T­he Imitation Game spends so much time setting up complex puzzles that it oversimplifies the life of its central enigma: Alan Turing, a British genius posthumously regarded as a father of the modern computer, a man who envisioned the possibility of artificial intelligence. Fortunately, Turing is played by Benedict Cumberbatch, who expertly fleshes out what’s left off the script in another one of his deeply cerebral performances as a tortured intellect who sits to the left of center of society’s norms.

Most of the film is set during World War II, when Turing, the self-described “top mathematician in the world,” was enlisted by MI6 to toil as a cryptanalyst. He joins a brilliant team racing to crack the supposedly unbreakable Nazi codes of Germany’s Enigma machine at a top-secret location in Bletchley Park—or, as Turing sees them, “half a dozen crossword enthusiasts in a small village in the south of London.”

Based on the way he places himself at odds with every character in the film (save another outsider played by Keira Knightley), Turing may have had a touch of Asperger’s. But it’s his hidden homosexuality that serves as the key piece to the puzzle constructed by director Morten Tyldum and screenwriter Graham Moore, who freely adapted a biography by Andrew Hodges, a mathematician and activist in the ’70s gay liberation movement.

As the film opens, it’s 1951, and Detective Robert Nock (Rory Kinnear of recent James Bond films) arrives at Turing’s home to investigate a burglary. Turing insists that nothing has been stolen, but the detective senses that the mathematician doth protest too much and begins investigating Turing instead. As Nock probes the mystery of this strange man, the film flashes back to a 1939 meeting  between Commander Denniston (Game of Thrones’ Charles Dance) and the socially awkward Turing at an office at the Bletchley Radio Manufacturing building, the HQ for the secret team of scholars, linguists, intelligence officers and chess champions in possession of an Enigma machine that was smuggled out of Poland.

The group is led by Hugh Alexander (Matthew Goode of TV’s The Good Wife), who tells Turing he believes he’s made some progress with the machine. “Even a broken clock is correct twice a day,” Turing replies, not making any friends as he calculates that there are “159 million, million, million possibilities” for code encryption within the Enigma.

It’s not long before Turing’s colleagues have filed a formal complaint against him with Denniston, who doesn’t much like him either. Turing’s response? “Get rid of them,” he tells Denniston, who bristles when Turing instructs him to use the potential savings to fund his hypothetical machine, “the bombe,” which could theoretically exploit tiny weaknesses in the Enigma’s coding.

“Who is your superior?” Turing demands to know when Denniston refuses to take orders.

“Winston Churchill,” comes his smug response.

The next time we see Denniston, he’s in his office, looking dour as he announces to the assembled team that Turing has been appointed the division’s new leader, usurping a very pissed-off Alexander. Calling Denniston’s bluff, Turing had given a letter to MI6 agent Stewart Menzies (Tinker Tailor Soldier Spy’s Mark Strong) to deliver directly into the hands of Churchill. Turing’s first order of business? Firing two of the group’s dead-weight members.

“You must have been popular in school,” Menzies dryly observes as the movie dips into one of many flashbacks-within-a-flashback, set at the boarding school where Turing spent his youth. Played in these scenes by Alex Lawther, the quiet loner obsesses over keeping peas and carrots separated on his dinner plate, only to have the food dumped on his head by bullies, when they aren’t busy trapping him beneath the school’s floorboards. But the young Turing has a protector: classmate Christopher Morcom (Jack Bannon), who gives him a book on codes and ciphers, becoming his sole companion  and the object of the ginger-haired outcast’s pubescent longings.

Back at Bletchley Park, Turing recruits Joan Clarke (Knightley), and the two form a close bond, even as (or perhaps because) she correctly pegs him as a “fragile narcissist.” Goaded on by this “woman in a man’s job,” he attempts to endear himself to his colleagues as they inch toward the breakthrough that would help save thousands of lives, telling jokes (terribly) and offering them shiny apples as treats.

Hodges’ book reveals that the real Turing had always been fascinated by the scene in Snow White and the Seven Dwarfs where the evil queen dips an apple into poison. Given the obvious foreshadowing with the apples handed out by Cumberbatch’s Turing, along with the cyanide he handles at one point, it’s a real head-scratcher why neither Moore nor Tyldum—who displayed a dark wit with his last picture, the Norwegian noir Headhunters—thought to end their film at what would seem to be its natural conclusion.

As a result of Nock’s investigation, Turing was sentenced for indecency as a “sodomite” in ’52. Onscreen text informs us of his tragic suicide at the age of 41, a year after he began government-mandated “chemical castration,” 57 years before he’d be issued a government apology and 62 years before he’d be granted a pardon. It’s a very conventional ending to a film about a complex, puzzling life.

What we don’t see is that Turing died of cyanide poisoning—a half-eaten apple beside him, foam coating his mouth, no prince to save him.

The Imitation Game ***

Starring Benedict Cumberbatch, Keira Knightley, Matthew Goode, Rory Kinnear, Allen Leech, Matthew Beard, Alex Lawther, Jack Bannon, Charles Dance and Mark Strong. Written by Graham Moore, based on Andrew Hodges’ book Alan Turing: The Enigma of Intelligence. Directed by Morten Tyldum. At Coolidge Corner and Kendall Square.