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As with most labyrinths with conjoined borders, there are age-old methods of solution, such as simply placing one hand on a wall and proceeding through the maze without removing it until you come to either the centre or the exit (right-hand-rule). Yet IIRC Harry neglects to try this, let alone address any traditional maze-solving methods at all during the Third Task.

I'm sure the case is different for enchanted mazes, but I was wondering specifically whether this was actually addressed canonically at all, either by J.K. Rowling or the directors/screenwriters (books or movies).

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    Because it's a magic maze. It moves around :-) – Valorum Jun 2 '16 at 12:16
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    He used a spell to have his wand point toward the center. I assume he considered that superior to any Muggle methods, if he even knew such methods existed. – Paul L Jun 2 '16 at 12:18
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    The right hand rule only works with a static, perfect maze. Which this one surely was not – Kalissar Jun 2 '16 at 12:49
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    Because he hadnt heard of it? It's not like Harry sat there reading puzzle books or Stackexchange. – Shantnu Tiwari Jun 2 '16 at 14:15
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    Because the right hand rule doesn't work for every maze, even if it's static and non-magical. – DJClayworth Jun 2 '16 at 15:17
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Not addressed anywhere in canon, as far as I'm aware. With that in mind, let's proceed with speculation.

Even if we ignore the fact that the structure of the maze may potentially be changing (because it's magic), there are several factors that would cause Harry not to use a maze solving algorithm:

  1. He's probably not aware of them. I can't imagine Harry did a lot of maze solving prior to coming to Hogwarts because the Dursleys aren't the kind of people who take him places. He's also not portrayed as the kind of person who would study maze solving algorithms for fun.

  2. The wizarding world - including Harry by his fourth year - have a tendency to prefer magical solutions over more mundane ones, to the point where they will completely overlook potentially better solutions that don't involve magic.

  3. The Triwizard Task was a race. A maze solving algorithm may eventually lead Harry to his goal, but eventually doesn't necessarily cut it when you have to be first.

  4. The Task wasn't designed to primarily test their ability to solve mazes. The primary purpose was on testing the Champions' ability to react to different situations and solve different types of problems. The maze aspect was ultimately secondary.

  5. Harry knew he needed to get to the centre of the maze, and therefore prepared a spell that would keep him on the right track (supposedly - I have doubts about whether what's described in the books actually makes any sense). Being constantly pointed in the direction you need to go is possibly more efficient than a maze solving algorithm.

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    #2 definitely. The wizarding world has huge "not invented here" issues. – Donald.McLean Jun 2 '16 at 14:13
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    The right-hand-rule works only for a very specific type of maze and any maze builder with an ounce of guile will make sure it doesn't work. Then there's the fact that it's magic... – Jim Garrison Jun 2 '16 at 15:54
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    Put the goal/exit in the middle inside an island of walls unconnected to the outer wall. – Jim Garrison Jun 2 '16 at 17:13
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    @Kyle: well, it works for two-dimensional mazes in which the goal is to get from one point on the boundary to another point on the boundary (or same, I suppose, but that's trivial). "Any maze where the goal is to get out" might be misinterpreted to include mazes where you start in the middle, in which case the rule doesn't necessarily work. – Steve Jessop Jun 2 '16 at 17:26
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    Being constantly pointed towards the end point is the basis of A* path finding algorithm. – ewanm89 Jun 2 '16 at 22:50
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First of all, maybe Harry just ignores these "scientific" methods. However, the participants had ample time to prepare for the final task, and I find it difficult to believe that Hermione did not know of these strategies:

Finally, in the last week of May, Professor McGonagall held him back in Transfiguration.

"You are to go down to the Quidditch pitch tonight at nine o'clock, Potter," she told him. "Mr Bagman will be there to tell the champions about the third task." [...]

"That's right!" said Bagman. "A maze. The third task's really very straightforward. The Triwizard Cup will be placed in the centre of the maze. The first champion to touch it will receive full marks." [...]

"There will be ostacles," said Bagman happily, bouncing on the balls of this feet. "Hagrid is providing a number of creatures ... then there will be spells that must be broken ... al that sort of thing, you know. [...]"

[Chapter 28 - The Madness of Mr Crouch]

Harry's nerves mounted as June the 24th drew closer, but they were not as bad as those he had had before the first and second tasks. For one thing, he was confident that, this time, he had done everything in his power to prepare for the task.

[Chapter 31 - The Third Task]

Then, the right-hand strategy assures that you either find the exit or return to the start, which is surely not a desirable outcome, especially as there are other contestants.

maze example

In this example using a right-hand or left-hand approach always gets you back to the entrance (and I guess the task maze would be more complicated!)

Even supposing we are in a situation where the right-hand rule would make you find the exit, the path could be extremely long and convoluted, again a disadvantage in a competition; a longer route, besides taking more time, also means a higher chance of bad encounters along the way.

a different maze example

In this example the cup is eventually reached using a right-hand approach, but only after having explored the dead end on the right side; turning left and then right makes for a much shorter path.

And I think this is the main argument against using a fixed strategy: the magic encounters. The right-hand rule can work only if you are always free to choose your direction at each fork, which is not the case when magical creatures are involved! When Harry meets a Skrewt he is forced to make a U-turn:

Then he rounded another corner, and found himself facing a Blast-Ended Skrewt. [...]

The Skrewt was inches from him when it froze - he had managed to hit it on its fleshy, shel-less underside. Panting, Harry pushed himself away from it and ran, hard, in the opposite direction [...].

And again, upon meeting the sphinx Harry is given a chance to pass her (which he is told it's the quickest way) only if he guesses the riddle:

If the riddle was too hard, he could keep silent, go away from her unharmed, and try and find an alternative route to the centre.

If you cannot always apply the right-hand rule, there is no point in applying it at all.

Finally, as user @damien-lavizzo already said, a scientific method could not be the right choice when magic is involved. However, re-reading Chapter 31 - The Third Task I could not find references to the paths shifting and changing, so that could just be a movie thing.

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    The point that the right-hand-rule doesn't apply to mazes where the goal is in the center of the maze (which is the case for the Triwizard maze) is really the fundamental point here. – Kyle Strand Jun 2 '16 at 16:52
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    @KyleStrand not necessarily: see the added image and explanation, where the cup is in the same position, but the walls are differently connected. – lfurini Jun 2 '16 at 17:56
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    @KyleStrand, the right hand rule applies unless it's an island maze or 3d maze. The solution to Hampton Court Palace maze, the classic right hand case, is actually to turn left first then it's almost a simple right hand rule to the centre (one deviation). Even though it has islands they're not significant to the problem. – Separatrix Jun 3 '16 at 19:09
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    @Separatrix "it's still the first thing you try on going into a new maze." Sure, but I'm not a wizard with a pathfinding spell running and a time limit. – Timbo Jun 3 '16 at 23:43
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    re: changing maze: Regardless of whether the maze actually changed (in the books), assuming that it wouldn't is unwise. When preparing ahead of time, Hermione probably rejected wall-following strategies for that reason. (And I completely agree that if anyone would be looking into maze-solving techniques, it would be Hermione on Harry's behalf.) – Peter Cordes Jun 5 '16 at 2:19
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The answer is extremely simple. It's part of lfurini's answer, but it's not mentioned until halfway through, so I fear people are losing sight of the core of the explanation:

The right-hand rule is not a guaranteed solution to this type of maze.

The right-hand rule ensures that if the participant starts at the edge of the maze, they will see every point along the outer boundary of the maze before (1) finding another exit or (2) returning to their starting point.* In the case of the Triwizard maze, the goal (the Goblet) was not along the outer boundary--it was in the center.

This has nothing to do with Harry's personality, the magical properties of the maze, etc.

* Right-hand rule explanation rephrased as per SteveJessop's comment on another answer.

  • I think you are wrong: I added a slightly different example with the goal in the same position: an additional wall (connecting the outer wall with the inner one) is enough to make the goal reachable using the right hand rule. – lfurini Jun 2 '16 at 18:01
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    @lfurini My statement is not meant to imply that the right-hand-rule never causes the participant to reach a goal that's not at the edge of the maze. It simply doesn't provide any guarantees, which makes it pointless. – Kyle Strand Jun 2 '16 at 18:02
  • You could avoid outer wall shenanigans easily by heading straight towards the nearest interior wall and applying the right-hand rule there, as that's the better guess for the "boundary" anyway. Conceivably there could be a series of disconnected walls within the maze, though it doesn't sound terribly hard to notice and algorithmically adjust to loops (basically, when found, head towards a different/more interior wall and resume right-handing it; Harry's little wand trick is convenient for identifying which wall is "more interior" if at all confused). – zibadawa timmy Jun 3 '16 at 1:12
  • @zibadawatimmy That's not really an algorithmic solution any more (let alone "the right-hand rule"), and it's still not guaranteed to work. – Kyle Strand Jun 3 '16 at 15:28
  • @KyleStrand Yeah, it took me a bit to realize that's basically just the Pledge algorithm and that there are scenarios where that fails to get you to the goal. – zibadawa timmy Jun 3 '16 at 20:22
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Because all the simple methods would probably fail.

There are many maze-solving algorithms, check this wikipedia page: https://en.wikipedia.org/wiki/Maze_solving_algorithm Now the problems with these ones on are either the necessary time (remember, this is a race) or the lack of information in Harry's case:

  • Wall following a.k.a. Right-hand-method

    As the others already mentioned, the Right-hand rule doesn't always work. I would add something to this: the rule provides a sure method if, and only if all the walls are connected to the maze's boundary, thus there are no loops; or on a maze with loops if both the entrance and the exit is on the same piece of wall. (See @lfurini's answer.)

    Harry could not know if this is the case.

  • Random mouse method

    This is the most basic one, but the drawback is obivious. The method is randomly choosing direction at every junction, and turning back at dead-ends. After an ernomous amount of time, this will with high probability lead to the exit.

    Harry would still be in that maze...

  • Pledge algorithm

    The algorithm is an enchanced wall following with a compass: you choose a direction as zero angle, and at each turn you add the (signed) turned angle to a sum. If the sum and the current direction are both zero, leave the wall and head forward to another wall. The problem is that the method, like the wall follower one, only works if the exit is on the outer wall of the maze. (The entrance could be anywhere though.)

    Again, this one requires information Harry didn't have.

  • Trémaux's algorithm

    From Wikipedia:

    This method requires drawing lines on the floor to mark a path, and is guaranteed to work for all mazes that have well-defined passages. A path is either unvisited, marked once or marked twice. Every time a direction is chosen it is marked by drawing a line on the floor (from junction to junction). In the beginning a random direction is chosen (if there is more than one). On arriving at a junction that has not been visited before (no other marks), pick a random direction that is not marked (and mark the path). When arriving at a marked junction and if your current path is marked only once then turn around and walk back (and mark the path a second time). If this is not the case, pick the direction with the fewest marks (and mark it, as always). When you finally reach the solution, paths marked exactly once will indicate a direct way back to the start. If there is no exit, this method will take you back to the start where all paths are marked twice. In this case each path is walked down exactly twice, once in each direction.

    This may be the only method that could work for sure. Drawing on the floor, however, is not so pleasable during a race... And considering the possible magical propertities of the maze, even this one could fail.

The other two methods, the Recursive and the Maze-routing are too complicated to expect Harry using them.

  • Nice analysis of the various algorithmic options available to Harry, but note that the goal of the Triwizard maze is not to escape but to find the Goblet. E.g. you mention the "exit" in your paragraph on the right-hand rule as if that were the goal. – Kyle Strand Jun 3 '16 at 15:33
  • @KyleStrand You are right, the world 'exit' is not the one that suits the best... but effectively the exit and the Goblet is identical: the exit is where the Goblet is. I focused only on the theory when writing the answer... – Neinstein Jun 3 '16 at 18:06
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    You're incorrect on your first point, it's not that all walls need to be connected, but that a wall associated with your target destination must be connected. Look at Hampton Court Maze as an example, you'll see two walls are unconnected but the right hand rule still applies. – Separatrix Jun 3 '16 at 19:14
  • @Separatrix You are right. I guess what I thought of was that it works for an arbitary maze for sure if it is simply connected, but not always if there are loops; but this is not what I wrote... so let's fix this. – Neinstein Jun 3 '16 at 20:27
  • Urm, Pledge algorithm may in fact be what Harry was using, only his compass didn't point to an outer wall (this is why pledge algorithm only works for outer wall exits). But to his target, the maze center, so he was using a different pledge depending on which side of the maze he was on. – ewanm89 Jun 6 '16 at 9:43
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I don't think it was addressed canonically in the books, but in the film the maze is nearly immediately shown to shift and change right from the get-go. Using the "right hand rule" largely wouldn't make sense when the maze is demonstrating that it changes itself nearly constantly.

2

Hardly any information about the maze was available. Even if it didn't actually shift on the fly, there was no way for Harry to know that. Or to know whether it was a type of maze that was solvable by any wall-following algorithm, let alone quickly.

As @lfurini's answer points out, Hermione was almost certainly aware of maze-solving strategies (either already or from research). During the ample time between the maze competition and when they first learned that it would be a maze, she probably at least mentioned the wall-following idea to Harry, in case things got to the point where he was lost.

She probably didn't take up much of Harry's time with more than the basics, because it's not a safe assumption that it would work at all. And remember, they have to plan according to what they know, not what we as readers learn about the maze during/after the competition!


I think it's a safe bet that Harry was at least somewhat prepared for maze-solving as its own problem, but it turns out that wasn't needed, so the book didn't mention it at all. (Not exactly riveting story material.) Or maybe he was mentally tracking a wall at least at first (but not planning to depend on that strategy), but again, not mentioned in the text.

Harry had magic to point him towards the centre, but that's not the same thing as pointing him along the shortest path to the centre. There could be dead-ends.


I suspect that we're giving maze-solving more thought than JKR did. While I enjoyed the books, logic was not always a high priority. (e.g. extremely powerful time travel exists in the world, but only gets used in one book, not all the time by the bad guys.) My impression was to enjoy the characters and storytelling, and don't think to hard about how the world works.

0

If I remember correctly, the maze, being enchanted kept changing. Walls would merge and be created while the champions were in the maze. So any standard rule would fail.

Also, time was of the essence. The triwizard champion would be the person who was first to find the cup. Hence a general direction to go in would be much more helpful than a heuristic rule which doesn't optimize the time taken, even if it is able to find a solution.

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    This doesn't seem to add anything to the existing answers. – Meat Trademark Jun 3 '16 at 16:51

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