I must have started reading this topic too late. I got the same answer as Half Fast after a couple of attempts. But I wasn't as devious as Mike D, who had to wait an extra 4 minutes.
Who boils their eggs for 9 minutes anyway? It must be very hard. I like my yolk runny!
If you start with a rectangle with no integer edges, then at least one of the smaller rectangles will also lack integer edges. This is heuristically true - but I can't prove it yet.
If my first statement can be proved, then it follows that the original proposition (If you have a rectangle and you partition it into smaller rectangles such that every rectangle has at least 1 edge of integer length, then the large rectangle has 1 edge of integer length) must be true.