B. Felgenhauer has found minimal solutions for dividing a rectangle into squares, up to size 300x300. http://int-e.eu/~bf3/squares/ http://int-e.eu/~bf3/squares/view.html#13,11http://demonstrations.wolfram.co...  An n×m rectangle can be tiled with a minimum k squares. Can rectangle of size 2(n×m) be tiled with k-1 or fewer squares? Conjecture-- No. That's an item at  http://math.stackexchange.com/questions/240277/   So far, no counterexample is known.   --Ed Pegg Jr