The Question

The actual question is as follows: The scheduling system for elective classes at Hogwarts, as presented in the books (particularly Prisoner of Azkaban) does not seem to make sense. I am seeking an explanation as to how it can actually make sense. The rest of this post will consist of my attempt to demonstrate that it doesn't make sense. Be warned: it will be long, tedious, and complex. If you don't care for that, feel free to stop reading now.

The Facts

There are several facts mentioned in the books that the analysis will be based on.

  • Beginning in Third Year Hogwarts has electives.

    From Chapter Fourteen of Chamber of Secrets:

    The second years were given something new to think about during their Easter holidays. The time had come to choose their subjects for the third year, a matter that Hermione, at least, took very seriously.

  • You can't drop any of the core classes.

    From Chapter Fourteen of Chamber of Secrets:

    "I just want to give up Potions," said Harry.

    "We can't," said Ron gloomily. "We keep all our old subjects, or I'd've ditched Defense Against the Dark Arts.

  • Hermione signed up for every possible elective.

    From Chapter Fourteen of Chamber of Secrets:

    Hermione took nobody's advice but signed up for everything.

  • Hermione took five elective classes.

    • Ancient Runes
    • Arithmancy
    • Care of Magical Creatures
    • Divination
    • Muggle Studies

    From Chapter Four of Prisoner of Azkaban:

    "Well, I'm taking more new subjects than you, aren't I?" said Hermione. "Those are my books for Arithmancy, Care of Magical Creatures, Divination, Study of Ancient Runes, Muggle Studies –"

  • Harry (and Ron) took two electives, and we can assume that was standard (or at least the minimum requirement).

    From Chapter Four of Prisoner of Azkaban:

    Most important of all, he had to buy his new schoolbooks, which would include those for his two new subjects, Care of Magical Creatures and Divination.

  • Beginning in Sixth Year students get free periods.

    From Chapter Nine of Half-Blood Prince:

    "I love being a sixth year. And we're going to be getting free time this year. Whole periods when we can just sit up here and relax.

The Analysis Part 1

Based on the above mentioned facts we can reach the following conclusions: If Hermione took all the electives, and she took five electives, then the total number of electives is five. If regular students choose two out of the five electives there would be a total of ten possible combinations. If there are no free periods and everyone has to take the core classes then that means that everyone has to be taking electives during the same period(s).1 I.e. no elective can be simultaneous with a core class (because then students in that elective would be missing a core class), and no elective can be simultaneous with a free period (since free periods don't exist until Sixth Year).

Now we can begin to see what the problem will be. During every period that an elective is given, every student must be present in an elective class. That leaves us with five possibilities to examine:

  • Only one elective is given at a time.
  • Two electives are given at a time.
  • Three electives are given at a time.
  • Four electives are given at a time.
  • Five electives are given at a time.

For simplicity's sake let's assume that in a particular year there would be equal distribution among the ten combinations of classes, i.e. ten percent of students are doing each combination.

  • We can eliminate the first possibility right off the bat. If only one elective was given during a period then 60% of the students would have to either be in a core class or in a free period, both of which are impossibilities (as per above).

  • We can also immediately eliminate the last possibility. If all five electives are given at the same time then no one would be able to take more than one elective.

So what about the other three possibilities?

  • If there are exactly two electives at a time then what would the 30% of students taking neither of them be doing? Again, they can't be in a core class and they can't be in a free period. Moreover, what about the 10% of students who would have signed up for both of those classes? They wouldn't be able to take both classes if the classes are at the same time.

  • How about if there are four electives at a time? At least in this situation every student would be in an elective class during that period (as everyone has to be taking at least one of the four classes). However, that would mean that the fifth class would have to be given at a different time. But only 40% of students would be taking that fifth class, so what would everyone else be doing? They can't be at a core class or a free period (as per above), and they can't be at an elective class because the other four electives were already accounted for. Thus, this option doesn't work either.

  • That leaves us with three electives at a time. Yet this also doesn't work out. Once again there will be 10% of students with nothing to do during that period. It would also mean that there would have to be a different period when the other two electives are given, but we already rejected that option above.

Thus, apparently none of the five options for arranging the elective classes are possible.

The Analysis Part 2

The above analysis was all theoretical. That is to say that without knowing which specific classes were at the same time as each other we eliminated every possibility. But in fact we do have some information about specific simultaneous classes, with which we can see that in actual practice it also doesn't work out. There are several passages in Prisoner of Azkaban where we are explicitly told about some overlaps.

In Chapter Six we have:

"But look," said Ron laughing, "see this morning? Nine o'clock, Divination. And underneath, nine o'clock, Muggle Studies. And" – Ron leaned closer to the schedule, disbelieving – "look – underneath that, Arithmancy, nine o'clock. I mean, I know you're good, Hermione, but no one's that good. How're you supposed to be in three classes at once.

In Chapter Twelve we have:

"Getting to all her classes!" Ron said. "I heard her talking to Professor Vector, that Arithmancy witch, this morning. They were going on about yesterday's lesson, but Hermione can't've been there, because she was with us in Care of Magical Creatures! And Ernie McMillan told me she's never missed a Muggle Studies class, but half of them are at the same time as Divination, and she's never missed one of them either!"

From Chapter Sixteen:

Harry and Ron had given up asking her how she was managing to attend several classes at once, but they couldn't restrain themselves when they saw the exam schedule she had drawn up for herself. The first column read:


9 o'clock, Arithmancy

9 o'clock, Transfiguration


1 o'clock, Charms

1 o'clock, Ancient Runes

Also from Chapter Sixteen:

Harry's and Ron's last exam was Divination; Hermione's, Muggle Studies. They walked up the marble staircase together; Hermione left them on the first floor and Harry and Ron proceeded all the way up to the seventh, where many of their class were sitting on the spiral staircase to Professor Trelawney's classroom, trying to cram in a bit of last-minute studying.

We learn several things from these passages:

  1. Arithmancy, Divination, and Muggle Studies were all at the same time. (First passage)

  2. Arithmancy was at the same time as Care of Magical Creatures. (Second passage)

  3. Half of the Muggle studies classes are at the same time as Divination. (Second passage)

  4. The Arithmancy exam was at the same time as the Transfiguration exam. (Third passage)

  5. The Ancient Runes exam was at the same time as the Charms exam. (Third passage)

  6. The Muggle Studies exam was at the same time as the Divination exam. (Fourth passage)

The most obvious problem is with Facts 4 and 5. In both of these cases there is an elective exam at the same time as a core class exam. This was not a problem for Hermione who was taking those electives, as she had a Time-Turner. But the other students taking those electives did not have Time-Turners, and thus would be unable to attend two exams simultaneously.

Before going on it might be useful to document the ten possible combinations of electives:

  1. Ancient Runes and Arithmancy
  2. Ancient Runes and Care of Magical Creatures
  3. Ancient Runes and Divination
  4. Ancient Runes and Muggle Studies
  5. Arithmancy and Care of Magical Creatures
  6. Arithmancy and Divination
  7. Arithmancy and Muggle Studies
  8. Care of Magical Creatures and Divination
  9. Care of Magical Creatures and Muggle Studies
  10. Divination and Muggle Studies

Now just from Fact 1 we can eliminate several combinations. If Arithmancy, Divination, and Muggle studies are all given at the same time then Combinations 6,7, and 10 cannot exist. So now there are only seven combinations left to choose from.

Fact 2 which has Arithmancy occurring at the same time as Care of Magical Creatures eliminates Combination 5. Now there are only six combinations left.

Here it gets a little tricky. If we put Fact 1 and Fact 2 together, what happens? On the face of it if Divination and Muggle Studies are at the same time as Arithmancy, and Care of Magical Creatures is at the same time as Arithmancy, then Care of Magical Creatures would also be at the same time as Divination and Muggle Studies. This would eliminate Combinations 8 and 9 as well, bringing us down to only four viable combinations. Moreover, these four combinations are the Combinations 1,2,3, and 4, all of which contain Ancient Runes. That would mean that every student has to take Ancient Runes, which would be both ridiculous (as it would then not be an elective) and demonstrably false (Harry does not take Ancient Runes).

However, it is possible that the period in which Care of Magical Creatures is at the same time as Arithmancy is not the same as the period in which Arithmancy is at the same time as Divination and Muggle Studies. I.e. the schedules might be variable, where Arithmancy sometimes coincides with Divination and Muggle studies and sometimes coincides with Care of Magical Creatures. Indeed this would be supported by Fact 3 which states that half of the Muggle studies classes coincided with Divination, implying that the other half didn't – i.e. the schedule is variable.

However, this doesn't even help. When Muggle Studies doesn't coincide with Divination, something else would have to. If we pick Care of Magical Creatures as the coincider we eliminate Combination 8 anyway (which incidentally is impossible, as that was Harry's actual combination). If we pick Arithmancy as the coincider then where are the Muggle Studies students during that time? They can't be at Arithmancy because Arithmancy coincides with Muggle Studies during other periods. If they are at Ancient Runes, then we are now saying that Ancient Runes sometimes coincides with Divination and Arithmancy, in which case we have now eliminated Combinations 1 and 2 as well (in which case we are still down to only four viable combinations). If we say that only Ancient Runes (and not Arithmancy) coincides with Divination when Muggle Studies doesn't then that means that there would be one period with Ancient Runes and Divination and another period with Arithmancy, Divination and Muggle Studies. That leaves nothing for Care of Magical Creatures, so we would have to then assume that there was a third period with Care of Magical Creatures and Arithmancy (Fact 2). However, that would eliminate combinations 3 and 4 unless we also assume that Ancient Runes is given simultaneously. But we still have a problem because there would be no time when there is a Muggle Studies class without a Divination class (Fact 3). So we would have to add in Muggle Studies to this period, leaving us with three different periods:

  1. Arithmancy, Divination, Muggle Studies2
  2. Ancient Runes, Divination
  3. Ancient Runes, Arithmancy, Care of Magical Creatures

But this also doesn't make sense because Arithmancy is then impossible to take, because between Period 1 and Period 3 it coincides with both classes offered during Period 2. And there isn't any other class we can add to Period 2. If we add Arithmancy then anyone taking Arithmancy would only be taking one class. If we add Muggle Studies then all the Muggle Studies classes would coincide with Divination (which contradicts Fact 3), plus the students would have to be in two classes at once during Period 1. If we add Care of Magical Creatures then the students would have be in two classes at once during Period 3. Also, this schedule would mean that the only two viable combinations are Combinations 4 and 8.

In order to save Arithmancy we would have to add Arithmancy to Period 2 and posit a fourth variable period. But that too is impossible since Arithmancy would already coincide with every other class during Periods 1, 2, and 3.

As you can see, the system seems untenable.

Hermione's Solution

One last problematic point relates to Hermione's schedule specifically. During Prisoner of Azkaban she was able to take every class because she had a Time-Turner. However, in the last chapter of Prisoner of Azkaban we have the following statement:

"I know," sighed Hermione, "but I can't stand another year like this one. That Time-Turner, it was driving me mad. I've handed it in. Without Muggle Studies and Divination, I'll be able to have a normal schedule again."

Yet even if she dropped dropping Muggle Studies and Divination she would still be taking Arithmancy and Care of Magical Creatures. and as per Fact 2 above, those classes coincided. Thus, Hermione would still have to be in two classes at once, and without a Time-Turner.


Given the premises mentioned above and the analysis thereon, there does not seem to be any way to work out elective scheduling. Thus, to reiterate the question: Is there any way that we can actually make sense out of this scheduling system? If so, what is it?

Note that the above analysis is somewhat confusing, and it's certainly possible that I have confused myself. Thus, an answer can point out any mistakes I might have made in the analysis, or attempt to refute some of the premises, or come up with additional information that would help explain things.


1. For the purposes of this question we can focus just on Gryffindor House without introducing needless complications involving different houses taking core classes separately.

2. If we would add Care of Magical Creatures to Period 1 then we'd just be back to the third paragraph after the list of combinations.

  • 8
    I don't know about Hogwarts - we're not provided with that background information - but in real US schools that offer what we call "departmental" schedules (where students move from class to class, individually depending on their courses), most courses are offered in multiple "sections" - that is, several different groups of students take the course at different times. Sometimes, but not always, different sections of the same course may be taught by different instructors. Commented May 3, 2019 at 18:25
  • 1
    I'm going to presume you've never had to sign up for classes at a US university. This is how the real world actually works. :(
    – KutuluMike
    Commented May 4, 2019 at 1:09
  • 3
    To me your question is clear, so instead of an answer: You have discovered that this doesn't make sense. Other comments mentioned that real schools teach the same elective at different times. While true that doesn't make sense for a small school like Hogwarts. The whole elective topic is just an excuse to explain the time turner that is later discovered as a Deus ex Machina to save Sirius.
    – user102803
    Commented May 4, 2019 at 6:03
  • 4
    Re-posting what's essentially the same question (with formatting changes but without altering the thrust of the question) in order to avoid downvotes/close-votes is heavily frowned upon; instead you should edit the original post and hope the votes turn around. I've edited and undeleted this post, and cast the 5th vote to reopen. I also merged across the comments and answer from the other question, so that nothing is lost.
    – Rand al'Thor
    Commented May 6, 2019 at 12:09
  • 3
    "taking those electives did not have Time-Turners, and thus would be unable to attend two exams simultaneously." When I was at school in the UK doing O levels such exam clashes did sometimes occur, one happened to me. Pupils were put into isolation under the supervision of a teacher even overnight if need be without access to radio. tv etc. until they could take the exam themselves.
    – Sarriesfan
    Commented May 9, 2019 at 6:01

3 Answers 3


I don’t think you will find it.

Remember, we are talking about a fiction. It seems clear enough to me that JKR did not take time, right from the start, to design a fully functional and accurate schooling system with a real schedule of classes that were to be strictly followed in her novels.


Because it has little relevance to the actual story.

That is to say, perception suffices. In Real Life™, we all go to school and Things Just Work™. This is only because a lot of people behind the scenes, all the way from the city council down to a school’s administrative governors, spend thousands of hours on the curriculum, which is, frankly, very tedious and boring. We normal people, experiencing it as students or outside observers, simply do not see the massive effort.

Which is exactly the point: all this administrative stuff in the background is not relevant to the story any more than the author explicitly gives it. As far as it matters, we can say that Hermione was following an impossible schedule, were it not for the Time Turner, and continue on as if that were true.

When you try too hard to look past the Suspension of Disbelief, you will invariably find it shattering. To quote a master of fiction, “Never let the truth get in the way of a good story.” JKR wisely spent her efforts crafting a good story instead of implementing a practical school system.

Now, to be clear, I very much like the Harry Potter series, and there are a few things in both the books and the movies that mess with my own suspension of disbelief. And, honestly, I have freeze-framed through quite a few movies and page-flipped back chapters (and books) myself. It’s fun!

But you should not let that destroy the immersive fun of the story itself.

Just pretend that there is more going on than Harry perceives, and that it works, and enjoy it.

  • 1
    Normally I would agree that the details of the scheduling are not relevant, but in this case it is the justification for Hermione having a time turner, which again is an important point at the end of the story.
    – user102803
    Commented May 6, 2019 at 17:06
  • 4
    The only relevance in the scheduling is that Hermione is observed to be doing something impossible with her schedule. A concrete exposition of the schedule is not.
    – Dúthomhas
    Commented May 6, 2019 at 17:10
  • 1
    I agree with Dúthomhas that Rowling did not have to craft a consistent background in order to create a convincing and enjoyable story. If the story hadn't been convincing on the surface, Alex probably wouldn't have bothered to write this impressive analysis. What we can carry away with us today is a bit of treasure: We now know a little more about how Rowling wrote, and what she didn't think was important. I rather suspect that she might have made Hermione's schedule inconsistent on purpose, simply to add to the hectic quality of the term. Commented May 6, 2019 at 18:58
  • 1
    @Dúthomhas, you're right about being careful not to destroy the fun of the story. However, it's also fun for many people including myself to look at the details and try to build a background story not contradicting to snippets from original art. Alex made a great and deep analysis, I can't wait for someone to come with an 'in-universe' solution! Commented May 7, 2019 at 17:40
  • 1
    @Demosthenes Clearly the teachers are using Time-Turners to teach multiple classes at the same time. (This isn't a serious suggestion, it would be spotted by the students far too easily to be practical.) Commented May 8, 2019 at 13:40

Electives aren't a problem

Let's assume for a moment that there are 3 slots in the timetable for Electives for each year. We will ignore how many Periods of classes these occupy for now, and we will also exclude the "special" students, such as Bill Weasley / Percy Weasley / Hermione Granger, who get given a Time Turner for purposes of taking 12 O.W.Ls

Let us also assume, for tidiness, that there is a hidden "bonus" Elective called "Study Hall", and you must take 3 electives (Taking "Study Hall" is equivalent to having 2 electives, without Free Periods) You may recall Snape whacking Harry and Ron on the head in film-version of The Goblet of Fire: students all busy studying in the Great Hall, in the middle of the day, while dressed in their School Robes. Sure seems like "Study Hall" to me!

We will label our Electives as A, B, C, D, E and X ("Study Hall")

There are then 20 different combinations of ways to select 3 out of 6 items (nCr = n! / (r! * (n - r)!), so 6C3 = 6! / (3! * (6 - 3)!) = 720 / (6 * 6) = 20)

If we exclude assign 4 classes per slot, with 2 classes excluded (meaning that each teacher will be only teaching each year for 2 of the 3 slots), we can achieve the following table:


You will see that while some classes are locked to a specific slot, most can be swapped or reordered (e.g. if you are taking Electives A, B and C, then Elective A must be slot 3, but Electives B and C can be either way around. If you take A, B and E, then you can take either 1=A;2=B;3=E or 1=B;2=E;3=A)

This means an Elective Teacher teaches 2 slots for 3rd, 4th and 5th year (a total of 18, 24 or 30 periods per week, depending on if each slot is 3, 4 or 5 periods of classes), while a typical school will have somewhere between 30 and 40 periods per week.

As Hogwarts is a Boarding School, and they seem to move straight from Classes to Supper, I suspect that it works on a 40-slot basis (8 per day, plus 2 short breaks and Lunch?)

However, this means that each of the 7 core subjects would have to be 3 slots per week for each pair of Houses (so, 6 total) for the first 5 years (total 30 slots), plus 5 slots for each of N.E.W.T. years - which leaves huge gaps in the students' timetables!

(And, also explains why Flying Classes only seem to be available for the first 2 years)

Of course, the simplest solution to that would be JK Rowling simply choosing not to show the 'boring' classes such as English, Maths or Geography that are mandatory parts of the UK's National Curriculum

(After all, it's the Ministry of Magic, like the Ministry of Justice or the former Ministry of Education, not a separate Government: the Minister for Magic would be a member of the UK Cabinet - but apparently never deigns to turn up for meetings!)


I think that with a bit of speculation we can attempt to make sense out of this. First let's try to address it in theory, and then let's see if we can actually make it work out in practice. The premise that's really causing much of the problem is that the schedule has to be able to support combinations of electives, so let's try to dispense with that premise.

On the face of it, it seems like a valid premise. If there are five possible classes and students take two classes, then mathematically there are ten possible combinations. But that assumes that there are no factors at play other than statistical randomness. However, there may in fact be various factors at play which would alleviate the need for ten combinations.

Let's begin by considering the classes themselves. These are the five classes:

  • Ancient Runes
  • Arithmancy
  • Care of Magical Creatures
  • Divination
  • Muggle studies

But are all classes created equal? Presumably not. Some classes are harder than others, some classes are more interesting than others, and some classes are more useful than others. If we look at difficulty level, we can surmise that Ancient Runes and Arithmancy are significantly more difficult than the remaining three classes. In Chapter Thirty-One of Order of the Phoenix we have this statement:

“Only!” said Hermione snappishly. “I’ve got Arithmancy and it’s probably the toughest subject there is!”

In that same chapter we have the following:

“How were the runes?” said Ron, yawning and stretching.

“I mistranslated ‘ehwaz,’” said Hermione furiously. “It means ‘partnership,’ not ‘defense,’ I mixed it up with ‘eihwaz.’”

“Ah well,” said Ron lazily, “that’s only one mistake, isn’t it, you’ll still get —”

“Oh shut up,” said Hermione angrily, “it could be the one mistake that makes the difference between a pass and a fail.

Here Hermione admits to making a mistake in Ancient Runes, and is worried that she did not do well on the exam. And this is not just a fleeting thought that she had in the moment; in Chapter Five of Half-Blood Prince she reiterates this:

"I know I messed up Ancient Runes," muttered Hermione feverishly, "I definitely made at least one serious mistranslation.

It may thus be fair to conclude that Ancient Runes is particularly difficult, as Hermione doesn't normally make such mistakes.

With this in mind, if you are a student choosing your electives (especially if you're the type of student like Harry and Ron and probably most young teenagers) you might easily decide to stay far away from Ancient Runes and Arithmancy. Why take difficult classes when you can take easier classes? Especially classes like Ancient Runes and Arithmancy which don't seem particularly useful to the average wizard.

Muggle Studies, while it doesn't seem to be particularly difficult, is probably not very interesting to most wizards. (See all the adult wizards throughout the series who haven't the foggiest idea about anything relating to Muggles, and don't seem to care.) Moreover, unless you are looking for a career in Muggle Relations (Order of the Phoenix Chapter Twenty-Nine) it doesn't seem to be particularly useful either.

That leaves us with Care of Magical Creatures and Divination. Unless you specifically don't like animals, or you know that Hagrid is a nut, there doesn't seem to be much reason to not want to take Care of Magical Creatures. You don't have to be stuck in a classroom, you can spend half the lesson chatting with friends, and to top it all off I don't think we ever saw them have homework under Hagrid's tenure. Additionally, a working knowledge of magical creatures is probably something that is useful to many wizards, even if they don't specifically go into a field that deals with them.

Divination (at least as taught by Trelawney) is decidedly not useful. However, it's possible that naive students who don't know any better think that they will actually learn how to see the future, which sounds kind of awesome. Alternatively, students may have heard from older veterans that Trelawney is "an old fraud", and that all you have to do is make up deadly predictions, thus making it a relatively easy class.

Additionally, students probably also take into account what their friends will be doing. If your friends are all taking one class and you don't have any strong preferences, you'd probably just follow them. Indeed this is what Harry did (Chapter Fourteen of Chamber of Secrets):

In the end, he chose the same new subjects as Ron, feeling that if he was lousy at them, at least he'd have someone friendly to help him.

(At this point it is not really important that we picked Care of Magical Creatures and Divination as the two most likely classes that students would want to take. For the sake of developing the theory we could pick any classes; Care of Magical Creatures and Divination just seem like the easiest ones to argue for.)

With this in mind, let's return to the original premise we are trying to address. Sure, if the students are evenly distributed among the ten possible combinations then we run into the problems described in the question. But why should we assume that there is an even distribution? Maybe most or all of the students choose to take a particular class or classes, and a bunch of the combinations are not used at all!

With this suggestion we can easily resolve the main problem in theory. The problem was that no matter how you set up the classes, there will always be a conflict for at least some students with other classes that they are taking, or there would have to be free periods. Not so if not all the combinations are used. If all the students take a particular class then that class can be the only class offered during that period and everyone can attend. The other four classes can then be offered during another period, and every student should be able to pick one of them to attend.

If we don't want to go so far as to say that there is one class that every student took, we can still apply the same basic idea with a minor variation. We can divide the classes into two groups (one group of two classes and one group of three classes) and say that every student is taking one one class from each group. As long as there is no student taking two classes from the same group, the classes from each group can be offered simultaneously and every student should be able to attend.

Of course, if this is the case there has to be a way to arrange the schedule based on student preferences instead of fitting student preferences into a schedule. And indeed there is a way. Let us contrast the first day of classes in Half-Blood Prince with the first day of classes in Prisoner of Azkaban. In Half-Blood Prince (Chapter Nine) we find the following:

After they had eaten, they remained in their places, awaiting Professor McGonagall's descent from the staff table. The distribution of class schedules was more complicated than usual this year, for Professor McGonagall needed first to confirm that everybody had achieved the necessary O.W.L. grades to continue with their chosen N.E.W.T.s.

The students are approved one by one and sent off to their first class. This means that the schedule already exists before the students get approved for the classes they want to continue. This doesn't cause any issues because at this point in their education no student will be starting new classes. They are merely deciding between continuing old classes or dropping them. If the schedules already worked out in previous years no new issues would arise.

However, in Prisoner of Azkaban it is a different story. There we have the following:

"New third-year course schedules," said George, passing them over. "What's up with you, Harry?"

Here everything has already been arranged beforehand. Indeed, if we look back to Chamber of Secrets (Chapter Fourteen) we see that the students already picked their electives the previous year:

The second years were given something new to think about during their Easter holidays. The time had come to choose their subjects for the third year, a matter that Hermione, at least, took very seriously.

Thus, there was ample time to collect the student preferences and create the schedule accordingly well before the classes ever started. Therefore, it is possible that the problems discussed in the question were avoided by the fact that the ten combinations might not have been equally desired.

Now let's see if we can apply this theory to the reality of Harry's year. We have immediate support for this from the fact that between Prisoner of Azkaban and Goblet of Fire every single named Gryffindor in Harry's year appears in both Care of Magical Creatures and Divination.

Additionally, as far as I can tell there is no other student named as taking Arithmancy or Ancient Runes besides Hermione, and no one besides Hermione and Ernie taking Muggle studies (not counting previous years where we are told things such as that Bill and Percy got 12 O.W.L.s, which presumably means they took those courses).

Therefore, we can suggest that the scheduling problem is resolved simply by having Care of Magical Creatures and Divination at two separate times. As long as those two classes never overlap, every Gryffindor has two electives accounted for. (This sort of relies on the idea that there are only ~10 students per year per house, which is not anywhere near an ironclad assumption)

This fits very nicely with the passage from Chapter Six of Prisoner of Azkaban:

"But look," said Ron laughing, "see this morning? Nine o'clock, Divination. And underneath, nine o'clock, Muggle Studies. And" – Ron leaned closer to the schedule, disbelieving – "look – underneath that, Arithmancy, nine o'clock. I mean, I know you're good, Hermione, but no one's that good. How're you supposed to be in three classes at once.

Here we see that there was one period in which Arithmancy, Divination, and Muggle studies were given, and presumably Ancient Runes coincided with Care of Magical Creatures during another period. This works out because Hermione has a time-turner, and the other Gryffindors can be assumed to not be taking both Care of Magical Creatures and Ancient Runes, or any two of the other three classes.

However, the passage from Chapter Twelve of Prisoner of Azkaban still seems to pose a problem:

"Getting to all her classes!" Ron said. "I heard her talking to Professor Vector, that Arithmancy witch, this morning. They were going on about yesterday's lesson, but Hermione can't've been there, because she was with us in Care of Magical Creatures! And Ernie McMillan told me she's never missed a Muggle Studies class, but half of them are at the same time as Divination, and she's never missed one of them either!"

The issue here is that now Arithmancy is suddenly simultaneous with Care of Magical Creatures, which would mean that anyone taking Arithmancy cannot take any other class. Furthermore, if only half of the Muggle Studies classes coincided with Divination what was going on during the other half?

Now the easiest way to address these difficulties would be to simply posit that Ron is not entirely correct. Perhaps it was not Care of Magical Creatures that Hermione was with them at when she was supposedly in Arithmancy, and really it was Divination. Indeed, I believe this is the only time in the series where Arithmancy is shown to be simultaneous with Care of Magical Creatures, whereas it is shown as simultaneous with Divination about a half-dozen times. Furthermore, Ron apparently heard them talking about “yesterday’s lesson” (and not “yesterday’s lesson at Time X). He may have known that Hermione was with them in every class, and Care of Magical Creatures may have just been an assumption/guess on his part. Likewise, Ron may not have been perfectly precise when discussing Muggle Studies, and in reality Muggle Studies was always at the same time as Divination.

However, we can potentially address the issues in a different way without having to resort to blaming Ron's ineptitude. As mentioned above, every named Gryffindor in Harry's year is shown to take Care of Magical Creatures and Divination. So perhaps there simply were no Gryffindors taking Ancient Runes, Arithmancy, and Muggle Studies. There thus might not have been a Gryffindor Arithmancy class, a Gryffindor Ancient Runes class, or a Gryffindor Muggle Studies class to begin with. If the classes weren't given to Gryffindor then there wouldn't be any conflict with the rest of the Gryffindor schedule, even if the classes vary their times. Since it would be inefficient to give an entire class just for Hermione, she might have simply been absorbed into the other houses for those classes. Thus, she was taking Muggle Studies with Ernie Macmillan, a Hufflepuff, and perhaps she was taking Arithmancy and Ancient Runes with other houses as well.

This suggestion has the added benefit of making it a bit more believable that no one caught on to Hermione's situation. If she was taking the other electives with fellow Gryffindors it would be much more likely for them to notice that she is apparently in more than one class at the same time. If she is taking her electives with students from other houses who are not as familiar with her general schedule, they would be less likely to get suspicious.

The question this leaves us with is how to fit the other houses into the equation. We have to assume the same basic idea for all the houses – that there were one or two classes that everyone took – though the popular classes need not be the same as the ones for Gryffindor. We know that Slytherin had Care of Magical Creatures together with Gryffindor. This would mean that every Slytherin was taking Care of Magical Creatures as well, or that there was at least one other Slytherin (or joint) elective where all the other Slytherins were. That is certainly possible as long as it's not an elective that we already know Gryffindor has at that time. Then, during another period (e.g. when Gryffindor has Transfiguration) the rest of the Slytherin electives could be offered.

The same would apply for Hufflepuff and Ravenclaw. Given that we already know that Ernie was taking Muggle Studies, it might be simplest to just assume that everyone else in Hufflepuff was as well, or that they had another elective (besides for Divination which the Gryffindors had) at the same time. And with Ravenclaw perhaps they were all taking Arithmancy or Ancient Runes.

In short, it seems quite possible to arrange the schedules for all four houses as long as only a few of the ten potential combinations are utilized. At most we would have to accuse Ron of making one not-entirely-coherent statement, which would hardly be a novelty.

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