(related to Problem: A Cutting-out Puzzle)
The below 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 $9\times1,$ or $16x\times1,$ or $25\times1,$ 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 $24\times1,$ 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.
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