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This interesting problem could have kept us busy for weeks, drawing rectangles and finding exceptions to the rule.
Our first rule was:
The number of broken tiles (T) = Length + Width - 1 OR T= L + W - 1
This works in most cases, except if the room is square (in which case T= L and T=W). Another exception is if the length and width have a common factor. In this case the answer is worked out by dividing both numbers (length and width) by the common factor, finding the answer using (L+W-1) and multiplying by the common factor. Or, in algebratic terms:

T = n ( L/n +W/n - 1), where n is the common factor.