Henry Ernest Dudeney/Puzzles and Curious Problems/Geometrical Problems/Dissection Puzzles

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?