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.
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
- Care of Magical Creatures
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:
Arithmancy, Divination, and Muggle Studies were all at the same time. (First passage)
Arithmancy was at the same time as Care of Magical Creatures. (Second passage)
Half of the Muggle studies classes are at the same time as Divination. (Second passage)
The Arithmancy exam was at the same time as the Transfiguration exam. (Third passage)
The Ancient Runes exam was at the same time as the Charms exam. (Third passage)
The Muggle Studies exam was at the same time as the Divination exam.
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:
- Ancient Runes and Arithmancy
- Ancient Runes and Care of Magical Creatures
- Ancient Runes and Divination
- Ancient Runes and Muggle Studies
- Arithmancy and Care of Magical Creatures
- Arithmancy and Divination
- Arithmancy and Muggle Studies
- Care of Magical Creatures and Divination
- Care of Magical Creatures and Muggle Studies
- 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:
- Arithmancy, Divination, Muggle Studies2
- Ancient Runes, Divination
- 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.
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.