The bombes were a development of an earlier Polish design (bomba). The thing to understand is that the engima machine worked by having several rotors which were set at the start of each message in some order and positions based on that day’s instructions. The rotors would then advance with each letter of the message.
The aim of the codebreakers was to find the starting order and positions for a given day, because then they could reverse the encryption and read all the messages for that day. They did this basically by brute force – trying lots of possible starting combinations and eliminating each one as it produced a contradiction. Specifically the codebreakers would produce a menu which was a sample of plaintext that was likely to appear in messages. This was run through the bombe, and a contradiction would occur for example if a letter was encoded as itself; something the enigma machine was unable to do.
The bombe itself is basically just lots of enigma machines combined so it can try lots of starting positions simultaneously.
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