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In math, the question is "why should it be true?" But if we can settle whether something is true or false at all, then it is only true if we can show a reason that works for all cases. For it _not_ to be true a single counterexample  such as the one you gave  is sufficient. In fact, cutting pieces and moving them around furnishes a valid way to prove the areas are different. That was actually the definition of equal/unequal area that Euclid used, and it's a very good one  though it doesn't generalize well to higher dimensions. So the answer is: they are not in general the same, and you know this because in at least one case you have proved it. A way to get a better intuitive feel for this is to consider extreme cases. Think about a 49x1 room, or even a 50x0 room in which the walls meet! There are some deep philosophical questions involved here, it is true  for instance, some things _are_ just true by accident. [Example: the number of regular convex polyhedra is the same as the number of ways to arrange the numbers {1,2,3} so they aren't in increasing order  five in each case.]  


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