Let's say the sides of the rectangle are $a,b$ and the cut out square has side $x$. We have $$2(a+b)=ab-x^2$$Clearly if the square was cut so that it shared 2 sides, then the perimeter would be invariant, which would imply that square has zero area. Also we can easily eliminate the option of the square sharing 4 sides. The square sharing 3 sides is also pretty unlikely, but we will not eliminate it at the moment. Assuming the square shares 1 side with the rectangle, we also have $$ab=2(a+b)+2x$$Combining the 2 equations gives $$x^2=2x$$$$\boxed{x=2}$$ If the square did share 3 sides, then we would have $$ab=2(a+b)-2x$$$$x^2=-2x$$$$x=-2$$This is not feasible.