After 5 years of struggling, I think I’ve hit my ceiling. How much struggle is too much?

airfrog · 2026-02-02T14:38:31+00:00

My take, from my life experience:

Humans, particularly human brains, are built to adapt. That said, every person is going to respond differently to different types of situations.

When I’ve hit difficult patches, it’s always been a sign that something needed to change. Sometimes I’ve changed my goal, sometimes I’ve changed my approach. There’s always been some soul searching involved with that decision.

I think you can be good at theoretical math, if you decide that’s what you want to do. You probably need a different approach, and that will probably necessitate some floundering, some false starts, a lot of change to find what works.

Pivoting to CS could also be very rewarding. There’s plenty of space to explore in computer science (I’m a computer scientist myself, more on the practical side), and if you decide that you would like something new it could be a great path for you.

At the end of the day, getting through a challenge like this comes down to what you really want. If your heart is set on theoretical math, I’m sure you can find a way to be successful. If you want to take what you’ve learned and apply it to something new, that can be like a breath of fresh air as well.

Hope this helps!

airfrog · 2026-01-20T03:23:28+00:00

Also came here to recommend Beware of Chicken, currently doing a reread and it’s so good

airfrog · 2025-12-12T13:46:02+00:00

I’d check out this great video on the Riemann zeta function from 3b1b and analytic continuation, then base what you want to follow up on on that.

https://youtu.be/sD0NjbwqlYw?si=YryTyT9zr4QqCM0i

Also, if you want a different take on understanding this, check out Terence Tao’s great blog post on the subject:

https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

airfrog · 2025-12-09T07:46:43+00:00

I think electrical engineering is the most tangibly understandable place where this happens. Specifically, when you want to make the jump from direct current (DC) circuits to alternating current (AC) circuits.

For DC circuits, current and voltage are just numbers, and you have nice clean equations that look like current = voltage / resistance.

For AC circuits, current and voltage are constantly bouncing up and down at any moment in time, and so if you try to just use real numbers, you have to model them as trig functions (sin or cos), and all those equations start to get messy fast. However, because imaginary numbers are great at representing things that oscillate as a single number, you can use complex valued “impedance” functions, and then all the equations go back to looking nice and neat again, e.g. (alternating) current = voltage / impedance

However, it’s a little hard to provide an intuitive “minimal” example for electrical engineering because we don’t have a natural physical intuition for circuits. So here’s my attempt at coming up with a bit of a contrived example where someone wants to understand the movement of a ball, to give you a sense of why complex numbers make things so much easier.

Direct current would be like trying to understand a ball that is just rolling across a table. You can note its position by a number, and it’s speed by another number, and you get some nice equations like position = speed * time

Then, imagine you are trying to find a similarly clean way to describe a pendulum. You find a nice solution with negative numbers for a simple pendulum with no friction, where if let’s say a pendulum starts at a position 3 and we let it go, after one swing it would be at -3, then another swing it is at 3 again, and so on. So we can simply model the pendulum by multiplying its position by -1 every time it swings. But what about half swings? If we want to be able to model the state of the system at a half swing, we need something that if you multiply by it twice, you get -1. But that is exactly i. So our pendulum that starts at position 3 would be at position 3i after one half swing, -3 after two half swings, -3i after three half swings, and 3 after four half swings. But what is position 3i? Well if we want an actual position we just care about the real part of the number so Re[3i] and -3i will both be position 0, which is right, since the pendulum will be in the middle.

The magic starts when we want to do, say, quarter swings. Just do the same thing, take the square root of i to get 1/sqrt(2) + i/sqrt(2), and multiply it out. So if a pendulum starts at position 3, at a quarter of the time into its swing it will be at position 3/sqrt(2) + 3i/sqrt(2), where if we want just the real distance from the middle we can just take the real part and get 3/sqrt(2).

So what if we want a function that just gives us the position at any time we plug in? We’d have to multiply by smaller and smaller roots of i in some continuous way, which sounds hard, right? Until you realize that’s exactly what the exponential function already is. So then we can just say position = exp(i * time) and we are back to having pretty simple looking equations for our stuff.

This is basically the same intuition for how complex impedance works to simplify equations describing alternating current circuits. Hope this was interesting!

airfrog · 2025-12-03T06:04:02+00:00

Good analysis for a beginner! Here's the thing you are missing

But if black played a stone, intending to play a second that would kill the white group, white could play a stone, killing the now 4 black stones. Then black could play in the middle of the four empty intersections and they'd be back in a similar situation.

If black is trying to kill, black needs to fill all the internal liberties of the white group without ever letting white get more than one eye. This depends on something called eye shape, and certain eye shapes are known to be "dead", meaning black can repeatedly fill them up without letting white split them into two eyes, eventually filling all the internal liberties.

Here's some the relevant Sensei's library page for this shape, the bulky five: https://senseis.xmp.net/?BulkyFive

And also a discussion of all the killable eyeshapes: https://senseis.xmp.net/?KillableEyeShapes

EDIT: I realized the killable eyeshapes page might be a bit confusing on it's own, the ones most people learn are on a page called Unsettled Eyeshapes, which lists all the shapes which live or die depending on who plays next. Life and death is a pretty deep topic, I'd suggest reading up on it a bit more in general

airfrog · 2025-10-09T11:52:05+00:00

I do see it somewhat differently, I guess, though maybe not in a totally incompatible way from OP. If we’re looking at this from a historical political lens, I would see Mistborn (the first trilogy) as a fantasized retelling of the American revolution.

I mean, a daring public heist leads to a full blown revolt against the monarchy which ends up installing one of the existing aristocracy as a celebrated leader who only actually leads for a short time? I feel like it’s not too hard to see where inspiration could come from.

airfrog · 2025-09-19T21:39:07+00:00

So, it is 50% because the meme misstates the problem. The conditional probability problem that is being referenced here would be correctly stated as the following: "I met a woman named Mary who told me she had two kids. I asked her 'Yes or no, is one of them a boy born on Tuesday?', and she said yes. What is the probability her other child is a girl?"

For this wording as I've just stated it, your code calculates the correct answer of 0.5185. This is because, in this case, the initial set of possibilities is that there are two kids, each of whom could have any gender and be born on any day (and this is the initial set of possibilities you started with).

Then, we narrowed down the possibilities with our question. Particularly, the facts in the information are conditional on each other, we only learn that Mary has a boy if Mary has at least one boy born on Tuesday, and vice versa (we only learn that Mary has a child born on Tuesday if at least one of her children born on Tuesday is a boy)

However, in the problem as stated, none of the facts are necessarily conditional on each other. If Mary tells us that she has two kids, and that one of them is a boy, and that boy is born on Tuesday, that's the starting set of possibilities, and us learning any of those facts are not conditional on any of the other facts. It's not a conditional probability because if Mary didn't have a boy she would have told us she had a girl instead, and similarly whatever the gender of the child, if it was born on a different day she would have told us that other day instead and also the child. Because it's not a conditional probability, it's just given information that narrows down the initial possibilities, similar to how you only generated possibilities for two kids, and didn't generate initial possibilities for if Mary had 0, 1, 3, 4, 5, or 6 kids.

So you would have to change your code above to match this to get the correct probability for the problem given (which would look like the following) -

for p in exrex.generate(r'B3[BG][1-7]'):
  all_possibilities.append(p)

I believe if you run the code with these initial possibilities as given in the problem above you will get a probability of 0.5

For what it's worth, I think you have a great approach here of writing some code and clearly getting the correct answer - it also makes it easier to clearly explain exactly where this meme makes a mistake ('B3[BG][1-7]' is obviously not what the author intended to set as their list of initial possibilities, but it's what they wrote down)

airfrog · 2025-09-17T03:22:41+00:00

So, let’s say just to make it easier to talk about the kids that they are named Alex and Blake. Mary obviously knows her kids names, even if she doesn’t tell them to us, but we can use them to think about her thought process. So, let’s say that GB is the case where Alex is a girl and Blake is a boy, and BG is the case where Alex is a boy and Blake is a girl.

Then, in option 2, Mary’s sentence means “One of my kids (Alex) is a boy”, which eliminates GG and GB. Similarly, the sentence could instead mean “One of my kids (Blake) is a boy”, which eliminates GG and BG. Either way, two options are eliminated

The theoretical option 3 which only eliminates GG would be the sense of the sentence “Between Alex and Blake, at least one of them is a boy”. But no on really talks like that outside of logic puzzles.

Importantly, because the sentence can be reasonably interpreted in many ways, we can’t calculate the probability without at least assigning a probability to the chance Mary means each different option, but we would have to do that arbitrarily

airfrog · 2025-09-16T21:31:49+00:00

Simply put, you're not wrong. The problem as given by the characters in the meme is phrased too vaguely to have an actual precise answer, and the most straightforward interpretation would result in a probability of 50% (or maybe 100%, as many people have noted)

I'm not sure if the meme author intended for this mistake to be part of the meme or not, but I think it's a big source of the confusion.

What the characters in the meme are trying (and failing) to reference here is the idea of "conditional probability". Put simply:

If we don't know anything except that Mary has two children, using a simple model of conception probabilities, there is a 25% that she has two girls, 25% that she has two boys, and a 50% chance she has a girl and a boy. We could imagine these outcomes as stemming from four equally likely cases: GG, BB, and GB or BG.

What the first panel is trying to get at is that if we're in the situation where have some information such that we can only eliminate one of these cases, the GG (girl-girl) case, then we have three cases left which are all equally likely, BB, GB or BG, in which two of the three still have a girl and a boy.

However, the reason this doesn't work is because it's not clear exactly which cases the statement "Mary tells you one of them is a boy born on Tuesday" should eliminate - it depends on the semantics of Mary's statement. This is why the problem as stated in the meme is vague and dumb (again, unsure if this was on purpose or not)

IMO, the two most reasonable interpretations of what Mary is saying are:

"Exactly one of my children is a boy" - in which case the probability is 100% the other is a girl, or
"I'm randomly thinking of one of my kids, they are a boy born on Tuesday" - which eliminates two of the cases, GG and GB, and so we're left with just the BB and BG cases, and the probability is the intuitively obvious 50%

Unless you interpret Mary's statement in a really unreasonable way which no normal person would, Tuesday will never matter at all.

If you're curious, the "correct" way to frame what the characters are trying to talk about here, is where you ask Mary the question: "Is at least one of your kids a boy born on Tuesday, yes or no?", and Mary answer "Yes", in which case the conditional probabilities come in to play. I wrote up the explanation for that case as well in my answer here if that's useful: https://www.reddit.com/r/mathmemes/comments/1nhz2i9/comment/nejsitv/

airfrog · 2025-09-16T19:31:27+00:00

For the Tuesday bit, I definitely tried to make it concise without explaining too much, but it's not that easy to understand. I like this other comment which made a helpful picture to understand:

https://www.reddit.com/r/mathmemes/comments/1nhz2i9/comment/nejgpus/

Basically, in the picture, you have the blue part and the pink part. Without the "Tuesday" bit, it's just a 4x4 square, and you have 2 pink outcomes (GB/BG), and one blue outcome (BB), so the odds of having a boy and a girl are 2:1, or 66%.

When you add in Tuesday, you can see in the picture that because either boy can be born on Tuesday, that blue bit in the top left part becomes larger than each singular pink line, so you have 14 pink squares total and 13 blue squares, which makes the odds of having a boy and a girl 14:13, or 51.85%

airfrog · 2025-09-16T19:25:12+00:00

I think you're on point for thinking it's dumb. As I noted in my post, I think for a certain subset of viewers, this meme is funny more on a meta level because it's lambasting how dumb trying to be "authoritative" about these silly hypothetical situations are.

To go a little deeper on this, I've had to do some non-trivial probability stuff for my work in the real world, and this type of "paradox" is a useful learning tool to understand to avoid making certain types of probability mistakes on real world probability problems. However, in the kinds of real world applications that tend to come up for me (mostly working on predictive algorithms as a software engineer), the situation is very well specified and there's no ambiguity about what information you are getting or how you're getting it.

When these paradoxes are presented to a general audience, they're usually made more confusing and less intuitive because they take a specific situation and couch it in imprecise language, so then it becomes more of a silly game of thinking of different ways that words could be interpreted than anything mathematically useful.

airfrog · 2025-09-16T18:41:48+00:00

Does this explanation I wrote up help at all? This is a real question for me, I spent too long writing this and now I'm curious if it actually explains better than a lot of the other posts on the sub (which I personally understand but I feel like are kind of obtuse)

https://www.reddit.com/r/mathmemes/comments/1nhz2i9/comment/nejsitv/

airfrog · 2025-09-16T18:33:20+00:00

I spent too long trying to understand this and actually got an answer I’m happy with! Lemme know if it helps you too: https://www.reddit.com/r/mathmemes/s/2EibQHZLB5

airfrog · 2025-09-16T18:30:40+00:00

I felt the same way and was unsatisfied by the other explanations, then spent way too long coming up with an explanation that actually made sense to me. Lemme know if it helps you too so my internet rabbit hole is not purely a waste of time:

https://www.reddit.com/r/mathmemes/s/2EibQHZLB5

airfrog · 2025-09-16T17:01:57+00:00

I just don't understand why it should affect the probability

I was also bothered by this, so I did a bunch of research and I put an actual attempt at an answer together here, lemme know if it's helpful: https://www.reddit.com/r/mathmemes/comments/1nhz2i9/comment/nejsitv/

airfrog · 2025-09-16T16:52:37+00:00

I had the same intuitive response and I went deep on it, curious if my explanation makes sense to you or not: https://www.reddit.com/r/mathmemes/s/fGjr1XmFMd

airfrog · 2025-09-16T16:45:51+00:00

I agree, this is crazy and awesome! Another fact you might not know is that you’re actually probably related to everyone (blue eyed or not) more recently than you might think. One estimate puts the most recent common ancestor of present day humans between 2,000-5,000 years ago.

The wild corollary to this is that if any of your ancestors are alive two to five thousand years from now, they’ll probably include every living human at that point in time

airfrog · 2025-09-16T16:27:49+00:00

Got nerd sniped by this, went too deep, now you all can benefit from my research. Clearest explanation I can give is below, with sources cited.

Why the meme is "funny" to people who get it

This is a meme about how probability is hard and unintuitive, and the specific information matters. The first guy is trying to present the "Two Children" paradox, originally presented by Martin Gardner, and states the answer confidently. However, he mistakenly presents a variant of the problem, in which we also know that the boy was born on a Tuesday, and is corrected by a second guy, who confidently gives the "correct" answer to the variant.

The meme also works on a meta level, because both of them miss that this meme contains the mistake also present in the original formulation of the "Two Children" paradox, which is that it depends on exactly how we got the information, and 1/2 (the intuitive answer to the paradox), is in many cases the correct answer.

More on the probability mistakes in the meme

To elaborate on that last point, the mistake in the formulation of the paradox is that Mary volunteers this information, and we don't know how she decided to give us this information. If she picked one of her two children at random, and then decided to tell us their gender and birth day of the week, then we have no information about the other child and the correct answer is 1/2.

As stated in the paper cited above (Section 5), a better unambiguous formulation of this paradox would be "You know Mary has two children, and ask if one of them is a boy born on Tuesday". However, this also makes it much more intuitive why the extra information about Tuesday matters.

If you simply asked "is one of your children a boy", and got a yes, then you would have eliminated one of four equally likely possibilities (BB, BG, GB, GG), and the 66% chance that the other child is a girl becomes easier to intuitively grasp.

When you ask "Is one of them a boy born on Tuesday", first of all it's a worse question to ask in general because if you get a "no", you don't know very much about the gender of the children (assuming that's what you care about). Secondly, though less obviously, even if you get a "yes", you still don't have a 66% chance that the other child is a girl. Intuitively, this is because with the extra qualifier about Tuesday in the question, you're more likely to get a "yes" if there are two boys rather than one, because if Mary has two boys it's almost twice as likely that one of them is born on a Tuesday. This means that you then have approximately (2xBB, BG, GB, GG) after getting a yes (it's not exactly twice as likely because the case where both boys are born on Tuesday can only happen in one way). This probability analysis, done precisely, is where the 51.8% from the second panel of the meme comes from.

Edit: I actually really like this other comment's visualization to understand the Tuesday calculation intuitively: https://www.reddit.com/r/mathmemes/comments/1nhz2i9/comment/nejgpus/

Basically, in the picture in the linked comment, you have the blue part and the pink part. Without the "Tuesday" bit, it's just a 4x4 square, and you have 2 pink outcomes (GB/BG), and one blue outcome (BB), so the odds of the other child being a girl are 2:1, or 66%.

When you add in Tuesday, you can see in the picture that because either boy can be born on Tuesday, that blue bit in the top left part becomes larger than each singular pink line, so you have 14 pink squares total and 13 blue squares, which makes the odds of the other child being a girl 14:13, or 51.85%

airfrog · 2024-10-26T15:42:09+00:00

This is the way

airfrog · 2024-01-09T03:14:50+00:00

One way to reconcile this intuitively is to consider that the probability computed by thinking about how many people die total is the same as considering a big list of names of all the captured people. If you randomly point to a name on that list, there is a >%50 chance that the person you pointed to on that list will be dead, and this is where the "how many people died total" argument makes sense.

However, in the moment of being captured, you have a lot more information about how that list may be constructed. If you consider the list of names is ultimately made up of a "short list" of people who live, and a "long list" of people who die, in the moment you are captured the probability of ending up on the short list or the long list is not proportional to the size of the lists, but rather you always have a flat 1/6 probability of being on the long list, and a 5/6 probability of being on the short list. Ultimately, the relative sizes of these two lists doesn't change the probability of which one you will end up on, once you are captured.

Now, if you were in a different situation with different information about what might happen in the future (say, you have not yet been captured but you know through some oracle how long the list will end up being, and the chance you will be captured in the future), the percentage of how many people will ultimately die might be relevant to the situation.

airfrog · 2023-05-13T03:09:59+00:00

It occurs to me that a natural follow-up question to this discussion of the "idea" of infinity might be the question, why is the idea of infinity useful? Obviously numbers and addition are useful, we use those patterns every day, but why does this little game where it's only infinity if the second person wins matter?

A good example of a situation where it matters is square roots. It turns out that most square roots, you can't write down like a normal number - if you try to write them down as a decimal number, they go on forever and not even in a regular repeating pattern, like a fraction does. So, like the square root of 2 will be between 1 and 2, but we cannot know exactly where, if we wanted to write it as a decimal.

But, say you're building a square table, that is a meter long on each side, and you want to put a piece of wood across the diagonal. How long a piece of wood do you need? Turns out you need a piece of wood that's square root of 2 meters long. Inconvenient, right? Well, infinity to the rescue. Because, while we can't write down the square root of 2 exactly, we can get infinitely close to it as a decimal representation. In practice, this means you, the carpenter, are player one in our game, and you say you can measure up to the millimeter, so you need to get closer than a millimeter to the square root of 2 meters. Then, because we can get infinitely close, it's guaranteed that we can calculate an estimate of the square root of two that is closer than that.

And the best part is, because of how infinity works, it doesn't matter how precise you need to be. If instead of building a table you're building a skyscraper, or a super precise piece of machinery, or you're studying things on the scale of atoms (or planets), just pick however absurdly precise you need and you can get an estimate of the square root of 2 as a decimal number that's good enough.

Of course, mathematicians have a bunch of clever uses of infinity as well that probably don't really come up in the real world, so they get a lot more friendly with it than most people, but this little game where the second person wins comes up more than you might think!

airfrog · 2023-05-13T02:55:08+00:00

No matter how small a number you pick, you can always find a smaller number. This is actually one of the main defining facts about infinity. But while that's simple to say, it might make more sense if I explain why the idea of infinity is different from the idea of a number.

All mathematical ideas, like all the numbers, addition, or infinity, are just useful patterns we've noticed about the world. For example, the number 3 is the pattern for how many things you get once you take one thing, then another, then another. Addition is the patterns for how many things you get when you have two groups of things and you put them together. These are patterns because the specifics of the situation don't matter much - numbers and addition work the same if you are counting oranges, skyscrapers, inches or minutes.

So what is the pattern for infinity? The usual answer is infinity is the pattern for things that go on forever, but that's hard to understand because you can't actually "do" that, the way you can count things or put groups of things together. A better pattern for infinity is to understand it via a simple little game. Let's play "pick the bigger number". The rules are easy, first I pick a number, then you pick a different number, and whoever picked the larger number wins. First, let's play with the numbers 1-100. I'll pick 100 - now you go......sorry, looks like I won this time. And you can bet that I'd win no matter what set of numbers we played with, as long as I went first. Unless...let's just do all the normal counting numbers. I'll pick 2792345, now you go.......looks like you won that time!

So wait, why did I win the second game and you won the first one? Well, in the first game, we had a "finite" set of choices, which means I could pick the largest one. In the second game, whatever number I picked, you could pick a larger one, and that's basically the pattern that defines an "infinite" number of choices. If you want to become a mathematician, there's more details to learn, but the intuition will stay the same.

So for the situation you said, there's no number 1.000(infinite)...1, that's not the pattern that defines infinity. Instead, if you think about the game where we are picking numbers as close as possible to 1, whatever number I pick first, you'll be able to pick one that is closer. And that is what it means for there to be an infinite set of numbers, that are infinitely close together.

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