eli5: Why could the Enigma Machine not map a letter onto itself?

Question

I know that that was why the codebreakers could break Enigma, but why?

In: Mathematics

Answer 1

It was just a byproduct of how it was designed, and the circuitry does not allow for that. There are two areas of note for the design, and you can examine [this diagram from Wikipedia](https://en.wikipedia.org/wiki/Enigma_machine#Electrical_pathway) to take a look at the internal workings.

There was a reflector in the mechanism, which connected to the last rotor and paired up the outputs (crucially, **all 13 pairs of them**, as will be explained in the next paragraph) before connecting back to the rotors, in reverse order, (and plugboard, etc.) again. This makes the code a self-inverse, so as long as you have the same settings, typing one of the plaintext or encrypted messages will give the other.

**The presence of the reflector itself does not inherently prevent mapping a letter to itself** (as you could pair only some letters and still preserve the self-inverse property). However, *the resulting simplification of the design that it permitted did*, as this simplification also meant that **all 13 pairs had to be made** in the reflector. The internal mechanism of the keys is such that that the operate a switch. When not pressed, the switch of a key connects the circuit from the encryption output to that letter’s lightbulb. When pressed, it instead connects from the battery to the start of the encryption process. (The wise among you may note that a letter cannot be both pressed and not pressed simultaneously, which is precisely the issue here.)

The simplification means that the resulting electrical pathway is always in the order: battery, depressed switch of pressed key, plugboard, rotors, reflector, rotors in reverse order, plugboard, switch of output letter, lightbulb. The fact that the switch cannot be in both positions – connecting to the battery, and connecting to the lightbulb, a design choice made with the reflector in mind, is the physical reason why a letter cannot map to itself. If pairing a letter with itself was allowed, then the circuit here would need to be redesigned.

Edit: It’s worth noting that it’s in theory this flaw could be fixed by having the switch connect both to the battery and lightbulb when pressed (using the other lit lightbulb as the encrypted letter and the pressed letter when no other bulb is lit), but even this could introduce issues such as with wire resistance.