Rectangles made entirely of squares can be wonderful problems. Bill Gosper: "Search the Web for 'squared rectangles'. If I were king, one of these diagrams (undimensioned) would appear daily in the newspaper puzzle pages." http://tweedledum.com/rwg/rectarith12.pdf http://gosper.org/Googebra.htm   (Solving techniques) I made a few thousand puzzles involving Squared Rectangles.  In each case, the dissection is "nowhere neat", in that no two squares share a full edge.  Also, all of the squares in each dissection have a size less than 100.  I give two different types of puzzle for each dissection. http://demonstrations.wolfram.com/MondrianPuzzles/   I also attach a large scale puzzle of this type, as a challenge.  The squared rectangle it produces is extremely tiny.  I'm willing to bet $20 that an asymmetrical, nowhere-neat, squared rectangle of an equal or smaller size and more squares is impossible.  A proof would be straightforward and impossible -- just run Bouwkamp processing on all gajillion planar graphs with 54 to 90 edges.  At 35 edges, there are already 5986979643542 planar graphs. http://www.numericana.com/data/polycount.htm More history and solving techniques at http://squaring.net/sq/tws.html  --Ed Pegg Jr http://www.mathpuzzle.com/