Problem: Suppose we have a house (with finitely many rooms) in which every room has an even number of doors. Prove that the number of doors from the house to the outside world is also even.
I will assume that there are no doors to cupboards. All doors go either to another room or to outside.
My logic on this is quite simple.
I expect you want me to look at your logic. Maybe I will do that afterwards.
Let 2n be the number of internal door SIDES.
Let k be the number of external door sides.
The number of door sides altogether is 2n+k
Every door has 2 sides so the number of doors is (2n+k)/2 = n+k/2
There must be a whole number of doors so k must be divisable by 2
therefore
If there is an even number of doors in each room then there must be an even number of doors going outside as well.