[Illustration]
The illustration shows how to cut the four pieces and form with them a
square. First find the side of the square (the mean proportional between
the length and height of the rectangle), and the method is obvious. If
our strip is exactly in the proportions 9 × 1, or 16 × 1, or 25 × 1, we
can clearly cut it in 3, 4, or 5 rectangular pieces respectively to form
a square. Excluding these special cases, the general law is that for a
strip in length more than n² times the breadth, and not more than (n+1)²
times the breadth, it may be cut in n+2 pieces to form a square, and
there will be n-1 rectangular pieces like piece 4 in the diagram. Thus,
for example, with a strip 24 × 1, the length is more than 16 and less
than 25 times the breadth. Therefore it can be done in 6 pieces (n here
being 4), 3 of which will be rectangular. In the case where n equals 1,
the rectangle disappears and we get a solution in three pieces. Within
these limits, of course, the sides need not be rational: the solution is
purely geometrical.