There have been a good few killings in the 73 "Maigret" books so far. Their publication in French began in 1931 and when "Maigret et l'homme tout seul" ("Maigret and the Loner") was first published in 1971, only the 74th and 75th would remain before Simenon retired from writing at the age of 69 in 1972.

After so many murder mysteries, it could explain why the Belgian author opens up this book in rather desultory fashion. We've been here before: a quiet Paris in August when many of the locals have left the city to go on holiday and their place has been taken by hordes of tourists. No surprises in the usual telling of the weather either, with Simenon this time offering a violent rainstorm each afternoon to end the heatwave of the day.

He's not wasting much effort on preliminaries and it's almost straight into the inevitable murder, introduced on only the second page. But at least the spilling of blood bucks up Simenon a bit and you can almost feel the lift in involvement as he starts to hit his stride. A small boy out playing in Rue de la Grande-Truanderie, a narrow street of derelict houses waiting to be demolished in the first arondissement of Paris, has discovered a slain homeless man shot three times in the chest while asleep.

The fascinating thing for Maigret is the tramp's startling appearance, with his ragged clothes and desolate surroundings in marked contrast to his aristocratic face. He is quite old and has long silvery hair with blusish streaks. His white moustache is slightly turned up at the ends with an equally white Richelieu-style goatee beard. These have been trimmed with care, obviously very recently.

Maigret has another surprise on discovering that the man's hands are well-tended and do not look like they have been sifting through rubbish bins. The hands are carefully manicured with poilished nails, and all in all the impression is of an old actor playing the part of a tramp.

The victim is on an iron bedstead with an old straw mattress and, oddly, the crumbling room contains an incredible jumble of totally useless objects, most of them broken: an old coffee grinder, badly chipped enamel jugs, buckets with dents or holes, a kerosene lamp with no wick or kerosene and unmatched shoes. Was he a mad man or an obsessive to have collected this rubbish?

It is the first time in his career that Maigret has seen such a scene. There is no water, gas or electricity, no identity papers. Who is the dead man? Photos of his dead face will be sent to the newspapers, with the whiskers and without, after the police get a barber to shave him. One headline says: "A dead man with no name."

Could there be a hairdressing school nearby, with aspiring manicurists, where the novice pupils practise on tramps and beggars? Will the murdered man's fingerprints be found in Criminal Records? Did he eke out an existence by helping out at night in the nearby Les Halles fruit and vegetable market, unloading lorries from the countryside? Who is he?

Maigret and his inspectors go on the trail, though at the start there is no trail. But slowly the facts emerge, and it turns out that the mystery had its roots a full 20 years ago. There is an abandoned wife and daughter. Another woman with two lovers. Police inspectors do the rounds of the bistros, hotels and a bank, and add to the growing number of revelations. By the time it's all over, Maigret will have solved not one but two murders, by different slayers. But when he nails the man the police have been hunting, how to actually prove that the suspect is the killer?

Simenon skillfully puts it all together fact by fact, event by event, puzzle by puzzle. He shoots through it at pace – nothing really unusual about that – with plenty of dialogue and not a lot of colour added. Maigret goes through his usual grumpy moods when the case isn't progressing and he downs quite a few beers and aperitifs along the way, with scant regard for his doctor's earlier caution.

His pipe smoking continues unabated: they can't take away all his pleasure just because he is nearing 55 years old. There are a couple of fleeting references to Meung-sur-Loire, where the Maigrets have bought a country property, but his impending enforced retirement at 55 isn't mentioned. Madame Maigret knows to tread tenderly while her husband is tense and preoccupied.

Examining Magistrate Cassure mentions Maigret's method, which distinguishes him somewhat from other famous fictional detectives. This is his reputation for "going everywhere yourself, questioning concierges in their lodges, artisans in their workshops, housewives in their kitchens or dining rooms". Maigret can only agree. This is how he understands human psychology.

Cassure's office is in a part of the police building at Quai des Orfevres that has not yet been modernised. With a black-painted desk with gashes in it, just like in an old school, and piled-up files and a clerk who appears to have been there since the previous century, it is like being in a novel by Balzac.

Simenon doesn't trouble to explain why the tramp collected all that junk, though it sounded like it might have been important when it was first mentioned. Presumably it was just an eccentricity. And despite the satisfying way in which the police dig and dig, turning up clue after clue, in the end it is an anonymous caller who suddenly provides the name of the murderer.

This is unexplained with no further mention. It is rather jarring and unsatisfying but Simenon simply waves it aside with the lazy comment that Maigret "knew from experience that many criminals are only arrested thanks to an anonymous phone call or a tip-off from an informer". Convenient, that!

And, in another odd plot turn, if the earlier murder, in 1946, was reported in the "Parisien libéré" newspaper, as Maigret discovers by going through the back issues, why wasn't it in police records, which he had asked to be checked earlier? Still, here we do at least get one of those rare snippets of Maigret history from Simenon: "He himself hadn't been in Paris in 1946. That was the time when he had done something to displease the then commissioner of the Police Judiciaire … He had been sent to Lucon, where there was very little to do, and in order to kill time he had played billiards almost all day long. He had moped there for nearly a year. Madame Maigret, too, had found it hard to adjust to life in the Vendée."

Book by book, month by month, murder by murder, Penguin Books continues its chronological reissue of the famed canon with only numbers 74, "Maigret and the Informer" ("Maigret et l'indicateur", 1971), and 75, "Maigret and Monsieur Charles" ("Maigret et Monsieur Charles", 1972) to come in December 2019 and January 2020. After that, some Maigret omnibuses or collections of the 28 short stories might be a nice idea.

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