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.
Thanks very much Heureka,
Max has asked me if I can help him to understand your answer so that is why I am posting here.
Firstly max, every normal door has two sides.
Here is a very simple example. Every room has 2 doors. Here, each room has 2 doors. So the sum of the doors in the rooms is 4. But as you can see, there are 3 doors. If I want to count both sides of all doors then I can do that by thinking of the outside of the house as one big room. There are 2 doors to the grand outdoors. So there must be 4+2=6 door sides.
That is there are 6/2 = 3 doors altogether in the house.
The number of door sides in any house is the sum of all the doors in all the rooms + the number of doors that are on the outside of the house. (you can think of the outside of the house as one big room)
And, the number of doors in any house is half the number of door sides (since every door has 2 sides)
the number of doors on the outside + the number of doors in each room = 2 * the total number of doors that the house has.
We are told that the number of doors in each room is even. And a positive even number can be expressed as 2N (where N is a positive integer) so
the number of doors on the outside + 2N = 2 * the total number of doors that the house has.
rearranging we get...
the number of doors on the outside = 2 * the total number of doors that the house has - 2N
the number of doors on the outside = 2 *(the total number of doors that the house has - N)
So if every room has an even number of doors the number of doors on the outside of the house is even.
As Heureka stated, closet doors do not count :)
Wait no.....we have no way of knowing that $n_0+n_1+n_2+...+n_r$ is even, bcause n_0 is included in the other n_i i>0. ANy more help?
The issue here is that the exterior doors are included in the count of how many doors a room has. Meaning, for example, if room 1 has some exterior doors, then they will be counted in n_1. Since i need to *prove* that the number of exterior doors in even, i can't assume that is even, and we're only told that every room has an even number of doors, so each n_i for i>0 is even. That means we have no way of knowing n_0 + n_1 + n_2 +...+n_r at this point that is even.