A rectangle ABCD is to be cut into three pieces
(with two straight line cuts) and the pieces re-
arranged into a square.

Define the endpoints of the cuts.

Note 1: |AB|/4 ≤ |BC| < |AB|.

Note 2: A straight line cut (as the name implies)
            is a cut along a straight line that
            divides a piece into two pieces.

(In reply to Full solution by Jer)

here is how I analized your solution. First some labeling.  In the first drawing, let the length of the horizontal (top/bottom) side by m and let the length of the vertical (left/right) side be n.  Also, I assumed that m<n, if m=n then any cut works as it is already a square.  Now for the diagonal cut, let end of the cut be x from the top.  Also, let the horizontal cut be y from the top.  Finally, let the length of the horizontal cut be z.  Now from similar triangles we have x/m=y/z thus z=my/x.

Now in the bottom drawing we have the left side as x and the right side as n-y.  Also both the top and bottom are given by m+z=m+my/x.  So for this to be a square we need both
x=n-y
x=m+my/x
substituting the first in the second we have
n-y=m+my/(n-y)  since y<n (otherwise there can't be a cut) we can multiply through by n-y to get
(n-y)^2=mn
y=n+-sqrt(mn)
nowhever since m<n then sqrt(mn)<n and since y<n we have
y=n-sqrt(mn)
which gives x=sqrt(mn)<n thus the diagonal cut can happen as well.

So from this analysis I conclude that your solution works for all rectangles.

Posted by Daniel on 2013-08-16 10:34:57