This problem can be generalized. Prove that in a network of computers there are two which are connected to the same number of computers. The proof goes by induction. Of course it is true for so let us assume it is true for network of computers. Say we have network. There are two cases: every computer is connected; there exists a computer not connected to the rest of the other computers. In the first case if every computer is connected then it means the # of connected computers for each computer is between 1 and k. But there are k+1 computers, so there exist two computers that have the same number of connections. In the second case isolate the computer not connected to anything else. Then we are left with a k-computer network. But then it follows by induction that there exists two computers in this network that are connected to the same number of computers.