A perfect squaring of a rectangle may be defined as a partition of the rectangle into finitely many squares all of which are mutually non-congruent. https://en.wikipedia.org/wiki/Squaring_the_square shows, among other things, perfect squarings of the *square* with provably least number of smaller squares.

Regarding existence: I don't know any non-square rectangle that has a perfect squaring. If such rectangles do exist, how does one

those rectangles that admit perfect squaring? Or is it the case that any given rectangle has some perfect squaring, maybe with a very large number of squares?**characterize**If such rectangles do exist, are there rectangles for which a perfect squaring can be done using a lesser number of smaller squares than is needed when a square is squared? This seems unlikely.

anytiling of the rectangle with squares. For a proof, see for instance the chapter "Tiling rectangles" inProofs from THE BOOK(4th edition or later, chapter number varies between editions), or Chapter 12 of Matoušek's "Thirty-three Miniatures". $\endgroup$