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Oh, Ludwig – how do I love thee? Let me count the ways.

Or at least allow me to hold your favourite biro as you scurry around Cambridge in a state of rumpled exasperation, like a Womble that’s just been chased out of Ryman following a misunderstanding over a hole-punch.

Missed the first series of this endearingly nerdish crime caper? Panic not. The second arrives bearing that most traditional of housewarming gifts – a recap. And very welcome it is too, given Ludwig’s fondness for befuddling conspiracies, impenetrable ciphers, and scenes in which corduroy-clad David Mitchell points at a graph while spluttering things like “the whole thing is dichotomous!”.

The recap, then: Mitchell is John “Ludwig” Taylor, a crossword-setting introvert who has been posing as his identical twin, James – a DCI – in order to uncover the reasons for the latter’s mysterious disappearance. Though swiftly rumbled by the fuzz, John’s extraordinary knack for solving low-stakes murders has seen the bumbler hired as a “crime scene consultant”.

Meanwhile, James has left cryptic clues re the covered-up murder of a journalist named Sinclair. The key to the conundrum, it seems, lies in Sinclair’s innumerable files, which currently reside in James’s wife Lucy’s (Anna Maxwell Martin) spare room.

And now? John is diligently organising the folders into what he calls, with touching fondness, “my piles”.

“Do this enough times,” he explains, “and we’ll find the one file that Sinclair died for and the reason for my brother’s disappearance!” A pause. “Probably!”

All clear? Splendid. And so to the theatre, where a narky pensioner has been stabbed by one of her daughters. But which one? “I did it,” honks Portia (Doon Mackichan). Oh no she didn’t, counter-honks her sister, Patricia (Rebecca Front). “It was me.” The reason? “Mummy wouldn’t let me have a bichon frisé.”

John is flustered. “This one’s really hard,” he splutters, his expression, as always, that of a mole startled to learn that the last UK branch of Tie Rack closed in 2013 and not, as he was convinced, 2021.

“Whichever one didn’t do it knows that the other one did and is protecting her. While the one that did do it knows we can’t prove that the one who didn’t do it, didn’t do it.”

There is a lot of this sort of thing in Ludwig. That is, John holding forth on the impossibly convoluted ins and outs of each preposterous case-of-the-week, oblivious to the bewilderment of all concerned. Frankly, there is not much room here for the armchair detective. Magnifying glasses will remain unwielded; pipes resolutely un-puffed. It’s simply too idiosyncratic. Too … Ludwig. Instead, the fun lies in the sight of John boggling at each low-stakes murder before dissolving into radiant, childlike grins as he realises he has – gosh, really? How marvellous! – solved said low-stakes murder all by himself. How? A combination, usually, of a) accident b) preternatural boffinry and c) shouting about his piles into a cretaceous-era Nokia.

Like all cosy crime series, Ludwig is up to its elbow patches in nostalgia. There is a tangible yearning for a time before the impertinences of modern drama; a time of corniced accents and iced-sponge values, of labradors in padded gilets upbraiding menials on gravel driveways. But Ludwig is as sharp as it is cuddly, its bobbly pullover concealing a script (by series creator Mark Brotherhood) that revels in wordplay and the infinite micro-horrors of social awkwardness.

Best of all is the ever-expanding friendship between John and the gloriously direct Lucy, who has finally and rightly been elevated from underused supporting character to giddily eccentric co-lead. And what fun Maxwell Martin is having; bounding around Cambridge in her dinky ankle socks, her schoolgirl bob bouncing with investigative zeal.

John and Lucy are clearly perfect for one another. So much so that you occasionally find yourself wishing James would remain buggered-off to let nature take its course (even if any exertion in the spare room department would play havoc with John’s piles).

How best to describe Ludwig’s blend of whimsical mystery and cerebral silliness? Midsomer Murders meets Monk? No, that would be awful. Jonathan Geek? If anything, that would be worse. Instead, let’s leave it at this: Ludwig is Ludwig, and hurrah for that. Long may it bumble.

• Ludwig aired on BBC One and is on iPlayer now