Dudeney's solution says:
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 x 1, or 16 x 1, or
25 x 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 n^2 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 x 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.
[*] An image of the solution is available on-line at the following links:
http://puzzles.50webs.org/a153.html
and
https://archive.org/details/amusementsinmath00dude/page/172/mode/1up
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