Answer :
If the squares had not to be all the same size,
the carpet
could be cut
in four pieces in any one of the three manners shown.
In each case the
two pieces marked A will fit together and form one of the three
squares,
the other two squares being entire.
But in order to have the squares
exactly equal in size, we shall require six pieces, as shown in the
larger diagram. No. 1 is a complete square, pieces 4 and 5 will form a
second square, and pieces 2, 3, and 6 will form the third—all
of exactly
the same size.
If with the three equal squares we form the
rectangle IDBA,
then the mean
proportional of the two sides of the rectangle will be the side of a
square of equal area.
Produce AB to C, making BC equal to BD.
Then
place
the point of the compasses at E (midway between A and C) and describe
the
arc AC.
I am showing the quite general method for
converting rectangles
to squares, but in this particular case we may, of course, at once
place
our compasses at E, which requires no finding.
Produce the line BD,
cutting the arc in F, and BF will be the required side of the
square.
Now
mark off AG and DH, each equal to BF, and make the cut IG, and also the
cut HK from H, perpendicular to ID.
The six pieces produced are
numbered
as in the diagram on last page.
It will be seen that I have here given the reverse
method
first: to cut
the three small squares into six pieces to form a large
square.
In the
case of our puzzle we can proceed as follows:
Make LM equal to half the diagonal ON.
Draw the
line NM and
drop from L a
perpendicular on NM.
Then LP will be the side of all the three squares
of
combined area equal to the large square QNLO.
The reader can now cut
out
without difficulty the six pieces, as shown in the numbered square on
the
last page.
