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Warp wrote:
The sum of the reciprocals of the squares of the natural numbers is π2/6. I'm wondering if this is just a coincidence, or if there really is an intuitive relation between this sum and the ratio between the circumference and diameter of a circle.
I was thinking about quickly replying with something to the tune of, "Probably not, because I can't think of any formulas in geometry that involve π2 rather than π alone." I decided that I should let more experienced mathematicians take a crack at it first. But I just had an epiphany in the shower and I'm a little more confident in saying there likely isn't any geometric significance.
Is there goemetric significance to every square number? (Or alternatively, every square root?)
As a very simple example, we know from physics that the kinetic energy is equal to 1/2mv2. Does that mean that lurking behind every moving object is a square with a side whose length varies proportional to the velocity? Or perhaps even more to the point, I know that when I drop an object from a height h, its speed will be v = sqrt(2gh) just before it hits the ground. Does that mean that "somewhere" there is a square I can draw with great physical significance that has a side of length sqrt(2gh)?
Probably not. Of course, I could arbitrarily introduce such a square, but I am having a hard time imagining some physical or otherwise intuitive reason for why it should have a side of that particular length. Velocity is squared in the equation for kinetic energy because, well, that's just how physics works. (You can also reason that kinetic energy should be proportional to v2 from symmetry arguments, but those don't invoke geometric squares either.)
This got me thinking-- and I may be far out of line in this conjecture-- that the insistence that equations be geometrically or otherwise significant may have been the lynchpin holding back mathematics for thousands of years. For example, I could imagine going back in time and telling a medieval geometer that kinetic energy is proportional to the square of the velocity and him replying, "Okay, but where is the square whose side is length v?" I would be unable to properly convey that the numbers are effectively confined to the chalkboard and upheld by experiment.
It seems that much of mathematical progress of the last 500 or so years has been related to the abstraction of mathematics and thinking of "numbers as numbers". A quick view of Wikipedia's page on the history of mathematics seems to confirm that math was essentially unused for anything but geometry and accounting all the way up to the time of Johannes Kepler (whose square-cube law represents perhaps the first empirical law based on the principles of geometry while lacking a geometric argument). (And as an aside, I also think it's fascinating that his first two laws are firmly rooted in classical geometry. I can imagine Kepler searching frantically for some squares and cubes that we know aren't there and just saying, "Welp, this is just the rule that it follows.")
If anyone has some knowledge of the history of mathematics and can shed some light on it, I'd be very interested in how close I am to being correct. But my official answer is that you won't find any circles or spheres that tell you why the sum of the squares of the integers is equal to π2/6.
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I think the funniest thing about this TAS is that the ring levels are the most entertaining. They come closest to "being Superman", flying through Metropolis with ease. The rest of the levels just look clunky in comparison.
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henke37 wrote:
Variable scope isn't a strange thing.
To you. I've never even heard of it. Wikipedia's article makes my head spin.
Warp wrote:
Bobo the King wrote:
What exactly is "me"? The player? If so, that seems redundant and confusing.
It's the name of the object that the function is taking as parameter. In the context of the program it's not explicitly clear what that parameter is, but it can be deduced that it's the object for which said function is called for (eg. if something collided with that object.)
Still not clear to me. I mean, "it can be deduced that it's the object for which said function is called for," is really hard to decipher. I get that all these little conventions make programming much more powerful, but it leaves people like me in the dust.
All this is reminding me how much I dislike object-oriented programming. It's bogged down in syntactical complexities and I never know (for example) whether the functions I call are acting on what's to the left or to the right of them.
Now get off my lawn, you crazy kids.
Edit: Here's my attempt at level 11:
Seems simple. Compiles. Even looks very similar to Bisqwit's solution. Execute the code and you get nothing. But Bisqwit's solution is syntactically foreign to me. It's a move command, but the argument starts with canmove, which should be a boolean. Ah, but it's followed by a question mark. Must be some kind of syntactic equivalent to an if-then statement. After that, it's 'right':'down' and I don't have a clue what that means.
I reiterate my claim that this game is great for learning JavaScript if you already know the tricks behind object-oriented programming and similar languages. And hell, knowing JavaScript itself outright doesn't hurt either.
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Bisqwit wrote:
Spikestuff wrote:
If I knew Java I would be doing this... but I don't.
It is not Java.
It is Javascript, which is a completely other language.
Additionally, as Patashu pointed out in the first post, if you know the principles of programming in general, you can always infer enough syntax from the surrounding code that you can get by without knowing any Javascript beforehand.
You'd think so, but I had a very hard time making significant progress in this game. I was able to reach level 9 on my own and get to level 10 by looking up level 9's solution. I got hung up on some very simple concepts. For example, many of your solutions (and some of the game's code) involve the object (?) "me". What exactly is "me"? The player? If so, that seems redundant and confusing.
I also noticed that your solution to level 7 was cut down to a few lines by use of an array. I didn't know how to construct arrays in JavaScript, so my solution was much less elegant.
On level 9, I copied "player.setPhoneCallback()" directly from the API and it gave me an error message:
TypeError: player.setPhoneCallback is not a function
Your code instead uses map.getPlayer().setPhoneCallback. Honestly, how was I supposed to know to use that? You also use an anonymous function, which I wouldn't confidently leap upon in a programming language I've never used before.
Don't get me wrong. I liked the game. It was fun and I was pleased to get as far as I did on my own and it did serve as a nifty introduction to JavaScript. I just don't think that it's as easy to pick up from scratch as you say.
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Warp wrote:
Bobo the King wrote:
Do we really need to have this argument every few months?
The very fact that the subject regularly comes up is an indication that the poll does not fully reflect what people want to vote.
I honestly cannot comprehend what the problem would be if the users would be given the option to vote what they want to vote. The only option given currently, if voting "yes" or "no" feels awkward (as I described earlier) is to abstain from voting, which in itself feels like some kind of protest against the system, which is not necessarily what one wants to do.
I really have come to the conclusion that this has become a question of pure stubbornness.
No, Warp, because the last time this came up (that I'm aware of), you were leading the charge, just as you're doing now. This isn't a whole bunch of users rising up to say, "Hey, this is a major problem with the site." Instead, it's mostly you trying to enforce your vision of what the poll should be.
Warp wrote:
I honestly cannot comprehend what the problem would be if the users would be given the option to vote what they want to vote. The only option given currently, if voting "yes" or "no" feels awkward (as I described earlier) is to abstain from voting, which in itself feels like some kind of protest against the system, which is not necessarily what one wants to do.
I really have come to the conclusion that this has become a question of pure stubbornness.
Yes, it is stubbornness-- on your part. You whine and whine and whine about how the non-binding poll is not uniquely tailored to suit your needs and are consistently told the same things: "judges aren't bound by the poll", "more valuable feedback can be (and is) left in the comments", "the poll provides feedback to the viewers, not the judges", and so on. And you seem to think you aren't being understood, or worse, actively ignored. No Warp, we all get you loud and clear. It's just that most of us (including the powers that be) think that there isn't a problem, or even to the extent that there is, it isn't worth making a fuss over.
Honestly, I think that this whole hullabaloo is a byproduct of working with very pedantic people-- programmers in particular. Everything is taken extremely literally and when faced with a choice of three options, none of which is quite what you're looking for, you think, "Program error!" and do the only thing you (think you) can and report it to the admins. When you're told, "We hear you. We don't care," you think, "But I still have the same program error!" and you just repeat yourself more loudly than before, insisting that it's a major impediment to you enjoying the site and you can't pipe down about it, regardless of how small the "program error" is.
Warp, you're so opinionated and so quick to dive into arguments, I'm often left baffled and want to know: Do you ever get tired?
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Do we really need to have this argument every few months?
Also, can anyone point to a specific example where judges made the "wrong" decision based on the results of the poll? I'd love to see one, since it would justify all the words we're spending on this.
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Twelvepack wrote:
Patashu wrote:
So it got renamed something like 'any% no memory corruption'
Then it was pointed out: "Isn't the chuck eat/orb glitch memory corruption? It puts something invalid into a memory address via an invalid process."
So it got renamed something like '11 exits'
No problem so far.
Patashu wrote:
But we all know the problem with that - you're looking at the thread dedicated to it!
The problem is not that this run was assigned to the '11 exit' branch, the problem, if anything, is that '11 exits' is a bad concept for a branch. Why not just pick 11 levels (not even including bowser's castle) then glitch to the credits? It would meet the requirements of the branch.
Pick a set of goals such that the run that completes those goals is entertaining to watch, not the other way around. (which would be finding an entertaining run and changing the goals so that nothing can ever obsolete it)
Anyway, whatever the outcome of the publication for this tas, thanks to masterjun for making it. You have done some fun and amazing things to this game.
Does TASVideos' formatting allow for very long branch names such as, "The specific 11 levels intended by the developer to comprise the minimum number needed to complete the game"? Because that's basically what you're suggesting.
And if you'll say, "Yes, that is what I'm suggesting," then fine. Do you really think that belongs on the front page? I'd rather have that stuff implied or at least behind the scenes.
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I would like to weigh in clearly in favor of rejection. What bothers me is that the TAS opens up arbitrary code execution and yet it's only used to alter the player's x- and y-positions. That means it's suboptimal as an "arbitrary code execution" run, which could instead jump straight to the credits (I assume).
On the other hand, it's presented as an 11 exit run, but that's problematic for the exact reasons Patashu gives. Basically, this run goes just far enough into glitchy territory to annihilate all records in the existing branch (and annihilating any entertainment value along with it), but not so far into glitchiness that it needs to be evaluated against similarbranches.
I try to evaluate TASes and not TASers, but I think Masterjun wants to have it both ways. This TAS isn't nearly as good as the other arbitrary code execution TAS, but it would obsolete the 11 exit TAS without any of the skills demonstrated there. Although we've obsoleted long runs before (I'm specifically thinking of runs like Chrono Trigger and Super Mario 64), that was through months or years of hard work, looking at the game's code, and hundreds or thousands of re-attempts optimizing a single task. It wasn't one runner sitting down and deciding he just wanted to use existing tricks from other runs to snatch up all the records for himself.
And for the record, I was entertained by the movie and I (along with just about everyone here) am a big fan of Masterjun's work. I even have a certain admiration for this run because it's close to what I was trying to do with my Carmen Sandiego and Beetle Mania runs: try to hit a sweet spot where the run becomes very difficult to evaluate under the current judging standards.
But my answer is still no.
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Wow! I'm so eager to sign up! I know the PayPal address isn't up yet, so can I just give you my credit card information directly?
Superfied Credit Union
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In Where in Time is Carmen Sandiego, you can search for the culprit by talking to an informant, talking to a witness, or using the scanner. The first option you choose never works. I never got around to checking the game's code for why that is.
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RGamma wrote:
Bobo the King wrote:
RGamma wrote:
...
I'm going to take a wild stab at this and say a + bx + cx2 corresponds to the first three terms of the Taylor series of ecos(x). I'll leave it to you to show that the limit approaches zero, especially since I have no clue how rigorous your analysis course is. (But the real reason is that it's been forever since I've done this kind of stuff...)
You were correct. A quick visual check with first three Taylor coefficients at x_0=0 (a:=e, b:=0, c:=-e/2). I'm honest to say I still don't really see the connection there or how you could see it at once. Tomorrow I'll be thinking more about this... too tired now. Thanks for the hint, I doubt I would have gotten a correct solution at all.
I obtained the same coefficients when I sketched out my answer.
As for how I did it so quickly... well, I'm not entirely sure myself. It's just that you do enough problems of this sort and you start to "read" them and see what kinds of points they're making. I think that's true of any subject. I still can't "read" a whole lot of mathematical proofs, nor can I "read" computer code. But Taylor series are incredibly practical, so I picked this one right up.
You have some function f(x). There is nothing to immediately suggest you should take the Taylor series of it, although when you do realize you should do a Taylor expansion, it helps to recognize that ecos(x) is analytic. (I.e., it is infinitely differentiable everywhere. Functions that only involve exponentiation, sines, and cosines tend to be analytic automatically and it is often unnecessary to rigorously prove that their Taylor series converge everywhere.)
From there, you provided a function g(x). The 1/x2 term out front is "bad" because it tends to infinity as x goes to 0. Let's disregard it for a moment and concentrate on making |f(x) - (a + bx + cx2)| go to zero. As you found out in your own attempt, it isn't enough to make the function go to zero at a given point (the 1/x2 screws everything up), so we really want to make it "as zero as possible". In other words, make it as close to zero as possible in the interval immediately surrounding x=0. Well, now we essentially have the condition that
f(x) ~ a + bx + cx2
for x approximately equal to zero, which is practically the definition of a Taylor series.
So we give the Taylor series a shot. Can we intuit that g(x) goes to zero as required? Yes. The (infinite) Taylor series of f(x) and a + bx + cx2 match up through the first three terms, so we are left with a polynomial that starts with the term x3. (In fact, the actual polynomial has coefficient 0 to the x3 term and it really starts with an x4 term.) That means that when we divide by x2 (the absolute value operation doesn't change things much), we are effectively left with a polynomial whose first term is no lower than x, meaning there is no constant term in this polynomial. That means g(x) goes to zero as x approaches zero.
Now, rigorously proving all this is another matter, and I'm forced to leave it to you for the reasons I gave above: I don't know how rigorous your course is and even if I did, I haven't done this stuff recently enough to be of all that much help. If your instructor is evil, they will make you do a proof involving epsilons and deltas (although if memory serves me, even that shouldn't be so bad for a function like this). If you've proved some theorems in class and are allowed to take "shortcuts", something like Taylor's theorem will be much help. Regardless, you should look at these problems as an exercise in building your intuition, not in memorizing a step-by-step series of instructions. As you can see, it is much more helpful to have a broad view of what is going on and to be able to "guess" a strategy rather than to embark on a direct proof from a completely naive perspective.
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RGamma wrote:
I don't want to turn this into a "help me with my Maths homework" thread or the like, but I've been stuck on this last task of an Analysis exam for quite some time and can't find a solution.. The task is as follows:
Let f(x) := e^(cos x). Choose coefficients a,b,c element R such that:
g(x) := 1/x^2 * abs(f(x) - (a + b*x + c*x^2)) --> 0 for x --> 0
Note: abs() is the absolute value function, --> denotes convergence.
My conclusion is that this task is impossible (and the way it is stated suggests there is a solution):
As x --> 0, f(x) --> e (because cos x --> 1). We get (g is > 0 and continuous on (0,infinity) so we can replace f(x) with e):
1/x^2 * abs(e - (a +b*x + c*x^2))
Because constants won't serve us a purpose, choose a := e:
1/x^2 * abs(-b*x - c*x^2))
= 1/x^2 * abs(-1 * (b*x + c*x^2))
= (b*x + c*x^2)/x^2
= b/x + c*x/x
= b/x + c --> infinity for x --> 0 regardless of b or c.
What am I missing here?
Edit: Imagine there is a lim_x-->0(..) on each line (and "= infinity" on the last) to obtain a more correct way of writing this down...
I'm going to take a wild stab at this and say a + bx + cx2 corresponds to the first three terms of the Taylor series of ecos(x). I'll leave it to you to show that the limit approaches zero, especially since I have no clue how rigorous your analysis course is. (But the real reason is that it's been forever since I've done this kind of stuff...)
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Warp, I like you. I really do. And although Tub's response to you was a bit acerbic, you are now in the process of derailing two threads in the Off Topic section. The common factor here is you. Drop it, please.
(Same goes for you, Tub, if you're tempted to reply.)
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I think it's funny how so many people cite "the run's original purpose" as if it were meant to be a particular way. The guy running Twitch Plays Pokemon can do whatever the hell he wants. He may have even had the democracy/anarchy system in mind when he started.
The system may not match your idealized view of the experiment, but your view is not special in any way.