ber. Note also in the diagram above that
not only are the opposite numbers on the outer ring complementary, always
making 9 when added, but that opposite numbers in the inner ring, our
remainders, are also complementary, adding to 17 in every case. I ought
perhaps to point out that in limiting our multipliers to the first nine
numbers it seems just possible that a short period circulator might give
a solution in fewer figures, but there are reasons for thinking it
improbable.

84.--_The Japanese Ladies and the Carpet._

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.

[Illustration]

[Illustration]

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.

[Illustration]

85.--_Captain Longbow and the Bears._

[Illustration]

It might have struck the reader that th