Henry Ernest Dudeney/Modern Puzzles/Geometrical Problems
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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,
- 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,
- 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,
$125$ - 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
$128$ - 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
$129$ - The Circle and Discs
- During a recent visit to a fair we saw a man with a table,
- 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,
- "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
- 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,
- 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?