Ah, yes, these late night posts: Well, if you were wondering, one of the reasons for people blogging at the oddest hours of day is because they're waiting for something that happens even later that night, like Euro 2008, or Tiger Woods being forced to play an 18-hole playoff with Rocco Mediate (previously ranked a 100-something in the world). But that's my reason, at least.
So, to the main post...
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(a mysterious narrator's voice is heard)
Previously on Scribbles...
The readers were given a problem. A problem titled "Rectangles and Integers". They had to prove a strange property regarding this basic quadrilateral, and it this was the challenge:
Prove that a large rectangle, consisting of smaller rectangles, all with the property of having either an integer width or height, also have this property.
(opening credits roll)
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OK, seriously, that's very TV series cliche, but I couldn't help myself.
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So, now on to the proof. It's surprisingly rather simple. Shade all the rectangles with integer width with one colour, (say white) and integer height with another, (say black). But leave a strip of the other colour along the the integer side. For instance, a rectangle with an integer width (white) will have a strip of black along the side that is the width, and vice versa, (see figure below). For any combination of black and white rectangles, one will be able to trace a line over a single colour (either black or white, or both) using the area of the rectangles and/or the strips. The reason why they are strips is because, going along vertical sides does not contribute to the displacement in that direction, since the strip is always perpendicular to the direction of displacement. But for any given rectangle, there will always be a single colour path.
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The example below shows the proof for a rectangle with an integer width, note how an path is drawn using merely white coloured rectangles and strips. That's it for this mathematical problem. Case closed. 
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