How did they get over the catch-22 that if they used the information that Nazis could guess it came from breaking the code but if they didn’t use the information there was no point in having it.
EDIT. I tagged this as mathematics because the movie suggests the use of mathematics, but does not explain how you use mathematics to do it (it’s a movie!). I am wondering for example if they made a slight tweak to random search patterns so that they still looked random but “coincidentally” found what we already knew was there. It would be extremely hard to detect the difference between a genuinely random pattern and then almost genuinely random pattern.
In: Mathematics
They did use the information; they just had to be careful with it and use it to help win the war in the bigger picture.
In any situation where it seemed like the Allies were aware of their plans, the Nazis generally thought that there would have been an information leak elsewhere. There was almost always an alternative explanation. They ultimately believed that Enigma was unbreakable and never stopped to think that it could have been broken.
It’s also important to know that the operations at Bletchley Park were highly secretive. Only the highest echelons of the British forces and government were aware of it. There was almost no Axis spy network in Britain, so there was no way for them to find out that Engima had been broken outside of conflict.
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