Puzzles

Add up the number of doors leaving each room; call the sum \(S\). As the number in each room is even, \(S\) will be even. Each interior door has been counted twice (as they can be seen in two rooms) and each exterior door has been counted once. Let \(I\) be the number of interior doors and \(E\) be the number of exterior doors. We have: $$S=2I+E$$ $$E=S-2I$$ \(S\) and \(2I\) are even, so \(E\) must be even.

If the number of doors in each room is odd, is the number of exterior doors odd or even?