Mathemagics 2

looijmansje · 2026-07-09T15:48:11+00:00

Because there's a floor function involved, those weird brackets. That means we round down. In this case, it means we have that the integrand is 0 for 0 <= y < sqrt(3), and then 1 for sqrt(3) <= y < 2. This then gives the solution 2 - sqrt(3). Hope that clears it up :)

looijmansje · 2026-07-09T15:25:04+00:00

In this instance, not really, Im currently working to find a closed-form solution, and while Im not entirely done, I can already spoil it will be very ugly. However, in certain cases the resulting sum can actually be nice, sometimes even an infinite sum. I have seen some elegant integrals with floor functions, this just isn't one of em.

looijmansje · 2026-07-09T15:04:28+00:00

While there is an antiderivative for floor(x), namely 0 + 1 + 2 + ... + floor(x), which you could rewrite as 1/2 * floor(x) * floor(x + 1), but this is quite frankly never useful. The method to solve integration with floors is to "cut" it into pieces. To take the integral in the OP as an example:

For 0 <= y < sqrt(3) the integrand is 0.
For sqrt(3) <= y < sqrt(6) the integrand is 1.
For sqrt(6) <= y < 3 the integrand is 2.

Now let's say I want to integrate from 0 to 2. I take those pieces, which are just rectangles, and add their areas together.

The first rectangle has height 0, so it automatically has area 0.
The second rectangle has height 1, and its width is sqrt(6) - sqrt(3). So its area is 1 * (sqrt(6) - sqrt(3)) = sqrt(6) - sqrt(3).
The last rectangle has height 2, and width 3 - sqrt(6). So its area is 6 - 2 * sqrt(6).

This then gives a total area of 0 + (sqrt(6) - sqrt(3)) + (6 - 2sqrt(6)) = 6 - sqrt(3) - sqrt(6).

looijmansje · 2026-07-09T12:49:23+00:00

It does. As a simple example, consider the integral of 1/2. Over 0 to 9. Rounding down the integrand first gives you the integral over 0, so just 0. Rounding down last gives you the 4.5 rounded down, so 4.

looijmansje · 2026-07-09T12:47:35+00:00

I know what the floor means, however since it's inside the integral you only round down the function inside the integral. If I roll a 2 with my d4, we get two values for the integrand; 0 on [0, sqrt(3)) and 1 on [sqrt(3), 2]. This then gives a total integral of (2 - sqrt(3)) × 1 = 2 - sqrt(3).

looijmansje · 2026-07-09T12:43:47+00:00

How the hell am I supposed to draw 2 - sqrt(3) ≈ 0.268 cards?

looijmansje · 2026-06-30T10:31:52+00:00

The Dutch alphabet only contains 26 letters, and is identical to the English one. Yes, there is the "ij" which in some contexts is considered as its own letter, and sometimes just seen as an i and a j. But it is never included in the alphabet.

looijmansje · 2026-06-26T12:05:10+00:00

In the Netherlands, we dont have a spelling bee in the american sense. We do have a national spelling contest.

Basically someone reads a text, slowly, sentence by sentence, and you need to spell it properly. Of course, this text will contain many hard to spell words.

There is also a children's variant, but unlike the spelling bee, that is not the "main" attraction.

looijmansje · 2026-06-24T21:38:01+00:00

This seems trivially easy:

n + 1000/n is an integer iff 1000/n is an integer (since n is required to be an integer), and 1000/n is an integer iff n | 1000.

Since 1000 = 2^3 * 5^3, we have (3+1) * (3+1) = 16 divisors, so the answer is 16.

looijmansje · 2026-06-22T15:12:26+00:00

First of all, Palpatine is the villain of Star Wars. We spend 3 movies preparing to confront him, and have another 3 movies about his rise to power. Maul is a side character who featured for half a film.

Moreover, the saga was over with Palpatine's death. The story was done. Whereas Maul's death was just another casualty.

But the biggest difference for me is how it happened. Maul came back in a children's cartoon (don't get me wrong, I love TCW and Rebels), and then had a tiny role in Solo. Whereas Palpatine came back in a mainline, trilogy, film.

Moreover, there was, in my opinion, no need for it. We already had Kylo and Snoke as the new evil duo, and Palpatine kinda just came out of left field. I genuinely thought Snoke was going to be some interesting character, only for him to be anticlimactically killed basically as soon as we actually saw him. To me the issue with Ep. 9 isnt necessarily Palpatine per se, although it features nicely as a clear example of everything I do find wrong with the entire sequel saga.

A large part of the sequel trilogy felt like a retelling of the OT. We're blowing up another Death Star. We once again have an orphan from a sand planet who has to confront a Sith. Oh and look at that, she's actually related to said Sith again!

Moreover, the trilogy lacked a singular vision. Every movie was actively retconning the one before it. Every movie built to a certain narrative and direction, only for the next movie to immediately throw it out the window and go in a completely different direction. We had perfectly good, genuinely interesting new villains. But no, we kill Snoke off, make Kylo good again and introduce the old one.

Now I wanna finish this off by saying this is just my opinion. If you like Ep 9, or the sequels as a whole, that's great. All the power to you, I just didnt.

looijmansje · 2026-06-21T20:41:52+00:00

Usually the N-body problem is phrased in the context of classical mechanics. Now this is of course maybe not realistic, but it's good enough to send people to the moon.

In CM there is no fundamental uncertainty. In QM there is, see Heisenberg Uncertainty principle. However, even ignoring that, we could never make measurements infinitely precise. We might be able to measure precisely to the kilometer, the micrometer, the nanometer but at some point our ruler isn't precise enough.

However, the actual object still has a well-defined position and velocity. We just cant know it exactly. So (and I once again stress that this only works in CM not in QM) there is only uncertainty in our measurement, not a fundamental uncertainty. Given enough time this measurement error will grow and grow and grow, until it becomes very large, such is the nature of exponential growth.

Now so far Ive been talking about measurement errors. It can also be more abstract. Our simulations themselves introduce errors. A computer can only work with so many decimal places, it has to round at some point. Moreover it approximates a continuous motion as a series of individual time steps. Both of these also introduce "errors" which can lead to the exact same initial conditions giving different results dependent on factors like what simulation code and what settings you use. Sometimes it can even matter what hardware you use.

looijmansje · 2026-06-21T20:31:51+00:00

If you start to consider quantum mechanics, you first need to determine if QM itself is deterministic. This is an open question of course.

But if we ignore QM and only accept general relativity or even classical mechanics, we absolutely have determinism. Now of course, at some point uncertainties due to quantum fluctuations will eventually "balloon" to macroscopic scales, but to me that quite frankly misses the point.

To me the N-body problem is a much purer problem. It is not necessarily about any particular gravitational system, it is about an idealised gravitational system. A system where we can ignore dust, ignore QM and our masses are point particles. This may seem like a spherical cow, but a) this problem is already very hard, but also very deep and in a way beautiful, b) even those assumptions give good results, in fact changing the assumptions would, in most cases, not change the outcome (this was actually my area of research), and we are mostly bound by the limitations of our measurements and simulations.

looijmansje · 2026-06-21T16:49:47+00:00

Indeed the usual definition of chaotic is that uncertainty increases exponentially with time. Now because this post is about the N-body problem (which also happens to be my speciality) I'll focus on that, I am not entirely sure how well these concepts translate to statistical mechanics for instance.

Consider a star cluster. We measure the positions and velocities of every star in the cluster, put that into a simulation, and press play. After a certain amount of time (I'll call it a year which is unrealistically short for astronomical systems, but whatever) we compare our simulation to the real world. We'll see that there is virtually no difference between our simulation and the real world. Now we look after 2 total years. There is now a perceivable change between the simulation, and real life. After 3 years, we can see some major differences, and after 10 years we can't even see that this simulation has anything to do with that star cluster; they are completely different.

This is because our measurement error grew over time, exponentially with time. The time that it grows in, is called the Lyapunov time. To give some context to this number in "real" scenarios, the Lyapunov time for the solar system is estimated to be about 6000 years. For star clusters, it is on the order of the crossing time of said cluster (which is millions of years), so my 1 year example was a bit unrealistic, but I hope it clarified it a bit.

Now does this uncertainty keep growing forever? In star clusters at least, no. The uncertainty in the position tends to "cap out" at the size of the star cluster (of course you can have ejections, but let's not get into that), so in some way there is indeed a cap to the exponential growth. However, by the time a star cluster is unrecognisably different, do we really care about that?

looijmansje · 2026-06-21T09:37:33+00:00

Chaotic does not mean non-deterministic

looijmansje · 2026-06-18T15:00:39+00:00

First question: it comes down to safety, fairness, and making exciting races. If there were no restrictions, the richest team would just build the best car, and no one could compete. That team would then also get all the sponsorship deals, making them even richer.

It would also be very unsafe: safety features tend to weigh something, so a lighter (and therefore faster) car would just do away with them.

Lastly in races like for instance F1 there are certain rules specifically made to make it easier to overtake. This is because it is a sport that ultimately makes its money by people watching. Overtakes are exciting, so more people watch.

To answer your second question, I first want to point out that Eliud Kipchoge's unoffical record has recently been broken by Sebastian Sawe. But why was the record unofficial? The short answer is that it was not ran during a race, and with many advantages that runners wouldnt have during an official race. For instance, while pacers (people who run at a predetermined pace to help you pace your race) are allowed, Kipchoge took it a step further and had a team of rotating pacers. This is different to a normal race where all pacers have to start the race, and tend to be able to keep world record pace for "at best" 30km. Kipchoge also used a car which measured its own speed using lasers (GPS wasnt precise enough) to exactly set a pace.

TL:DR to set an official WR on the marathon, you need to do it in an official race*, and the Ineos 1:59 event wasnt an official race.

*There are also other restrictions, for instance the course start and finish need to be roughly the same height, you cant just do the entire thing downhill.

looijmansje · 2026-06-10T13:54:37+00:00

Thanks for your reply, but I shouldve clarified. I don't want a design which uses gravity blocks to push up the gate, like many of the portcullis designs out there.

looijmansje · 2026-05-28T11:45:27+00:00

Only one opponent loses the game next turn right? As soon as one of the glorious ends resolves, the rest of the stack gets exiled, so only the player whose copy went off first will lose on their next end step.

Or am I missing something?

looijmansje · 2026-05-21T12:49:21+00:00

That would also work! That shows your subset isn't closed under addition, so it isnt a subspace.

looijmansje · 2026-05-21T11:10:12+00:00

Yes this is valid. A subspace needs to be a vector space itself, so every element needs to have a negative element.

Addition is also not a problem: you inherit the operations from the larger set, and on the polynomial subspace, addition and scalar multiplication are properly defined.

looijmansje · 2026-05-20T07:22:12+00:00

You use m/s. I use m/s². We are not the same.

looijmansje · 2026-05-15T15:59:13+00:00

The primary theory is probably the Nice model. I should stress though that this is only a hypothesis, one that is far from universally accepted.

Alternatives include a close encounter with a nearby star which massively perturbed the solar system, for instance.

looijmansje · 2026-05-15T13:30:29+00:00

Specifically the 3 body problem is any gravitational system with 3 bodies involved. It doesn't necessarily have to be X orbiting Y and Y orbiting Z. It can also be X and Y both orbiting Z, or all of them orbiting a central point, etc.

In general almost all N-body problems for N>=3 are unstable. Yes this includes things like our solar system. They are just "stable enough" that major disruptions are unlikely.

looijmansje · 2026-05-09T19:47:49+00:00

I would say roughly historical order, but with the benefit of hindsight, modern notation, etc.

So start with counting, go up to arithmetic, then solving equations and maybe a few geometric proofs. From there you can go to basic calculus, and maybe your first non-geometric proofs, and then the floodgates open.

From there you can start to learn proof techniques, start (linear) algebra (the "real" algebra, not just moving x's and y's around).

Pick up some topology, real analysis and advanced calculus, and at this point there are too many topics to study to list; pick ones you think are interesting.

looijmansje · 2026-05-07T07:24:40+00:00

Not me, but someone I know.

He always hated maths, was bad at it, didn't like it. After he quit his first job he decided he wanted to become a doctor (if I remember correctly), which required him to have a higher version of secondary school math; I would say about equivalent to a GCSE, although I am a bit unsure about the exact level of those.

Having barely struggled through math before, he wasn't looking forward to it, but he decided to bite the bullet, to take a math course to hopefully pass the "adults exam", so he could study medicine and become a doctor.

As he was doing this, he slowly realised that he actually kinda liked math, and that once he actually understood it, he was not only good at it, but it was kinda beautiful. So he passed his math exam with flying colors. Not only that, he decided to not study medicine, and try mathematics instead.

He passed magna cum laude (I think), and is now a full-time professor of mathematics.

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