Here there be spoilers:
In the early ‘50s, Detective Robert Nock (Rory Kinnear) is called to the house of eccentric and phlegmatic mathematician and Cambridge teacher Alan Turing (Benedict Cumberbatch), to investigate an apparent burglary. But Nock finds Turing uninterested in the crime and eager to rid himself of the visiting law. Realising Turing deliberately insulted him and other policemen to this end, and fascinated by the sprawl of mysterious technology Turing has set up in his parlour, Nock begins to investigate, hoping to catch a Cambridge Five-level spy. He follows instead a thread that uncovers Turing’s astounding, entirely secret role in breaking the Nazi’s Enigma code on the way to learning Turing’s rather different secret. Flashback to 1940, when the socially inept and often obnoxious Turing is interviewed by Bletchley Park chieftain Denniston (Charles Dance), turning into a duel of brusque dismissal between the two men, before Turing is reluctantly hired and placed on a team of code breakers and linguists captained by Hugh Alexander (Matthew Goode). When Alexander nixes Turing’s idea to build a proto-computer to unravel the staggering possible variants in the Enigma code because of the price tag attached, Turing gets MI6 interlocutor Stewart Menzies (Mark Strong) to pass on a letter to Winston Churchill explaining his proposal, and is rewarded with leadership of the team. Turing immediately stirs dislike by casually firing several team members and labouring for months on his experimental boondoggle. He harvests new talent for the project by placing a puzzle in the newspaper and offering a job to anyone who can solve it in under five minutes, and turns up mathematician Joan Clarke (Keira Knightley), whose talents and education are evidently up to the job but whose fusty parents are perturbed by the offence to decorum the job might entail.