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These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Fridge Logic: In the first story, when Flambeau asks for the package, he reveals that he actually swiped it some time ago in the next sentence. Why would he then stay with Father Brown and ask for it at all?
Flambeau likes to get fancy with his crimes. For example, at one point he lists various types of people that can be profitably robbed or swindled, along with the appropriate setting for the crime in each case. He certainly seems to have enjoyed The Reveal in this particular instance...
Spoiled by the Format: The first mystery, "The Blue Cross", is told through the Sympathetic P.O.V. of the great French detective, Valentin. So when the unremarkable, seemingly bumbling priest (who isn't even named Father Brown untill near the end) solves the mystery, it would've been a huge shock to readers when it was first published in a magazine. But the most likely place for a modern reader to pick up this story isn't titled The Complete Detective Valentin Stories, so it ain't that much of a surprise.
Values Dissonance: Chesterton's racial and national attitudes were actually very moderate for the early twentieth century, but some will often strike a sour note for modern readers in the midst of his most enjoyable works, as for example in "The God of the Gongs." His religious views, on the other hand, were entirely conscious, and will strike the reader as either refreshingly forthright or offensively aggressive, according to taste.