Henry Ernest Dudeney/Puzzles and Curious Problems/Geometrical Problems

Henry Ernest Dudeney: Puzzles and Curious Problems: 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.