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 9x1,
or 16x1, or 25x1, 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 24x1, the length is more than 16 and less than 25 times the
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