686 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



of handling only 1,250 calls at one time we find 



H = 22,000 bits, 

 6,666 



.015 or larger 



# switches = 6,666 

 ^ functions = 10 



# contacts = 10^-*^^ or more for all but a fraction 10 ^°^'^ * of the functions. 



Although these numbers of contacts appear incredible at first sight, there 

 is no reason to expect the number of contacts for almost all switching func- 

 tions to be a good indication of the number of contacts needed for the 

 switching functions of practical use. This phenomenon has been discussed 

 in detail by Shannon for the case of two- terminal networks. For iV- terminal 

 networks there are at least two other factors which may be mentioned. 



Almost all switching functions can assume states which are not typical of 

 the functions encountered in practice. For example, it can be shown that of 

 the <i>{N) possible ways of distributing N different things into parcels, al- 

 most all of them use a number of parcels which is near iV/log^ N, Thus, in 

 a typical state of a typical switching function the terminals are connected 

 together in groups which average about logeiV terminals per group in size. 

 Telephone switching equipment ordinarily connects terminals together in 

 pairs or in small groups. 



A big difference between the design of two-terminal networks and of A'- 

 terminal networks is that, in the former case, one wants to obtain one 

 specific switching function while, in the latter case, one is usually satisfied 

 with a network which can produce certain desired states. There are many 

 switching functions which all produce the same states but for different 

 settings of the switches. For example, if the 2* desired states are all dif- 

 ferent, the designer will be content with any one of (2^) ! different switch- 

 ing functions. We believe that it actually would require something like 

 jQ6,67i contacts to build a central ofl&ce if the designer first listed all the 

 desired states at random 5i , ^2 , • • • , S^h and then required the oflSce to 

 be in state ^i for switch setting (1, 1, • • • ,1). in ^2 for the switch setting 

 (1,1, ... ,2), etc. 



Number of Selector Switch Rotors 



Since our estimate (13) of the number of contacts is the same whether 

 the memory H is stored in two-position switches or in larger selector switches, 

 one might hope that the selector switch circuits could be built using fewer 

 rotors than the corresponding two-position switch circuits. We believe that 

 this is not true. A typical node in one of the graphs constructed by the 

 process of theorem VI has only about three or four branches connected to 



