$ to $B$ is $12$ units,

and the point of the star, $D$, contains $3$ square units.

Patchwork Puzzles

$117$ - The Patchwork Quilt

Here is a patchwork quilt that was produced by two young ladies for some charitable purpose.

When they came to join their work it was found that each lady had contributed a portion that was exactly the same size and shape.

It is an amusing puzzle to discover just where these two portions are joined together.

Can you divide the quilt into two parts, simply by cutting the stitches, so that the portions shall be of the same size and shape?

$118$ - The Improvised Draughts-Board

Some Englishmen at the front during the Great War wished to pass a restful hour at a game of draughts.

They had coins and small stones for the men, but no board.

However, one of them found a piece of linoleum as shown n the illustration,

and, as it contained the right number of squares, it was decided to cut it and fit the pieces together to form a board,

blacking some of the squares afterwards for convenience in playing.

An ingenious Scotsman showed how this could be done by cutting the stuff in two pieces only,

and it is a really good puzzle to discover how he did it.

Cut the linoleum along the lines into two pieces that will fit together and form the board, eight by eight.

$119$ - Tessellated Pavements

The reader must often have noticed, in looking at tessellated pavements and elsewhere,

that a square space had sometimes to be covered with square tiles under such conditions that a certain number of the tiles have to be cut in two parts.

A familiar example is shown in the illustration, where a square has been formed with ten square tiles.

As ten is not a square number a certain number of tiles must be cut.

In this case it is six.

It will be seen that the pieces $1$ and $1$ are cut from one tile, $2$ and $2$ from another, and so on.

Now, if you had to cover a square space with exactly twenty-nine square tiles of equal size, how would you do it?

What is the smallest number of tiles that you need cut in two parts?

Paper-Folding Puzzles

$120$ - The Ribbon Pentagon

I want to form a regular pentagon, but the only thing at hand happens to be a rectangular strip of paper.

How am I to do it without pencil, compasses, scissors, or anything else whatever but my fingers?

$121$ - Paper Folding

Suppose you are given a perfectly square piece of paper,

how are you going to fold it so as to indicate by creases a regular hexagon,

as shown in the illustration, all ready to be cut out?

$122$ - Folding a Pentagon

If you are given a perfectly square piece of paper,

how are you going to fold it so as to indicate by creases a regular pentagon,

all ready to be cut out?

$123$ - Making an Octagon

Can you cut the regular octagon from a square piece of paper without using compasses or ruler,

or anything but scissors?

You can fold the paper to make creases.

Various Geometrical Puzzles

$124$ - Drawing a Straight Line

If we want to describe a circle we use an instrument that we call a pair of compasses,

but if we need a straight line we use no such instrument --

we employ a ruler or other straight edge.

In other words, we first seek a straight line to produce our required straight line,

which is equivalent to using a coin, saucer of other circular object to draw a circle.

Now, imagine yourself in such a position that you cannot obtain a straight edge --

not even a piece of thread.

Could you devise a simple instrument that would draw your straight line,

just as the compasses describe a circle?

$125$ - Making a Pentagon

How do you construct a regular pentagon on a given unit straight line?

$126$ - Drawing an Oval

It is well-known that you can draw an ellipse by sticking two pins into the paper, enclosing them with a loop of thread,

and keeping the loop taut, running a pencil all the way round till you get back to the starting point.

Suppose you want an ellipse with a given major axis and minor axis.

How do you arrange the position of the pins, and what would be the length of the thread?

$127$ - With Compasses Only

Can you show how to mark off the four corners of a square, using the compasses only?

$128$ - Lines and Squares

With how few straight lines can you make exactly one hundred squares?

Thus, in the first diagram it will be found that with nine straight lines I have made twenty squares

(twelve with sides of the length $AB$, six with sides $AC$, and two with sides of the length $AD$).

In the second diagram, although I use one more line, I only get seventeen squares.

$129$ - The Circle and Discs

During a recent visit to a fair we saw a man with a table,

on the oilcloth covering of which was painted a large red circle,

and he invited the public to cover this circle entirely with five tin discs which he provided,

and offered a substantial prize to anyone who was successful.

The circular discs were all of the same size, and each, of course, smaller than the red circle.

he showed that it was "quite easy when you know how," by covering up the circle himself without any apparent difficulty,

but many tried over and over again and failed every time.

It was a condition that when once you had placed any disc you were not allowed to shift it,

otherwise, by sliding them about after they had been placed, it might be tolerably easy to do.

Let us assume that the red circle is six units in diameter.

Now, what is the smallest possible diameter for the five discs in order to make a solution possible?

$130$ - Mr. Grindle's Garden

"My neighbour," said Mr. Grindle, "generously offered me, for a garden,

as much land as I could enclose with four straight walls measuring $7$, $8$, $9$ and $10$ rods in length respectively."

"And what was the largest area you were able to enclose?" asked his friend.

Perhaps the reader can discover Mr. Grindle's correct answer.

$131$ - The Garden Path

A man has a rectangular garden, $55$ yards by $40$ yards,

and he makes a diagonal path, one yard wide, exactly in the manner indicated in the diagram.

What is the area of the path?

$132$ - The Garden Bed

A man has a triangular lawn of the proportions shown,

and he wants to make the largest possible rectangular flower-bed without enclosing the tree.

$133$ - A Problem for Surveyors

A man bought a little field, and here is a scale map that was given to me.

I asked my surveyor to tell me the area of the field,

but he said it was impossible without some further measurements;

the mere length of one side, $7$ rods, was insufficient.

What was his surprise when I showed him in about two minutes what was the area!

Can you tell how it is to be done?

$134$ - A Fence Problem

A man has a square field, $60 \ \mathrm {ft.}$ by $60 \ \mathrm {ft.}$, with other property, adjoining the highway.

For some reason he put up a straight fence in the line of three trees, as shown,

and the length of fence from the middle tree to the tree on the road was just $91$ feet.

What is the distance in exact feet from the middle tree to the gate on the road?

$135$ - The Domino Swastika

Form a square frame with twelve dominoes, as shown in the illustration.

Now, with only four extra dominoes, form within the frame a swastika.

$136$ - A New Match Puzzle

I have a box of matches.

I find that I can form with them any given pair of these four regular figures, using all the matches every time.

This, if there were eleven matches, I could form with them, as shown, the triangle and pentagon

or the pentagon and hexagon, or the square and triangle (by using only three matches in the triangle);

but could not with eleven matches form the triangle and hexagon,

or the square and pentagon, or the square and hexagon.

Of course there must be the same number of matches in every side of a figure.

Now, what is the smallest number of matches I can have in the box?

$137$ - Hurdles and Sheep

A farmer says that four of his hurdles will form a square enclosure just sufficient for one sheep.

That being so, what is the smallest number of hurdles that he will require for enclosing ten sheep?

$138$ - The Four Draughtsmen

The four draughtsmen are shown exactly as they stood on a square chequered board --

not necessarily eight squares by eight --

but the ink with which the board was drawn was evanescent,

so that all the diagram except the men has disappeared.

How many squares were there in the board and how am I to reconstruct it?

I know that each man stood in the middle of a square,

one on the edge of each side of the board and no man in a corner.

$139$ - A Crease Problem

Fold a page, so that the bottom outside corner touches the inside edge and the crease is the shortest possible.

$140$ - The Four-Colour Map Theorem

In colouring any map under the condition that no contiguous countries shall be coloured alike,

not more than four colours can ever be necessary.

Countries only touching at a point ... are not contiguous.

I will give, in condensed form, a suggested proof of my own

which several good mathematicians to whom I have shown it accept it as quite valid.

Two others, for whose opinion I have great respect, think it fails for a reason that the former maintain will not "hold water".

The proof is in a form that anybody can understand.

It should be remembered that it is one thing to be convinced, as everybody is, that the thing is true,

but quite another to give a rigid proof of it.

$141$ - The Six Submarines

If five submarines, sunk on the same day, all went down at the same spot where another had previously been sunk,

how might they all lie at rest so that every one of the six U-boats should touch every other one?

To simplify we will say, place six ordinary wooden matches so that every match shall touch every other match.

No bending or breaking allowed.

$142$ - Economy in String

Owing to the scarcity of string a lady found herself in this dilemma.

In making up a parcel for her son, she was limited to using $12$ feet of string, exclusive of knots,

which passed round the parcel once lengthways and twice round its girth, as shown in the illustration.

What was the largest rectangular parcel that she could make up, subject to these conditions?

$143$ - The Stone Pedestal

In laying the base and cubic pedestal for a certain public memorial,

the stonemason used cubic blocks of stone all measuring one foot on each side.

There was exactly the same number of these blocks (all uncut) in the pedestal as in the square base on the centre of which it stood.

Look at the sketch and try to determine the total number of blocks actually used.

The base is only a single block in depth.

$144$ - The Bricklayer's Task

When a man walled in his estate, one of the walls was partly level and partly over a small rise or hill,

precisely as shown in the drawing herewith, wherein it will be observed that the distance from $A$ to $B$ is the same as from $B$ to $C$.

Now, the master-builder desired and claimed that he should be paid more for the part that was on the hill than for the part that was level,

since (at least, so he held) it demanded the use of more material.

But the employer insisted that he should pay less for that part.

It was a nice point, over which they nearly had recourse to the law.

Which of them was in the right?

$145$ - A Cube Paradox

I had two solid cubes of lead, one very slightly larger than the other.

Through one of them I cut a hole (without destroying the continuity of the four sides)

so that the other cube could be passed right through it.

On weighing them afterwards it was found that the larger cube was still the heavier of the two.

How was this possible?

$146$ - The Cardboard Box

If I have a closed cubical cardboard box, by running the penknife along seven of the twelve edges (it must always be seven)

I can lay it out in one flat piece in various shapes.

Thus, in the diagram, if I pass the knife along the darkened edges and down the invisible edge indicated by the dotted line, I get the shape $A$.

Another way of cutting produces $B$ or $C$.

It will be seen that $D$ is simply $C$ turned over, so we will not call that a different shape.

Now, how many shapes can be produced?

$147$ - The Austrian Pretzel

Here is a twisted Vienna bread roll, known as a Pretzel.

The twist, like the curl in a pig's tail, is entirely for ornament.

The Wiener Pretzel, like some other things, is doomed to be cut up or broken, and the interest lies in the number of resultant pieces.

Suppose you had the Pretzel depicted in the illustration lying on the table before you,

what is the greatest number of pieces into which you could cut it with a single straight cut of a knife?

In what direction would you make the cut?

$148$ - Cutting the Cheese

I have a piece of cheese in the shape of a cube.

How am I to cut it in two pieces with one straight cut of the knife

so that the two new surfaces produced by the cut shall each be a perfect hexagon?

$149$ - A Tree-Planting Puzzle

How do you plant $13$ trees so as to form $9$ straight rows of $4$ trees each?