Jo's Math and It's Effect on the HP Books

JKR has said that math is not her strong point. In fact, it seems that situations where she might have to deal with numbers make her uncomfortable/stressed. I thought it might be interesting to look at some of the instances where her (lack of) math skills has affected the books.

Of course the first and most commonly discussed math problem is the number of students at Hogwarts. When asked in an interview how many students there are at Hogwarts, she replied “There are about 1,000 students.” But upon inspection of the text in the books, this number is difficult to back up.

1,000 divided by 4 houses = 250 per house
250 / 7 years = 35-36 per year
35-36 / 2 sexes = 17-18 per dorm
That doesn’t match the books.

Conversely,

5 students per dorm x 2 sexes = 10 students per year
10 x 7 years =70 students per house
70 x 4 houses = 280 students total
And that doesn’t match what Jo said.

I’m sure that, when Jo envisioned Hogwarts, she imagined a lot of students. I’m also sure that she imagined only a few in each dorm when the number of overall students was broken down by house, year and sex. The problem is, she didn’t actually do the math. I think she left it up estimations. When she pictured all the students together (Great Hall, Quidditch pitch, etc) , she estimated what it looked like to her and came up with “about 1,000”. Perhaps then she thought “1,000 divided by four houses, seven years and two sexes would probably be about five per dorm”.

I’m not a teacher, or a psychologist, so I can’t pretend to know how the brain of math-challenged person works. But it seems to me that part of the problem would be not being able to “see” what a number (of objects) would look like; not being able to closely estimate how many of something there are. If I’m wrong about this, feel free to correct me.

The account of the sorting ceremony supports the 280 number much better than the 1,000. One of the things in favor of that is the length of the ceremony. With 280 students, only 40 would have to be sorted. If you allow a conservative two minutes per student, the ceremony would last an hour and twenty minutes. With the 1,000 number, they would have to sort about 142 students, making the ceremony around four hours forty five minutes-while students are waiting to eat!

Also, 70 students (per house) would fit much better into a house common room than 250 would.

Since we can't resolve this discrepency, I guess all we can do is imagine a large number when students are toegther, and a small number in more intimate settings, along with the images in Jo's mind.

Wizard money: 29 knuts = 1 sickle, 17 sickles (or 493 knuts) = 1 galleon. This is a system that makes no sense; it’s totally illogical and confusing. Is this indicative of the way Jo feels about numbers and math? If muggle math is confusing to her, why shouldn’t wizarding math be equally or even more so? This may also be why there are only three types of coins—three coins were probably confusing enough to think up; any more would have required more thought about numbers (something that’s avoided by people who hate working with numbers).

When Hagrid explains the coins to Harry, he says “it’s easy enough”. Well, no, it isn’t. Yet we never hear Harry, Hermione, or anyone else who is used to muggle money, question or complain about the complexity of the currency system.
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It was a horrible sight. Twelve feet tall…

It was holding a huge wooden club…

He (Harry) took a great running jump and managed to fasten his arms around the troll’s neck from behind.

The club flew suddenly out of the troll’s hand, rose high, high up into the air, turned slowly over – and dropped,…on its owner’s head.

The ceiling of the girl’s bathroom would have to be unbelievably high in order for a ‘huge wooden club’ to rise ‘high, high’ above a twelve foot troll and ‘turn slowly over’. If the troll is 12 feet, and we estimate the length of the club at a conservative 3 feet, that would put the ceiling at 15 feet, not including the ‘high, high’ above the troll’s head. And for it to knock the troll out when it dropped on him, it would have to be quite a distance above him.

Then we have Harry jumping up and grabbing the troll around the neck. If we figure the troll’s neck is approximately 11 feet off the ground, and estimate Harry’s height at..say..5 feet, he would have to jump at least 6 feet into the air to reach the troll’s neck. That’s a pretty big jump.
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It was a magnificent mirror, as high as the ceiling…

If this ceiling was as high as the ceiling in the girl’s room, that is a very tall mirror.

And if every floor has ceilings that high, I'd hate to have to climb even one flight of stairs. Which brings us to:

Students seem to think nothing of climbing 5-7 (and even more to get to dorm rooms) flights of stairs on a regular basis. And hey, wouldn't this make the Gryffindors and Ravenclaws more physically fit than the Slytherins and Hufflepuffs?

[Defending mode]
Now, this is the pot calling the kettle black. Don’t you Americans still believe that 12 inches = 1 foot, 3 feet = 1 yard and 1760 yard (or 5280 feet) = 1 mile is easy enough, although the metric system would be so much more reasonable?

Wizard money is a parody of the not so ancient British Muggle money (12 pennies = 1 Shilling, 20 shillings or 240 pennies = 1 Pound), which was only abandoned in 1971. It fits perfectly into the Potterverse, where wizards do anything not magical in an oldfashioned way.

The troll in the girl’s bathroom is pure magic. It should not even have fitted through the door, so you are not supposed to think about it. But if you insist on a rationale approach, you should know that cars and tents may be bigger inside, and this is probably also true for buildings or single rooms – just remember the Room of Requirements.

Even if the girl’s bathroom is 15 feet high, neither this floor nor the other floors need to be of the same height. Also, I wouldn’t expect that this room is always 15 feet high or even higher. It probably adapts magically to whatever space is needed to allow the persons in the room to do what they need to do.

Harry jumped at least 6 feet into the air to reach the troll’s neck? Harry apparated to a roof in primary school. After only some weeks at Hogwarts he may still not have learned to control his instincts and do only intentional magic.

Student numbers are tricky. Jo did not only say "There are about 1,000 students" in an interview, there are also small tables for about 1000 persons at the yule ball. Subtract students from Beauxbatons and Durmstrang, staff and other guests. Add students who where too young and not invited by an older partner. The estimated number of students at Hogwarts is "at least 1000", and the common rooms are not as cosy as we like to imagine.

Since we have only ever seen one dorm, the common beliefe that there are "5 students per house, year and gender" is not really backed up in canon. The Sorting Hat doesn’t need to sort equal numbers of male and female students to all houses because there isn’t a standard size for the dorms; they adapt magically to the number of students.[/Defending mode] (Thank you for reminding me that something similar must have happened in the troll incident, so this ability is not limited to the  Room of Requirements.)

Okay, there is evidence that the sum of Gryffindor and Slytherin students in Harry’s year is twenty, and that the sum of Gryffindor and Hufflepuff students in Harry’s year is about twenty as well. But we don’t know anything about the Ravenclaws or students in any other year.

It’s weird enough that we did never see all the Gryffindor girls in Harry’s year (supposed that there are more than three). Not ever seeing all boys who share Harry’s dorm (and couldn’t be considered just supernumeraries when they are so close to Harry) would be even weirder. On the other hand, keeping track of too many characters and their backstories would be difficult. Thus I believe that Jo made the number of students in Harry’s year deliberately smaller than the average number of students per year. If you need to rationalize this, there may be a reason for the birth rate being at a minimum when Harry was born: There was a war going on and the Dark Lord was winning, until he made the fatal mistake to attack Harry. People who didn’t still believe that they could win may not have wanted to make their children live under Voldemort’s reign when they could avoid to have children at all.

You may have a point with the duration of the sorting ceremony, but I don’t understand where you get the two minutes per student. Jo says on Pottermore that the Sorting Hat needing more than five minutes for a student is a rare event which may occur perhaps once every fifty years. She also says that the Sorting Hat spent nearly four minutes on Hermione and on Neville, but at least for Hermione this is not actually shown in the book. For most of the students the Sorting Hat seems to know their house nearly immediately. From what I still remember of being eleven years old, most children, being eager to know their house, were probably running to the stool, und would have become quite impatient if McGonagall had waited for the previous student reaching their table before she called the next name. (I should really look this up rather than imagine a turbulent and exciting scene, but I’m not at home.) I would say that, with the exception of some difficult cases, sorting takes about ten to twenty seconds per student. With 150 students to sort, this gives us less than an hour of turbulence and excitement, which seems to fit Harry’s experience better than your hourlong boringly slow procedure for 40 students.

Jo may not have been able – or wanting – to do the maths, but she found many clever ways to avoid having to do it. Or she may have done the maths to be sure that Muggles would deem something "impossible" and expected us to understand that there is magic in this world even when we aren’t explicitly told that a spell was casted.

Interesting discussion.

When I read the HP books for the first time, I assumed there were other boys in Harry's year sleeping in other dorms besides those five (why would everyone in the same year sleep in the same room?), but now I realize we never get to see them. We can conclude that they don't exist - or insist that they still might. It is possible that Jo envisaged 1000 Hogwarts students scattered around in lots of dorms, but she simply didn't need most of them. Actually, handling 1000 students on an individual basis would probably be difficult for most writers. Then again, we could at least see some other Gryffindor students of Harry's year hanging around sometimes - even if not mentioned by name - as Harry probably had some interaction with them - provided they existed. 

Anyway, as far as I can tell, the problem is that Jo's answer to an interview question doesn't support what we see in the books. In such cases, I tend to give priority to what we can find in the books, because that's what she purposefully included there. Answers to interview questions are "secondary canon" in my opinion. After all, how do we know that Jo had ever thought of a certain problem before she was asked about it during an interview, where she had to give a a quick answer? (In this case, she may have done some quick maths - or none at all.) If the answer is contradicted by the books, it may simply be the wrong answer. Or, if we conclude that "crowd scenes" (Quidditch, Great Hall) support the 1000-student theory, then we just have to accept either that Harry never noticed most of his housemates or that Gryffindors were disproportionately few in comparison with the other Houses, perhaps because true courage is a rare quality, and it was much easier to qualify for Hufflepuff, for example, than for Gryffindor. 

Wizard money... I think it's a comic touch when Hagrid explains it to Harry and says it's easy enough. One might even think the Dursleys may have a point when they describe wizards as lunatics, LOL. It also emphasizes how exotic the wizarding world is for someone who has grown up with the Dursleys, who are perfetcly normal, thank you very much.  I also like the idea that's it's a parody of the penny - shilling system. But it must also have something to do with numbers and mathematics - I notice that both 29 and 17 are prime numbers. I don't know why it's interesting, but I think it is. 

Yet, money does cause difficulty later on in the books. Here's an interesting entry in the Lexicon: [You must be registered and logged in to see this link.] I definitely see some mathematical problems there.

The troll in the bathroom... It is true that mathematically it's difficult to imagine, but then there is so much magic everywhere at Hogwarts that I wouldn't have noticed this discrepancy myself. It nicely underlines Ron's amazing achievement when he can lift that heavy club with the spell he hasn't been able to use on a feather with any success before. We know that at Hogwarts doors and staircases can change in many ways, so why not the ceilings? 

Harry jumped at least 6 feet into the air to reach the troll’s neck? Harry apparated to a roof in primary school. After only some weeks at Hogwarts he may still not have learned to control his instincts and do only intentional magic.

I agree, except that this magic in the bathroom can't have been Apparition, which was impossible at Hogwarts. 

Students seem to think nothing of climbing 5-7 (and even more to get to dorm rooms) flights of stairs on a regular basis. 

Oh, students think of the stairs they are climbing only when they have nothing better to think of, even in the Muggle world.   I have always thought that those stairs replace physical education at Hogwarts. (They only have that broomstick riding in the first year and later Quidditch for only seven students per house.) 

And hey, wouldn't this make the Gryffindors and Ravenclaws more physically fit than the Slytherins and Hufflepuffs?

Not at all. Gryffindors and Ravenclaws climb upstairs when they return to their dorms, while Slytherins and Hufflepuffs climb upstairs when they come out of their dorms. I suppose the amount of climbing is about the same. Moreover, Slytherins and Hufflepuffs may have to run upstairs in the morning to avoid being late for class, while Gryffindors and Ravenclaws can climb upstairs at a leisurely pace after their classes. Except that they probably find some other reason to run.  

It’s so obvious now, but I would never have thought about 29 knuts and 17 sickles being prime numbers. The purpose is probably to make it impossible to have half-sickles or quarter-galleons or any other fraction of a sickle or a galleon. And that serves to make Muggle money ridiculously complex from a wizards point of view. One cent, two cent, five cent, ten cent, twenty cent, fifty cent, one euro, two euro, how do these weird continental-european Muggles remember all the different coins and their distinguishing features? A knut is a knut, it doesn’t need to have a number carved into its surface.

I agree that interviews are dubious canon at best, But the problem with the 1000 students is that Joanne didn’t  just say it in an interview. There are about 1000 attendees at the yule ball.

The assumption that Harry never noticed most of his housemates or that Gryffindors were disproportionately few in comparison with the other Houses, doesn’t work well, because we have seen that there were only twenty broomsticks at the first Flying lesson (Gryffindors and Slytherins together). Since we know seven Slytherins by name, there can not be more than thirteen Gryffindors in Harrys year,  and since we know eight Gryffindors by name, there can not be more than twelve Slytherins. Also, there can not be much more than twelve Hufflepuffs, because there were only about twenty earmuffs at the first Herbology lesson in CS.

But disproportionately few students in Harry’s year is possible, because we don’t know numbers from other years.

The few number of teachers is a real problem, and from [You must be registered and logged in to see this link.] it seems impossible to have more than thirteen of them (including Dumbledore), which by the way is another prime number.

Although I’m sure that it’s even harder with thirtyfive students in one class, the 1000 students are not really the problem here, because the classes have to be held anyway. There are only about ten Gryffindors in Harrys year, but they don’t join students from another house for Transfiguration. Minerva would have to teach twenty class periods per week just so each student in years one through five could have one class period of Transfiguration every week.

There are students from all houses in Professor Snape’s Advanced Potions, because he only takes the best of the best. I don’t remember what we have seen of Advanced Transfiguration, so Minerva would have to teach at least two and up to eight more class periods per week just so each student in years six and seven could have one class period of Transfiguration every week. I didn’t hunt for quotes, but I’m quite sure that students get two Transfiguration lessons per week every year. This makes at least 44 and in the worst case 56 periods for Minerva (I don’t see how Julia got 64). All this is true even if there are only 280 students in total (but it may become worse if classes with more than thirty students from one house and year have to be split up).

The few number of teachers is a real problem, and from [You must be registered and logged in to see this link.] it seems impossible to have more than thirteen of them (including Dumbledore), which by the way is another prime number. (HG)

Ah ... 13 sitting at a table... every single day.  

I don’t see how Julia got 64 (HG).

I got 64 because I calculated with a hypothetical 1000 students. Taking 1000 students as our starting point, I estimated that in a given year maybe 800 would be taking Transfiguration. That means there are 200 in the 6th and 7th year who are not taking Transfiguration in this particular school year (and I have a feeling that this would be one of Minerva's luckier years). I calculate on an average of 25 students per class (35 seems a huge number to me, in fact, 25 seems quite big, too) - no way to have everyone belonging to the same house and year in the same class. 800 divided by 25 is 32 - the number of Minerva's lessons per week if each of these students takes one Transfiguration class a week. If they take twice as many, that would be 64 weekly lessons for Minerva. It may be a somewhat smaller number for Snape if he takes only the best students in the 6th and 7th years, but in essence he would have a similarly impossible workload. (Of course, if they have smaller classes, the number of lessons per week increases.) 

Official Muggle calculations estimate that actual teaching is only three fifth to two third of a teacher's complete workload (but teachers can work a lot more than what is officially estimated). Teaching 64 class periods per week would mean a regular 100-hour working week at the very minimum, spying and medicinal potion-making not included.

As for your calculation, you got 44 or 56 because you didn't calculate on 1000 students, only on 4 houses and 7 years. These are also awfully big numbers, and they would be big even if the professors didn't have other duties besides teaching. 

Although I’m sure that it’s even harder with thirtyfive students in one class, the 1000 students are not really the problem here, because the classes have to be held anyway. (HG)

10 students per class or 35? That's a big difference to any teacher! 1000 students are a problem even if the number of classes is not, because more students mean more assignments to mark, more exams to conduct (more boiling cauldrons to watch). 200 students are diffcult enough to handle and they probably need some magic for that - but 1000 not being a problem? It would be a huge problem!

So the teachers' schedule may indeed be a point where Jo got it wrong (even if we calculate with 250 students) or perhaps we may have to wonder whether there are more joint classes than we are aware of. We see everything through Harry's eyes, and Harry always notices when Gryffindors share a class with Slytherins for obvious reasons, but what if they share classes with Hufflepuffs some other times, but Harry simply doesn't pay attention to them? 

I admit I haven't checked the books for that possibility, so it's mere speculation on my part, and I can be totally wrong... but I wonder.

At least now I understand why every single Hogwarts teacher seems to be unmarried, unattached and living exclusively for his / her job.   

Also, there can not be much more than twelve Hufflepuffs, because there were only about twenty earmuffs at the first Herbology lesson in CS.

The problem with this argument is that the only reason why we assume that everyone belonging to the same house and the same year is necessarily in the same class is that we see too few students to think otherwise. If there were more students, there would be more classes as well, but 20 earmuffs per class would still be enough. Don't get me wrong, I don't think Hogwarts could have 1000 students, but anyone who assumes there are 1000 of them must also assume a lot of students "invisible" in the books (except when there is a ball), and in that case we should have the necessary number of "invisible" classes as well.

Of course, the teachers couldn't cope with such numbers.

By the way, 1000 people at the Yule ball?    But that would mean even more students for the whole of Hogwarts as, if I remember correctly, the ball wasn't open to the first three years (although they could be invited). Of course, there were guests and teachers, too, but how many of them? Or, LOL, did all the younger students get invited somehow?

Okay, I looked it up:

The house tables had vanished; instead, there were about a hundred smaller, lantern-lit ones, each seating about a dozen people.

Which gives us a quite vague "about 1200".

Of course this may have been a case of being prepared for the worst, and if not all the younger students did get invited somehow, some of these seats may not have been in use.

I agree that an average of 35 students in one classroom is far to much, but  isn’t splitting and mixing the students to get classes of a reasonable size slightly contradictory to sorting them into houses?

Of course this may have been a case of being prepared for the worst... 

LOL, they were certainly overcautious!   Even if they did have 1000 students and even if all of them attended the ball, they only had about a dozen teachers, and a few dozens of guests in addition to that, which still doesn't make 1200. Perhaps organizing a major even in the wizarding world entails providing a lot of extra space in case someone throws in a Gemino Curse or something equally bad.  

I don't suppose participants were able to invite guests from outside, were they?

I agree that an average of 35 students in one classroom is far to much, but  isn’t splitting and mixing the students to get classes of a reasonable size slightly contradictory to sorting them into houses?

Oh, but perhaps Sorting is magically compulsory (thanks to dear old Godric), and they may be trying to alleviate the harm it does by mixing the students in classes.  

By the way, just to play with ideas a bit more, 1000 students would make about 140 students per year on average. However, if we suppose that in Harry's year there are fewer students than in other years, i.e. 8 Gryffindors and not much more Slytherins, Hufflepuffs or Ravenclaws (let's say about 40 or 50 in total) so that each class in that year is of a comfortable size, it means that the number of students in the other years must be considerably higher than 140. Supposing that the remaining 100 students are equally distributed in the other years and in the four houses, we end up with classes of 40 rather than 35 in the other years. If there are further differences in the other years, we may end up with a much higher number in a given year and house and consequently in a class, unless, of course, these large classes are split, which would in turn increase the number of lessons teachers have to teach. It's a rather horrible prospect. 

If we calculate with only 50 class periods per week for a given teacher, it still means 10 periods per day, which literally lasts all day, especially if there is a lunch break. Except, of course, if these teachers use Time-Turners all the time. Another solution is that they have very short class periods (like half the length normal in the Muggle world?) - but even then, they still have to deal with 1000 home assignments, essays, exams on a regular basis.

Oh, but perhaps Sorting is magically compulsory (thanks to dear old Godric), and they may be trying to alleviate the harm it does by mixing the students in classes.

This might have been a good idea, and I’m aware that absence of evidence isn’t evidence of absence, but to believe it, I would have needed to see more mixed classes than just Herbology with Hufflepuffs and Care for Magical Creatures (and one Flying lesson) with Slytherins and Advanced Potions with everybody. Or I would need an implication who “they” may have been. Even Dumbledore said only that Sorting might happen too early, not that it is nonsense and should be abandoned if this were not impossible.

Also, Gryffindors have Potions with Slytherins every year prior to the O.W.L. exams. 

Even Dumbledore said only that Sorting might happen too early, not that it is nonsense and should be abandoned if this were not impossible. (HG)

Organizing the students in four houses may make sense in itself - I think all schools organize their students in smaller units than the whole of the school in one way or another, and I think it has definite advantages. Of course, if it is done, it must be done at the beginning of their studies, not when the students leave school. What Dumbledore means, however, is that they sort their students according to their personalities too soon. Was that the first time Dumbledore, with a teaching career of several decades behind him, had thought of this? Had no else ever thought about it? Was Snape the first ever student whose personal development had not been finished by the age of 11? I can hardly believe that. 

Also, I think a healthy rivalry between houses may be inspiring, but the rivalry between Gryffindor and Slytherin is not exactly healthy any longer, and putting the kids into the same classes could be a way of trying to solve this problem.  

Still, it was only an idea on my part and I don't claim that there is any strong evidence for such an opinion at Hogwarts. The most likely explanation for the central problem in my opinion is that there weren't 1000 students at Hogwarts, and then we don't need to calculate with mixed classes all the time (though they do have at least some mixed classes, as we know). Even the problem of the 1200 seats at the Yule ball is easier to resolve, I think, than all the questions and problems a Hogwarts with 1000 students raises.

And how did I never notice this:

He had hardly spoken to anyone about this, least of all thirty eagerly listening classmates.

How are there thirty students in Umbridge’s Defence Against the Dark Arts class? If she gathered all students of Harry’s year in one class, there should be more than thirty, right? If there are two classes, they should be of equal size, right? Are there about ten Gryffindors, about ten Slytherins, about ten Hufflepuffs and about thirty Ravenclaws in Harry’s year? I’m really confused now.

(And if this is a glitch, why is it not reported as "to be corrected in future editions"? Were all readers so enraged about Umbridge that nobody ever noticed this?)

It seems to me that whenever JKR uses the word "dungeon" in the HP books, she's referring to the underground classrooms. Now if I'm wrong about this, correct me. So I've always assumed that Nick's deathday party was held in an empty classroom. But now, I'm not so sure about that. And if that's not the case, just what was that underground room?

"It was an incredible sight. The dungeon was full of hundreds of pearly-white, translucent people, mostly drifting around a crowded dance floor, waltzing to the dreadful, quavering sound of thirty musical saws, played by an orchestra on a black-draped platform." CoS, ch 8.

This would have to be a pretty large room to be big enough for a dance floor, a platform big enough to hold thirty people playing musical saws (which is done while sitting on chairs), a long table of food, and "hundreds" of people. This sounds to me more like a ballroom. But an underground ballroom? And if so, why didn't she say that?

"Hundreds" is plural. So it must mean at least 200 people (though it seems to imply more; otherwise she would have said "a couple hundred"). That's a really big birthdeathday party, don't you think? Do you think Nick even knows all those people?

I always assumed that everything underground is “The Dungeons”. This includes the kitchen, which is

an enormous, heigh-ceilinged room, large as the Great Hall above it (GF 21),

so I’m not surprised that Nick found another big underground room. It probably wasn’t built to be a ballroom. It may have been meant to be a large storage room, like a wine cellar.

I've always imagined Hogwarts as a large enchanted castle, so I have no problem with secret, magical places of any size or number, especially in the dungeons. (Perhaps the ghosts have their own RoR there.) Besides, most of the guests at Nick's party are ghosts, and they probably need less space than living people.

Regarding how many people Nick may know, 500 hundred years is a long time. Ghosts seem to have plenty of free time for socializing with each other and even with the living.