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According to Harry Potter Wiki, there are 48 National Quidditch teams. This seems like an extraordinarily high number considering the low number of known schools of Wizardry that include Quidditch in their curriculum. Are we to believe the 432 players, 7 per team plus 2 reserve players (unless the big leagues have more reserves), all came from the 3 Magic institutions mentioned in the Goblet of Fire?

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    Isn't it an established canon fact that JKR doesn't know maths? :) – DVK-on-Ahch-To May 17 '14 at 17:13
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    What makes you think there are only three schools in the whole wizarding world? – Valorum May 17 '14 at 23:12
  • @Richard There is a huge amount of reference material dedicated to the books and films that delves into just about every aspect of the stories, but they don't provide information on any other schools. What canon proof do you have that there are other schools? – Major Stackings May 18 '14 at 8:16
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    The quidditch world cup had 100,000 visitors. Taking into account the longer lifespan of the average wizard and assuming an average birth-rate of 1.0 children per wizard, there would need to be at least 20-50 schools the size of Hogwarts to educate their children. And that's not taking into account all the people who didn't go. – Valorum May 18 '14 at 8:57
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    Quidditch teams need not have started from a school. You're asking about 'National teams', but you tie that into schools; which are unrelated. – Möoz May 19 '14 at 2:10
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From this answer, we know that the total population of wizards is around 300K to 1 million globally. As you mentioned, the team size is 9, times 48, so less than 500. That seems perfectly reasonable. Just remember that there are more schools than just the 3 mentioned, those are merely the schools for Europe.

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    An excellent answer. Note that "Quidditch Through the Ages" says that there are 13 professional UK teams. Assuming it's similarly popular the world over, that makes around 5000 professional players in the world. – Valorum May 17 '14 at 23:24
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    13 UK teams is a bit harder, as there's only 3K-10K wizards in the UK per estimates. So that'd mean that there's a sizable part of the wizard population in Quiditch. But I don't really see a problem with that, after all, what else do wizards really have to do? – PearsonArtPhoto May 17 '14 at 23:29
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    Could some of the UK’s professional teams be made up of players who originally hail from other countries, similar to our football? (And so aren’t part of the 3–10K estimate) – alexwlchan May 18 '14 at 22:37
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    @Richard I see no problem with the Wannabe teams if you think about Sunday league football(soccer) teams most males from 9-40 will play in those and it seems that almost everyone loves quidditch and it can be mixed genders. – CandiedMango May 19 '14 at 9:47
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    @Richard Right, totally misread that then. Also don't forget wizards can be home schooled. Draco also mentioned that he wanted his dad to send him to Durmstrang which stands to reason you could send your child away for school. – CandiedMango May 19 '14 at 12:15
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The Harry Potter Wikia mentions that there are more schools in existance than the three in the increasingly-innaccurately-named Triwizard Tournament. We know the American one is the Salem Witches' Institute... which indicates that there is also a _____ Wizard's School Somewhere in America (as magic users are, evidently, as sexist as us Muggles). The Wikia states there is a mention of a South American school as well, possibly in Brazil, but I don't recall that myself. Assumably there are one or more in Asia, Africa, and at least one Down Under (Edit: NOPE).

But you also have to realize you also slightly contridicted yourself - The 3 Magic institutions mentioned in Goblet of Fire are Schools, and there are 48 National Teams. That's like being confused why there are so few members of the New England Patriots are from the University of Miami. It's comparing Apple to the US Government.

EDIT: JKR has now released the names of the other Schools (again, not nations). They are the Uagadou School of Magic (Africa), Mahoutokoro School of Magic (Japan), Koldovstoretz (Russia), Ilvermorny (North America), Castelobruxo (South America). She further indicated that Durmstrang Institute is in either Sweden or Norway (maybe on the border?) and services Northern & Western Europe whilst the Beauxbatons Academy covers Eastern Europe (everything east of the Western border of Belgium/Luxembourg/the Netherlands).

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    The Brazilian school is known because Bill (Weasley) had a pen-friend at a school in Brazil once. He wanted Bill to go on a student exchange trip, but Mum and Dad couldn't afford it. - Goblet of Fire – ssell May 22 '14 at 16:39
  • Not sexist, and not limited to the Wizarding world. There are plenty of gender specific non-magical boarding schools still in operation in the US. It also stands to reason that each European country has its own magical school. And given population densities, it is also likely that the US would have about half as many as the EU. – Dave Johnson May 22 '14 at 16:55
  • Rowling has said several times that there are eleven wizarding schools. – ibid Feb 2 '16 at 22:11
  • Yes, but not the names nor locations, which is why this post from 2014 was updated. – Vogie Feb 3 '16 at 13:32

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