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boyoboon Is the room open?

The factors of 27 are 1,3,9 and 27. That’s four factors. That's why door 27 is closed at the end of the open/closing process - employees 1,3,9 and 27 touch the door and so the final state of the door is the same as the initial state of the door.

Maybe you are talking about the number of prime factors. This quantity is irrelevant to door open/closing problem.

boyoboon Is the room open?

Simpler explanation for why

perfect square <=> odd number of factors

Proof: For every divisor d of n, n/d is another, different, divisor. This gives a pairing of all the divisors. The only exception is if n/d = d.

The exceptional case n/d=d only happens when n=d^2, i.e. when n is a perfect square.

Thus, when n is not a perfect square, there is a pairing of all its divisors, hence the number of divisors is even.

If n is a perfect square, the pairing pairs up all the divisors except for sqrt(n). Thus the number of divisors is odd.

boyoboon The “Windmill” Problem on the 2011 International Mathematical Olympiad [video]

I am giving an alternate proof of your implicit statement "There exists a line which has the same number of points on each side". You did this by computing duals of cell arrangements. I am arguing that you don't need to do that.

The proof I outlined will work for any point. Initially it might have the wrong number of points on both sides but for some rotation it will have the correct number of points on both sides.

boyoboon The “Windmill” Problem on the 2011 International Mathematical Olympiad [video]

> Check out William Bricken's "Iconic Math" http://iconicmath.com/ or "Proofs without Words" https://en.wikipedia.org/wiki/Proof_without_words or the other 3Blue1Brown videos for that matter.

Yes, those references demonstrate that some mathematical facts can be demonstrated easily, given the right presentation.

I agree that there is large gap between current mathematical presentations and the optimal presentation.

I am curious how effective the optimal presentation is. How easy can we make math? We have to be careful of the trap that the 3b1b video warns us against - when we understand something it is very difficult to put yourself in the shoes of a beginner. Something that looks like an elegant and clear presentation may seem like gibberish to the beginner (look at the YouTube comments on the 3b1b video).

I tried to google "Alan kay teaching calculus smalltalk" and didn't find anything. I am curious to see how much he actually taught the kids. It's clear that the typical college student, after taking a typical calculus class, doesn't really get the point of calculus. They may be able to follow some algorithms for differentiating and integrating but I don't think they understand when they can apply calculus.

boyoboon The “Windmill” Problem on the 2011 International Mathematical Olympiad [video]

Are you trying to say that math is hard because the teachers are withholding all the tricks?

> I am questioning their utility as an educational device.

Puzzles like this aren't found in mainstream math education contexts. As you acknowledged in your post, they are only found in math competitions. What do you mean?

boyoboon The “Windmill” Problem on the 2011 International Mathematical Olympiad [video]

Just pick a point and rotate a line around that point continuously. Keep track of the number of points on the left. Since this count is essentially continuous (the jumps are of size 1), at some point during this rotation you will have an almost-balanced configuration (same number of points on left and right, up to parity error).

boyoboon The “Windmill” Problem on the 2011 International Mathematical Olympiad [video]

> > say $150,000 to not teach 2 classes for a year

Is that why she is teaching two classes in her first semester, including one which is lower division?

Not really sure what you're saying. The fact that she's teaching just means that she isn't using (or doesn't have) a research grant to `buy out' of teaching.

> Usually you don't put the crappy teachers in the lower division classes, you give them graduate seminars.

Again, I don't know where you got this idea from. Usually (in a math research department such as the one at Stanford) whoever's arranging the teaching assignments doesn't look at an instructors teaching credentials at all, unless they are egregiously bad. So all we can conclude is that she isn't absolutely awful at teaching.

The fact of the matter is that many researchers (due to their incentives) view teaching, especially lower division courses, as a chore, so really anyone in the department who wants to teach such a course is not going to get much opposition.

boyoboon Interested in improving your relationships? Try Nonviolent Communication

Uh-oh, I think I might be doing this (as a college lecturer). Do you have some references, or some examples of how this might arise in a classroom?

boyoboon The exponential function is a miracle

If we just throw our hands up in the air and declare everything a "miracle" we never learn anything.

boyoboon The exponential function is a miracle

I would say it's not as interesting because a high-schooler (or Pythagoras) could easily verify the fact AND they would easily be able to explain how you came up with that sum.

One of my metrics for determining if a fact F is interesting is if P(F) is much larger than V(F).

Here P(F) is, vaguely speaking, the amount of effort required to "explain" or "generalize" the fact, and V(F) is the amount of effort required to verify the fact. In the case of e^{-x}, each individual example can be verified by an elementary schooler so V(F) is very small. On the other hand, I don't see how you could explain the phenomena of convergence to 0 without teaching the elementary schooler a large part of calculus, so P(F)/V(F) is large.

boyoboon The exponential function is a miracle

I'm not sure exactly what you mean here but there I can't think of an interpretation that makes your statement close to being true.

e.g. 1+4-9+16-25+36-49=-26

boyoboon The exponential function is a miracle

i= sqrt(-1)

https://en.wikipedia.org/wiki/Euler%27s_identity

boyoboon The exponential function is a miracle

Ah, I didn't realize that's what you were trying to say with the explanation at the beginning. Now your post makes more sense.

boyoboon The exponential function is a miracle

  Let's just let each other be excited about our favorite math, okay?

I don't think it's malicious, we're just trying to narrow down exactly what makes something interesting. In the process, we get to understand our favorite math more deeply.

  Nothing is surprising if you've seen it before.

On the other hand, if you can find a theory that makes a miracle look mundane, then the theory is probably interesting. So finding reasons why something is `not surprising' is a good guide to finding interesting math.

boyoboon The exponential function is a miracle

Fair enough. I do see how you could think that it's intuitive. I've even used your explanation when I'm teaching. Although I can't help but think there's some sort of circularity going on here, at least for me. My intuition on the subject is shaped by my knowledge of the theorem, and the functions that I tend to work with.

boyoboon The exponential function is a miracle

Everything you said seems to be valid if you replace 'exponential' by 'f' where 'f' is any real analytic function.

boyoboon The exponential function is a miracle

Your explanation is a nice restatement or interpretation for what the equation f(x)=f(0)+f'(0)x+.... is saying, but it doesn't give any insight for why the equation is true.

     and as there is no additional change that comes "out of nowhere", so to speak,

The whole explanation rests on this key point, and in fact this is the unintuitive part. In fact, it's not even always true. It's a "miracle" that it sometimes is true.

boyoboon The exponential function is a miracle

Analytic functions are usually defined to be those functions which can be represented by their Taylor series. This is what I meant by `tautological'.

boyoboon The exponential function is a miracle

Let's take "analytic" to mean "determined by its maclaurin series". It's not surprising that polynomials are analytic. Look at the definition of a polynomial. It's already in "power series" form, in fact there are only finitely many terms!

As for the other examples, here is one explanation for why they are ubiquitous in physics. Essentially, the class of analytic functions is closed under "solving differential equations".

So that's why sin/cos/exp/bessel are analytic - they are solutions to differential equations with constant/polynomial coefficients (we already know that constants and polynomials are analytic). That's why many functions found in physics are analytic - they are created from other analytic functions via a differential equation.

https://math.stackexchange.com/a/190167

P.S: Regarding your last statement "almost all smooth functions are almost polynomial", something much stronger is true: Every continuous function is almost a polynomial.

https://en.wikipedia.org/wiki/Stone–Weierstrass_theorem

boyoboon The exponential function is a miracle

Write down the first 50 terms of the series for e^{-x}. Go back in time and tell Pythagoras to plug in x=2,3,4,5,6,7,8,10,...

Each of the terms is a rational number, and the each of the sums will be an extremely small rational number (numerator will be tiny compared to denominator).

Don't you think Pythagoras would be impressed and mystified? Would he be able to come up with an explanation for this `coincidence'?

Now try the same with with x-x. That's not going to impress Pythagoras.

boyoboon The exponential function is a miracle

This seems pretty tautological to me.

  this is not true of all functions... But it is true for very large classes of functions

Which classes of functions? It sounds like you're saying that a function is determined by its derivatives if it's determined by its derivatives.

boyoboon Python Is Eating the World

I am a novice programmer who started with python and have recently been using javascript, where I use auto-indent-on-save.

Coming back to python is annoying because it's harder to make sure code is at the right indent level than to use { }.

In fact writing this comment inspired me to find this: https://github.com/mathialo/bython

boyoboon Information Theory for Intelligent People (2018) [pdf]

Ironically for me, the humanities people that it was aimed at often took it amiss. Dan Dennett told me as much. throwawayjava thank you for the suggestion; I'll use it in a footnote in the next iteration!

Yes, I thought this article was a very well written introduction, but I held back on sharing this link with some non-math friends because I didn't want them to feel dumb if they find it difficult to get through.

boyoboon Information Theory for Intelligent People (2018) [pdf]

It's an introduction to the fundamental concepts of information theory (entropy, KL divergence, conditional entropy), using 20 questions as an ongoing source of example applications and interpretations of these concepts. The author also sprinkles in various interpretations coming from their own research focus (the social sciences).

Definitely not a deep dive. The details can probably be found by chasing down the citations.

boyoboon I was a 10x engineer and I’m sorry

The problem is that we haven’t defined what we’re measuring.

E.g. Usain bolt makes far more than 10x money from running than the average healthy adult.

boyoboon YouTube bans content “showing users how to bypass secure computer systems”

I don't see how this solves the problem. What happens when your 'link aggregator'? decides to ban certain content?

boyoboon I turned my interview task for Google into a startup

How about this: People shoot in a random order (now everyone has the same chance of making it to the next round. Is it 1/e?)

  import numpy as np


  class Shooter:
      def __init__(self):
          self.dead = False
          self.right = None  # The person to the right


  def simulate(n):
      # Simulates a round with n shooters
      # Returns the ratio of survivors

      shooters = [Shooter() for i in range(n)]
      for i, shooter in enumerate(shooters):
          shooter.right = shooters[(i+1) % len(shooters)]

      np.random.shuffle(shooters)

      HIT_PROBABILITY = 1/6
      survivors = n
      for shooter in shooters:
          if not shooter.dead:
              if np.random.random() < HIT_PROBABILITY:
                  shooter.right.dead = True
                  survivors = survivors - 1
      return survivors/n

Simulations suggest that the survival probabilities do not converge to 6/7 if you add shuffling. This makes sense, since the survival probability for the random shuffling version must be strictly less than if you are guaranteed to go first.

  >>>sum([simulate(10000) for j in range(1000)])/1000
  0.8463075999999998

boyoboon Google .dev domain early access

In an idealized economic model, "rich developers" will get their desired domain names anyway as they can offer a large sum of money to the "poor developers" to give up their domain name.

https://en.wikipedia.org/wiki/Coase_theorem