In the context of a Clos network, each boy represents an ingress switch, and each girl represents an egress switch. A boy is said to know a girl if the corresponding ingress and egress switches carry the same call. Each set of "k" boys must know at least "k" girls because "k" ingress switches are carrying "k"×"n" calls and these cannot be carried by less than "k" egress switches. Hence each ingress switch can be paired off with an egress switch that carries the same call, via a one-to-one mapping. These "r" calls can be carried by one middle-stage switch. If this middle-stage switch is now removed from the Clos network, "m" is reduced by 1, and we are left with a smaller Clos network. The process then repeats itself until "m" = 1, and every call is assigned to a middle-stage switch.