Plugboard, here swapping A with J, and S with O. Photo Bob Lord (CC BY-NC-SA).
As you may know, the German military made a significant addition to the original Enigma design by adding the plugboard, or steckerbrett. The board has twenty-six sockets, one for every letter in the alphabet. The operator would use ten cables, each with a plug at each end, to ‘pair up’ twenty letters. This would have the effect of cross-wiring the keyboard before it’s electrical wires entered the rotors. So, a cable paired the A socket to the B socket would mean you had effectively pressed B, not A. This added significant complexity to the Enigma cypher.
One question I am often asked on tours is ‘why ten pairs?’. Surely it would make more senses to plug up every letter and completely scramble the keyboard? Well, not so fast. Adding the concept (and therefore probability) of an empty socket increases the odds significantly.
The maths itself is surprising. However, I’m going to hand over to PJ Bryant who explains how the optimal number of pairs is actually 11.