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In Harry Potter and the Goblet of Fire, Harry finds out several weeks before the second task of the Triwizard Tournament that he would be required to survive underwater for an hour.

Harry, Hermione, and Ron all proceed to spend hours and hours researching a solution to this problem, even engaging the librarian's help:

So Harry, thinking that he would soon have had enough of the library to last him a lifetime, buried himself once more among the dusty volumes, looking for any spell that might enable a human to survive without oxygen. However, though he, Ron and Hermione searched through their lunchtimes, evening and whole weekends -- though Harry asked Professor McGonagall for a note of permission to use the Restricted Section, and even asked the irritable, vulture-like librarian, Madam Pince, for help -- they found nothing whatsoever that would enable Harry to spend an hour underwater and live to tell the tale.

Goblet of Fire -- Bloomsbury -- chapter 26, The Second Task

Hermione, in particular, has a charm in mind as the most probable solution:

"I think your best chance is some sort of charm."

Hermione -- Goblet of Fire -- Bloomsbury -- chapter 26, The Second Task

Despite many hours spent searching, and despite Hermione's formidable researching skills, they are unable to find any solution.

However, in the second task itself, both Cedric Diggory and Fleur Delacour use a Bubble-Head Charm to survive underwater:

Miss Fleur Delacour, though she demonstrated excellent use of the Bubble-Head Charm, was attacked by Grindylows as she approach her goal, and failed to retrieve her hostage.

...

Mr Cedric Diggory, who also used the Bubble-Head Charm, was first to return with his hostage, though he returned one minute outside the time limit of an hour.

Goblet of Fire -- Bloomsbury -- chapter 26, The Second Task

The Bubble-Head Charm doesn't appear to be particularly difficult to perform, as it is in widespread usage among students by Harry's fifth year:

Dungbombs and Stink Pellets were dropped so frequently in the corridors that it became the new fashion for students to perform Bubble-Head Charms on themselves before leaving lessons, which ensured them a supply of fresh air, even though it gave them all the peculiar appearance of wearing upside-down goldfish bowls on their heads.

Order of the Phoenix -- Bloomsbury -- chapter 30, Grawp

Why did Harry, Hermione, and Ron all fail to discover the Bubble-Head Charm in advance of the second task, despite spending weeks specifically researching a charm to survive underwater, and despite both Cedric and Fleur learning that charm?

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    It became the fashion after they'd seen it used at the Tri-Wizard cup. – Valorum Mar 15 '17 at 12:56
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    It's called a plothole. Rowling is a very good author but sometimes makes some mistakes, as every author out there. It's extremely hard to write so many books in such short time without messing up one time or another. – T. Sar Mar 15 '17 at 20:40
  • I just had the weirdest flashback, I could swear I've read this question AND its answers before... – Weckar E. Mar 16 '17 at 8:45
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    @TSar - no kidding. Barty drops a book on Longbottom then when that doesn't work arranges to discuss gillyweed with McGonagall in such a way that Dobby can overhear, then counts on Dobby to steal the stuff from Snape's stores to give to Harry, knowing, somehow (was it mentioned in the book?) that Dobby would do anything (like steal from Snape) to help Harry. Or ... he could just surreptitiously set off a dungbomb in a corridor and then "conveniently" show up and teach some students the Bubble-Head Charm, knowing that it would obviously go viral. Not so much a hole as a contrivance. – davidbak Mar 16 '17 at 9:35
  • @T.Sar Barty Jr actually explains how proud Harry was; his biggest fear was that Harry was too proud to ask for help. That's why he had to stage what led to Dobby suggesting to Harry the gillyweed. Not at all a plot hole. It's called it's more complicated than that. I believe also it's explained that Krum - who was a lot more advanced, remember - didn't quite do it right (or was that the conjunctivitis on the dragon?). So who's to say that Harry would have managed it in the first place. – Pryftan May 19 '18 at 1:17
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The correct answer is, We don't know, unless Rowling confirms. However, there are a number of plausible explanations for this.

My personal explanation for this was that the Charm was not intended to be used for underwater purposes but rather a sort of comforting purpose like filtering out smells, toxins in the air only.

Breathing underwater maybe an explanation that experienced wizards such as Karkaroff or Madame Maxime may have been thought of, but it was not possible for inexperienced kids such as our protagonist trio to think of. I feel that there are a number of instances where Hermione fails to apply her knowledge in practical purposes. The first instance I think of is the one with Fire and Devil's snare.

Additionally, Wikia states (without reference)

It's unclear why Harry Potter was unable to find information on this spell while researching for the second task. It is possible that the spell was a relatively new invention not covered in many books as of 1995, or that the spell was not commonly used for breathing underwater, but for other purposes.
Source: Bubble-Head Charms

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    Another alternative (speculation only) is that the text says they were looking for ways to survive without oxygen. As less experienced students, they may have been blinded by their first idea to the alternative idea of his bringing oxygen with him. – Paul Sinclair Mar 16 '17 at 0:16
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    @PaulSinclair You should make that an answer - basically it's a typical case of the infamous XY problem – Zommuter Mar 16 '17 at 6:53
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    @PaulSinclair right, though this seems like something two wizards raised by muggles and thus aware of scuba equipment should have been more likely to come up with than Fleur or Cedric. – leftaroundabout Mar 16 '17 at 11:31
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    @leftaroundabout - actually, I find it quite common that when students learn some new tool, they often get so wrapped up in it they don't think of the tools they already know. Over on Math SE, questions equivalent to "how do you use calculus to show 1 + 1 = 2" are a staple. Even when told that is not needed, there are some who are so stuck on the idea, they still seek the "calculus approach". – Paul Sinclair Mar 16 '17 at 23:56
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    @delinear So..you think Harry having a compass in a maze isn't 'breaking the rules'? I don't think it is and I don't think Hermione would either. Just like 'taking his own oxygen supply' wouldn't in her mind be in that category. Besides that: he was wrongly placed into the tournament, he had to face a dragon and now he had to be underwater for some time - and worse to come. You actually think she would care more for 'rules' than the safety of her friend - esp when said friend is at great risk and clearly something was going on? – Pryftan May 19 '18 at 22:24
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I think the Trio were aware of the Bubble-Head charm, but due to their inexperience; Harry probably couldn't keep it active for a full hour. He very well could have produced a Bubble-Head charm, but perhaps only lasted like 15 min or so. I think to be able to produce that charm for a full hour took real inquisitive skill that Cedric and Fleur possessed. Since he didn't have too long to hone his skill in producing that charm, I believe all three dismissed it relatively fast, and were looking for something else.

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    A reasonable theory. Can you provide any evidence? – Blackwood Jun 12 '17 at 3:38
  • @Blackwood Although not directly stated (should everything be? That'd make a very long, very distracted story at the very least) it is known that both Cedric and Fleur used it and they were more advanced. Does this also count as human transfiguration? If so that's another reason - Minerva did say very clearly in the beginning that it's a very difficult art and you're not supposed to mess around with it as it's dangerous (or some such). – Pryftan May 20 '18 at 17:40
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    If this was correct would it not have said that they found a way but it was too difficult, rather than they couldn't find a way at all? – Alex Dec 25 '18 at 6:54

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