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John Buck: Deer Detective
Mentioned in Hypnosis Doesn't Work Like That!, John Buck is an animated movie released a few years before which seems to have made a bit of a media splash. It's successful in that every supermarket had shelves filled with tableware, bed linen, lunchboxes, and stationery bearing pictures of the characters for nearly a year.
The main characters seem to be a deer with a deerstalker and magnifying glass, and a dog in dark glasses. The deer occasionally has trouble getting his antlers through doors, which is mostly played for laughs. One sad moment in the movie is when an otter is locked up, but it's not mentioned why. There is a little adult humour in the movie, designed to keep the parents entertained, but it would mostly go over the kids' heads.
The antagonist is 'Detective Doug', a duck who turned to a life of crime after he got sick of everyone misreading his name and assuming he was going to be a dog (stereotyped as being the best animals for solving mysteries)
There is a series "Deer Detective on the Road" (mentioned in Unexpected Daddy chapter 16) following on from the movie a few years later, which takes a different tack. It is mostly computer generated, and features John Buck and his love interest Ruby Rabbit travelling around the country solving crimes. It is a cartoon for older children (or at least older than the target audience of shows like Paradise Pen and Bibi Babi Miko), where there are enough clues for a child to be proud of solving the mystery before the detective reveals the answer. The opening sequence features higher quality animation than most of the show, and features Buck racing along a country road in a red sports car, while Ruby lies across the back seat playing the harmonica (the opening theme starts with his solo).
It's mentioned in Unexpected Daddy that John Buck uses a whole notebook on each case he solves. Not mentioned yet, but I can imagine a pile of them in the footwell of his car, where he throws them when he's done and never gets around to typing up his notes as a memoir. Sometimes Ruby is seen writing them out, which may be the "source" for there being a TV series about him (which exists in-universe, but is more a docudrama thing). Also mentioned that the squiggles in his notebooks, if you zoom in on the relevant frames, are all critical path and segmented tree diagrams (nothing to do with detective work) labelled with 'Englishman/Irishman/Scotsman' jokes transliterated into old Germanic runes. Oswyn is probably the only viewer who has ever decoded them.