Question

5. A computer network consists of six computers. Each computer is directly connected to at least one of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of other computers.

Answer 1

Let the degree of a computer denote the number of other computers to which that computer is directly connected. The degree of any computer can be at most 5, since there are only 6 computers in total. Therefore, if each computer had a different degree, one would have degree 5, the next one 4, and so on. The last computer would have degree 0. However, this was excluded by the question.