Henry Ernest Dudeney/Modern Puzzles/Geometrical Problems/Various Geometrical Puzzles

Henry Ernest Dudeney: Modern Puzzles: Geometrical Problems: 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?