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I'm with you, Warp. The Wii soured on people after a year for being "gimmicky" and I'm supposed to believe that VR is going to disrupt the industry? Gamers get ridiculously hyped for things all the time (Superman 64 and Lair immediately come to mind). They fail to recognize that core gameplay principles (not graphics) and an affordable price point will drive console sales and then are shocked at their disappointment. Then they do it all over again. Call me a wet blanket, but I don't think I've been hyped up about anything-- video game or otherwise-- for the better part of a decade. It only ends in pain.
Tagging for spoilers because I don't want to ruin other people's fun. FYI, I've marked the spoilers paragraph by paragraph, so you can follow along or get hints as need be.
Let the bread be 2 units by 2 units. Place the origin of the coordinate system at the bread's center and consider only the first quadrant. In fact, I'll only consider the portion of the first quadrant below the line y=x.Use polar coordinates to define the boundary between the eaten and uneaten bread. Because we are below y=x, the right edge is clearly going to be closer to a point on this boundary than the top edge. The boundary is such that the distance to the origin (r) is equal to the distance to the right edge (1 - r*cos(theta)).We have r = 1 - r*cos(theta). This can be rearranged to obtain r = 1/(1+cos(theta)).One-eighth of the area that's eaten is equal to the integral of this curve with respect to theta from 0 to pi/4. Because we're finding the area in polar coordinates, the integrand is 1/2*r*r*dtheta = 1/2*r^2*dtheta.We have the integral from 0 to pi/4 of 1/2*r^2*dtheta which is 1/2 times the integral from 0 to pi/4 of 1/(1+cos(theta))^2 dtheta.Plug this integral into Wolfram Alpha. What? Did you seriously expect me to do that integral myself? See for yourself. It's a monster and it requires an obscure u substitution of tan(theta/2) to put it into a solvable form.The definite integral (excluding the factor of 1/2) is (4*sqrt(2) - 5)/3.To find the fraction of the area that's eaten, multiply this by 1/2, then multiply by 8 because we only found one-eighth the area eaten, then divide by the total area of 4. This is 1/2*8/4 = 1, so just multiply by 1.The fraction eaten is (4*sqrt(2) - 5)/3 which is approximately 0.21895.
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Nach wrote:
Bobo the King wrote:
We can (and have) run experiments to show that there are no underlying variables.
That show there are no underlying variables? Or that there is no explanation yet which accounts for the possibility of underlying variables?
Further all the experiments I read about said they only proved no local variables. Has there been more tests?
No underlying variables.
You can quibble about potential non-local hidden variables, but you need to get up to speed with the theory first. Non-local hidden variables are generally scoffed at because they are non-falsifiable and violate special relativity.
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Nach wrote:
Bobo the King wrote:
That's the thing about quantum mechanics: the Heisenberg uncertainty principle says that you cannot know all the variables!
And although the uncertainty principle is relatively easy to grasp to the point that high school students are regularly exposed to it, it is actually a consequence of deeper principles at work. For the sake of this argument, however, if you believe Heisenberg, you've already refuted your own notion of determinism.
Your logic doesn't follow for me, unless you're saying because we cannot know the variables, therefore they do not exist.
Just because we cannot know how something is determined does not mean nothing is determining it.
This is exactly what people keep trying to hammer home for you and is the central point of Bell's theorem. There are no hidden variables. Your phrasing that "they do not exist" is troublesome because it's not clear what you mean. Measurable quantities exist (i.e., knowing a particle's position does not mean its momentum "does not exist", it just means it is undetermined) but having information about one quantity certainly affects how much we can know about other quantities simultaneously. You cannot know the position and momentum of a particle simultaneously and you cannot know the spin states of a particle along two axes that aren't colinear.
As was described in the Wired article earlier, this argument is distinct from the old "If a tree falls in a forest..." adage. I am not simply saying that we cannot know the underlying variables and so they might as well not exist. I am saying that if such variables existed, their predictions would be inconsistent with the predictions of quantum mechanics. We can (and have) run experiments to show that there are no underlying variables.
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I was working on a longer reply to your previous comment but accidentally refreshed the page and lost that work. I'm afraid I won't be retyping it, but please let me know if you would like my input on any particular points. In lieu of that, let me address your latest post.
Nach wrote:
Derakon wrote:
Then please explain why you have any reason to expect that physics be deterministic at any level.
Because based on everything I've ever seen and experimented with myself: If all input variables are known and the algorithm is known, then the exact output is always calculable there is no room for "surprising" results.
Then you have never seen or experimented with quantum mechanics. In it, there are limits to how much we can know about a system and the output is probabilistic in nature. This probability is not governed by hidden variables or the chaos seen in complex systems, rather it strongly appears that it enters into the theory "on the ground floor" so to speak.
Nach wrote:
I can't even comprehend how exactly we can know all the variables for something, and all the algorithms involved, and not know what the result would be. It even leads to all kinds of questions how that something even operates being known through and through.
That's the thing about quantum mechanics: the Heisenberg uncertainty principle says that you cannot know all the variables! And although the uncertainty principle is relatively easy to grasp to the point that high school students are regularly exposed to it, it is actually a consequence of deeper principles at work. For the sake of this argument, however, if you believe Heisenberg, you've already refuted your own notion of determinism.
Nach wrote:
It would essentially mean that something has free will, which this topic is about. I have a hard time believing humans have free will, let alone minute particles that cannot be subdivided further. If we then postulate those particles have free will to do whatever, why aren't very strange things happening occasionally?
Despite what New Age hucksters will tell you, quantum mechanics has essentially nothing to say about free will. It does have plenty to say about determinism and it is strong indication that we live in a probabilistic, not deterministic universe. But if your own desires are governed by the laws of probability and not by some underlying volition on your part, does that somehow reignite free will?
The one thing that quantum mechanics does is free us from the bounds of strict determinism, which is what you have talked about throughout much of this thread. It does so by swinging the pendulum too far in the other direction, saying that probability governs the universe and there is nothing within the theory that suggests that particles (or human beings) have anything resembling free will. From a very optimistic viewpoint, there may be some sort of new theory that either subsumes quantum mechanics or perhaps bridges the quantum and macroscopic worlds and this new theory might allow for the intuitive notion of free will, but that is purely speculative. No such theory exists-- not while passing the muster for scientific rigor, at least.
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Nach wrote:
When did I say I fully understand the physics behind QM?
I seem to have misinterpreted your previous statement. I did, however, find this comment of yours earlier in the thread:
Nach wrote:
Bobo the King wrote:
Nach wrote:
Which is probably just wishful thinking if not outright baloney. Every perceived possibility for randomness is simply inability to understand the variables or algorithms involved.
This is strongly suspected to be false. To fully understand why, you'll need to thoroughly study RGamma's link on hidden variable theories as well as things like quantum entanglement and the Bell inequalities. We have strong reason to believe that probability naturally arises out of quantum mechanics and attempts to explain quantum mechanical phenomena as a consequence of probability theory results in predictions that are not upheld by experiment.
I've read up on all this material, I find it entirely unconvincing. All the experiments can prove is that so far there is much phenomena that cannot be explained by existing tools. I would go so far as to say that determinism is unfalsifiable, anything which proves non-determinism can be viewed in a different perspective to prove lack of knowledge.
Your last two sentences demonstrate that you have essentially no understanding of Bell's theorem.
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Nach wrote:
OmnipotentEntity wrote:
It's that for all intents and purposes, we have no reason to believe that there is determinism underneath. And absent of a very good reason to believe otherwise, all you are bringing to the table right now is wishful thinking. An argument from personal incredulity.
My reasoning is very simple, based on every study I've conducted with physics I fully understand, I see that "If all input variables are known and the algorithm is known, then the exact output is always calculable there is no room for "surprising" results" always holds true. Most of those studying QM want to claim this is not true of QM, despite being true for everything else. Perhaps, but they have 0 proof that all input variables are indeed known and that the "algorithm" is all known. There is still much study in the field of QM leaving the question of "algorithm" open, and the fact scientists are still occasionally proposing new variables to look for shows that there is no proof yet that all variables are known. Until there is proof that all variables are known, instead of proof against proposed variables, and knowledge of the field by all scientists is deemed complete, I find all the arguments entirely unconvincing. It is wishful thinking to assume everything is known, when there is no proof for that.
Ugh, Nach. Please stop. OmnipotentEntity is right. You really don't know what you're talking about and you're doing yourself no favors by insisting that you "fully understand" the physics behind it when you clearly do not.
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Derakon wrote:
Is it possible that nondeterminism is fallout from some equivalent to Goedel's Incompleteness Theorem as applied to reality?
I don't honestly know if what I just said is complete gibberish or a legitimate question. It does kind of feel like "it is impossible to make a consistent Theory of Everything for physics" is a statement of similar strength as "it is impossible to make a consistent, complete system of axioms".
From my limited understanding, no, that's not complete gibberish. To my knowledge, physicists, mathematicians, and philosophers have tried to connect Godel's incompleteness theorem to the physical universe from around the time it was published.
There are some issues with attempts to do so, if I'm not mistaken:
The universe is non-deterministic. As you point out, this potentially upsets application of Godel's incompleteness theorem because mathematics tends to be concerned with deterministic truths while the universe is probabilistic in nature.
The universe may be finite. Godel's theorem deals with very large numbers, so perhaps the universe is simply not large enough to contain a self-contradictory system. Hey, I suppose that even with an infinite universe, cosmic inflation may still prevent large enough systems from emerging.
Can a one-to-one correspondence be established between mathematical axioms and physical laws? I believe this is the biggest open question. I do not know if basic mathematical operations such as addition and multiplication or even the successor function can be written as a direct consequence of physical laws. How can the universe violate Godel's incompleteness theorem if axioms do not meaningfully exist in our universe? This question would need to be addressed before assuming that the universe is non-deterministic specifically because of the incompleteness theorem.
If I somehow haven't emphasized it enough, I have a relatively poor understanding of this subject, so don't take my word as gospel.
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Warp wrote:
People having an actual discussion about a topic is an embarrassment to this website... Why, exactly? Because the subject is feminism?
Because it is full of straw-man arguments from people who seem to know next to nothing about feminism aside from what they read on Men's Rights forums.
Warp wrote:
I think that's called white knighting?
And if a woman takes a stand on your behalf when it comes to certain men's issues (which they should!) is that also white knighting? Ask some women what they think about "white knighting". Yes, advocacy can be taken too far, but I'm simply arguing that you should gently speak out when you see something that's unfair to women. Don't talk to me about it. Talk to important women in your life.
Warp wrote:
Criticizing the part of the movement that has the most influence in society, the part that has the loudest voice and which can do the most harm to society, is completely warranted. "Not all X are like that" is not an excuse to deter criticism of that loud minority.
You have a point here. The segment of feminism that you refer to certainly deserves criticism for lowering the discourse. But what you have done is made blanket statement after blanket statement, insisting that all feminism is like that. Be the bigger person, raise the level of discussion.
Warp wrote:
That's the portion of the movement that causes changes in policies and laws, changes that are often veering towards the unconstitutional, and are against the fundamental human rights. The part that promotes and engages in gender and racial discrimination, and would want to enforce it by law. Criticizing this portion of the movement is completely warranted, and even urgent, even if "not all" feminists are like that.
I have, in fact, some time ago written a more or less formal statement of why I oppose modern "progressive" feminism, and rather than repeat it here, I'll just link to it: http://grindedgear.blogspot.fi/2016/01/why-i-oppose-progressive-feminism.html
I find your views to be thoroughly out of line with reality. Look past Tumblr and other superficial sources and try to find actual feminist discussion. Yes, there are some real scoundrels with simplistic worldviews there, but there are also very good and thoughtful people who are trying very hard to do what's right. And that's all I'm advocating for you-- try to be thoughtful, do what's right, and make the world a better place.
Eh, whatever. I tried.
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This thread is an embarrassment to this website.
I'll spare you all a long diatrabe and I'll try not to divulge my own viewpoints, which are actually quite nuanced. I'll just say that all of us-- men, women, whatever you identify as-- should work to make the world a more equal place for everyone. Men (since there seem to be quite a few vocal men in this thread), please heed my advice: stick your necks out just a little bit for women. Try to recognize some of the difficulties that women face in your communities. Ask them what you can do to help. Get out of your comfort zone and make a small but earnest effort to address women's issues. It doesn't have to be about the wage gap, which I know is quite controversial. It could be about reproductive rights or rape or anything else.
And you don't need to identify as a feminist either. I think feminism gets a bad rap these days, but some criticisms towards it are quite valid. I myself find the movement to be nebulous and unfocused, but most of the critiques I see here (I'm looking at you, Mitjitsu and Warp) are far off-base or paint it with too broad a brush. There are extensive discussions within the feminist movement about where it should stand on just about every issue and although you don't see them and they're not integral to Tumblr feminism doesn't mean they aren't hotly debated.
I don't want to go too off topic, but permit me to make an analogy. I recently saw a popular article about a Super Mario World speedrun that achieved code injection to jump to the ending. The comments in that article came mostly from non-speedrunners and they said something to the effect of, "It's neat, but I wouldn't say it actually beat the game. Rather, it reprogrammed the game to jump to the ending." Of course they're entitled to their opinion, but I wanted to scream to them that what does or does not constitute beating a game is very thoroughly debated within speedrunning communities and their input isn't particularly valuable.
That's what you're doing when you assert that feminism collectively believes one thing or another.
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I'd argue the game industry has been in a pitiful state for the last 20 years (I point to Sony's entry) and it is getting worse. There's far too much for me to talk about here, so I'll just touch upon a few simple themes.
First and foremost, I've grown up and the video game industry hasn't. This is part of what happens whenever anyone pines for the "good ol' days". All of the criticisms that follow have been a part of the video game industry since well before the modern era. Having said that, I also think that there has been an "erosion of values".
The video game industry is alive and well if...
You like first person shooters.
You like fantasy games.
You like MMORPGs.
You like multiplayer games.
You value graphics above all else.
You are a white male.
And look, that's not to specifically criticize white men or anything, but there are plenty of other demographics-- women especially-- who would love to get into video games but are repulsed by the community that flocks there. The video game industry used to be about creativity, creating new genres and carving out niches in its fanbase. For every Contra, there was a DuckTales or Dr. Mario or EarthBound. A lot of the whimsy has left the industry in favor of super-high-budget games that cater to the hypermasculine crowd. The whole "gaming culture" has turned incredibly toxic. If you go to gaming subreddits, you'll find circlejerking over the latest releases (I suspect much of this is planted promotional material), miscellaneous depravity, and blatant racism and sexism that self-identifying gamers refuse to acknowledge. It's the Mad Max culture come to life.
I'm an avowed Nintendo fanboy. In the face of increasing violence and sexism, Nintendo has realized that they cannot compete directly with Sony and Microsoft and I applaud them for it. They've consistently tried to do new things with games while keeping most of their library appropriate for all ages and, most importantly, fun. I bought a Wii on the day it was released and I don't care what anyone says, it was a damn good console and I enjoyed the hell out of it. I recognize, however, that Nintendo is struggling to stay afloat at this point. With the previous generation of consoles, they at least got a mention against the X-Box 360 and PS3. Today, whenever anyone talks about console wars, it's always Sony versus Microsoft and I think that's doing a terrible disservice to a major player that is actually attempting to do new things in the industry. Unfortunately, they're also suffering from sequel-itis and Nintendo seems to be financed by perennial Mario, Zelda, Smash Bros., and Mario Kart games. That's just as true of other systems, though. How many Call of Duty or Madden games do we need at this point?
My point is that Nintendo is running out of steam but Microsoft and Sony just disgust me outright. If Nintendo leaves the console market, then yes, I think we're headed towards a crash. Even with them in the industry, they need new talent and new ideas and I just don't see where they're coming from at this point.
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Nach wrote:
Which is probably just wishful thinking if not outright baloney. Every perceived possibility for randomness is simply inability to understand the variables or algorithms involved.
This is strongly suspected to be false. To fully understand why, you'll need to thoroughly study RGamma's link on hidden variable theories as well as things like quantum entanglement and the Bell inequalities. We have strong reason to believe that probability naturally arises out of quantum mechanics and attempts to explain quantum mechanical phenomena as a consequence of probability theory results in predictions that are not upheld by experiment.
As for the posed question, I am not the least bit religious but this is one instance where I have "belief" of a sort. I do believe that free will exists, not because I have any evidence for it (to the contrary, RGamma outlines a decent argument against it) but because the alternative is just too damn horrifying for me to accept. There is so much suffering in the world and the ideas that it was all predestined to occur and/or that we can exert effectively no control over our lives gives me the heebie-jeebies.
I realize that the universe doesn't care how much I'm creeped out by it. I also understand that there is much evidence that free will almost certainly does not exist for short timescales-- scientists can accurately predict in controlled lab settings what test subjects will do seconds before they believe they have made up their minds. Nevertheless, I think that there's something deeper going on that will take a long time to understand if it even can be understood. The presence of consciousness is not predicted by science or mathematics and I think there may be some injection of free will there, although I cannot offer any sort of mechanism. In fact, I'd argue that there are hints that free will might even lie outside the realms of mathematical modeling. Math does a great job of answering deterministic questions and a pretty good job of answering probabilistic ones too (although there is a surprising amount of philosophy in probability theory). As far as I know, math has never so much as touched the subject of desire. If free will exists, it is an expression of our (whoever "we" are) wish to interact with the physical world in a particular way. That's not modeled by the deterministic mathematics, nor by the probabilistic stuff. Instead, it's as if a math theory were to state, "two plus two wants to be four," which doesn't make a whole lot of sense to me.
That's about all I can contribute on the subject.
This is correct, although I'd like to emphasize that you are unlikely to use this information early in your studies of TASing and/or reverse engineering games. Once you start working in assembly language, which could be anywhere from weeks to years from now (or never), zero page addressing becomes significant. Even then, it remains mostly of academic importance, except to say that the addresses in that range are the most frequently called upon in the game's code.
For now, I would focus on the basics of RAM searching/watching and not get bogged down in ROM structure (which I still understand relatively poorly). Nevertheless, if you have any questions on anything from basic to advanced, other users and I will be happy to answer them.
I'd like to encourage you because so far you've asked some good questions; a lot of newbies to the site dive right in and produce sub-par work because they fail to study the material in advance.
Edit: By the way, if you're still interested in finding hitboxes and are comfortable with Lua scripting, you might use this tool, made by yours truly and FatRatKnight some years ago. I never got around to improving it up to the point I wanted and you may even need to download an old emulator to make it work, but if you can get it running, it should sniff out hitboxes very quickly.
Regardless of whether you use the tool, you should know more or less what to look for when conducting RAM searches. For example...
• Is the value signed or unsigned? That is, can it take on negative values or is it strictly positive?
• Is it something "exotic" like binary coded decimal? BCD is frequently used for on-screen displays.
• Where is the corresponding address likely to be located relative to other addresses? Hitboxes are a great example of this because once you find the x- or y- coordinates of one corner of one hitbox, you are likely to find nearby (within, say, 20 bytes) not only the corresponding y- or x- coordinates of the same corner but also the coordinates of the opposite corner. You'll also often find things like enemy health, armor, weapon, various flags (invisibility, invincibility), etc. in the same vicinity. Taking things yet further, you'll likely find corresponding data for all enemies in a big table that can be pretty simply represented.
• How is the value expected to behave? Again, hitboxes are great to search for with this because you can fairly easily track them. When the enemy moves left, its x-coordinate almost certainly decreases. When it moves right, its x-coordinate increases. If you conduct your search carefully based on those principles (search for RAM values with the "less than" or "greater than" conditions), you'll often find the address you're looking for very quickly. It's worth noting that in 2-D games, the vertical axis tends to be flipped from how you're used to it in your math classes. Values increase downwards and decrease upwards. This might explain some difficulty you may be having in finding hitboxes.
And I'm sure there's plenty more that I'm overlooking right now.
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DaTeL237 wrote:
Amaraticando wrote:
The lines are tangent and the angle is not given, as it is determined by R and h. The "towers" are perpendicular to the "ground".
I'm not sure if I'm interpreting this correctly... I have the impression that if the two 'towers' are the same height (h) and the lines are tangent to the circle then a=b=0
If you don't have trouble imagining it in three dimensions, the question is basically asking...
Suppose you are standing at the top of a tall tower and can barely see the top of an identical tower some miles away. If you were to climb a spire on the top of your tower, how far down the other tower would you be able to see?Amaraticando wrote:
The lines are tangent and the angle is not given, as it is determined by R and h. The "towers" are perpendicular to the "ground".
I see that now. I'll add one more line to my solution above.
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Masterjun wrote:
Quick reminder that we have [sup][/sup] and [sub][/sub] for neat superscript and subscript for anyone that likes to polish up their replies.
I've used them before, but I'm in kind of a lazy mood. This problem has too much going on in it for me to put in subscripts and superscripts everywhere, although I may include them in my final answer.
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Amaraticando wrote:
I was thinking of flat earth bullshit and this problem came to my mind. The answer that I got was pretty big and complicated for a seemingly simple problem, let's see if some of you finds out.
Click to expand.
Are those two lines both tangent to the circle?
Also, are we given theta? It seems very likely that the answer depends on the angle subtended by the line segments (the "towers"?).
Edit: After a few minutes of work, I've applied the Pythagorean theorem twice and the law of cosines once to obtain
c_1^2 = (r+h+a)^2 - r^2
c_2^2 = (r+h-b)^2 - r^2
(c_1 + c_2)^2 = (r+h+a)^2 + (r+h-b)^2 - 2*(r+h-a)*(r+h-b)*cos(theta).
The first two equations can be derived by considering a line from the center of the circle to the point where the red line intersects. If we are given theta, this constitutes three equations and three variables (c_1, c_2, and b) and so the rest is "trivial". I'll do the algebra anyway. Expect another edit in five or ten minutes.
Edit 2: So here's my answer, hastily derived in Notepad:
v = u*[cos(theta) +/- sqrt(cos^2(theta) - 1 + u^2*sin^2(theta))]/(1 - u^2*sin^2(theta))
where
u = (r+h+a)/r,
v = (r+h-b)/r
and so the above expression for v is written entirely in terms of known quantities. To find b explicitly, simply solve for v above, then take
b = r+h-r*v.
I may do a little more algebra to see if there's any more simplification to be done.
Edit 3: Last version of my solution. I hope this one is more readable than the previous one.
v = u*(cos(theta) +/- sqrt(2*w + w2))/(cos2(theta) - (2*w + w2)*sin2(theta))
where
u = (r+h+a)/r,
v = (r+h-b)/r,
w = (r+a)/r.
As before, solve for u and w (both written in terms of known quantities), then plug both into the right hand side of our expression for v. After finding v, take
b = r+h-r*v.
By the way, I'm not sure whether to take the positive or negative square root of the expression. It's very likely that one solution is non-physical.
Edit 4: I should have the correct answer now. It's the same as before, except with
theta = 2*cos-1(r/(r+h))
There is likely a way to remove all trigonometric dependence. I guess I'll work on that...
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FractalFusion wrote:
All this problem is, is what Bobo the King said above with the ratio tending to 3:2:1, and the negative binomial distribution applied to the expected number of total drops required to get one of each gem.
If probability of success is p and probability of failure is q=1-p, then the expected number of trials required to obtain success is 1(p)+2(qp)+3(q2p)+... = p(1+2q+3q2+...) . Using the identity 1/(1-x)2 = 1+2x+3x2+... , the expected number is p(1/(1-q)2)=p/p2=1/p. Ex. The expected number of rolls of a 6-sided die required to roll a 1 is 1/(1/6)=6.
To calculate the expected number of total drops required to get one of each gem, condition on the order in which the types of gems are first encountered. That is, we calculate:
1 + P(Common)*{E(# of drops to get Uncommon or Rare) + P(Uncommon|Uncommon or Rare)*E(# of drops to get Rare) + P(Rare|Uncommon or Rare)*E(# of drops to get Uncommon)}
+ P(Uncommon)*{E(# of drops to get Common or Rare) + P(Common|Common or Rare)*E(# of drops to get Rare) + P(Rare|Common or Rare)*E(# of drops to get Common)}
+ P(Rare)*{E(# of drops to get Common or Uncommon) + P(Common|Common or Uncommon)*E(# of drops to get Uncommon) + P(Uncommon|Common or Uncommon)*E(# of drops to get Common)}
=
1 + (1/2)*{2 + (2/3)*6 + (1/3)*3} + (1/3)*{3/2 + (3/4)*6 + (1/4)*2} + (1/6)*{6/5 + (3/5)*3 + (2/5)*2}
=
1 + (1/2)*7 + (1/3)*(13/2) + (1/6)*(19/5) = 1 + 7/2 + 13/6 + 19/30 = 30/30 + 105/30 + 65/30 + 19/30 = 219/30 = 7.3
Since the ratio of the types of gems tends to 3:2:1, the expected number of common, uncommon, and rare gems, respectively is 7.3/2 = 3.65, 7.3/3 = 2.43333... , and 7.3/6 = 1.21666... . This is consistent with the numbers given by OmnipotentEntity's program.
Bravo, FractalFusion! I knew it had to do with the negative binomial distribution but I couldn't quite put all the pieces together. Your explanation is clear and it appears to be correct.
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rhebus wrote:
The answer cannot be 3:2:1. You will always get at least one of each gem, but you have a one in 36 chance of having 2 rare gems from the first two monsters.
This puts a lower bound on the expected number of rare gems at (1/36 * 2 + 35/36 * 1) = 37/36. The exact number will be higher than this, but it's enough to disprove the 3:2:1 answer.
I'm not so sure about that. A standard argument in favor of 3:2:1 goes as follows:
Line up several players. Name them A, B, C, etc. for clarity.
Instruct player A to continue killing enemies until they have at least one of each gem, then stop.
After player A is done, instruct player B to continue killing enemies until they have one of each gem, then continue on to player C and so on.
After this has been done many times, examine their totals. Surely some will have ratios different from 3:2:1, but as an aggregate, their procedure was no different from one player collecting gems consecutively without restriction. Because this (imaginary) player would surely collect the gems in the ratio 3:2:1 and our individual players are indistinguishable from one another, surely the individual ratios are also 3:2:1.
Someone please fill in any gaps in my proof or correct me if I've made a mistake.
Edit: Okay, I now see that two different things are being said. The ratio will be 3:2:1, but that doesn't mean the player will pick up 1 rare gem on average. It seems no one has (correctly) determined the expected number of gems to be picked up. I'll think about it for a bit, but someone else is likely to figure it out before I do.
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This looks like a parody of a video game. As in, it looks like something a non-gamer would imagine a video game to be like in the mid- to late-'90s. Even the credits are headache-inducing.
That's not a critique of the run itself, which looks fine. I'm just making a stray remark.
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Masterjun wrote:
Randil wrote:
2. Is s(n) bounded above or below? My intuition says no, but I cannot formally prove it.
3. Consider the histogram of the values s(1), s(2), ..., s(n), for ever increasing n. I suspect this should converge to a normal distribution, but again I cannot formally show this. If it does converge to a normal distribution, what are the parameters of said distribution?
Sounds like this problem would be easier to solve if we'd know whether pi is a normal number or not.
Wikipedia wrote:
it is widely believed that the numbers √2, π, and e are normal, but a proof remains elusive.
Yeah, that makes generalizing a proof to other irrational or even transcendental numbers almost impossible.
For example, a number like
0.90900090900000909000909000000090900090900000909000909000000000909...
is non-repeating and therefore irrational (possibly even transcendental) but increases without bound. Its histogram also looks nothing like a normal distribution.
We could play a similar game and construct this number:
0.110110011000110000110000011000000110000000110000000011...
which is certainly irrational because it never repeats. With the 1s always consecutive, s(n) always takes the values of -1, 0, or 1, so it is bounded from above and below.
Now, if we were talking about normal numbers I'd have no idea how to proceed, although I suspect that you could construct similar numbers that would be bounded from above and/or below as you please.
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This is a very minor point, but I'm still curious. You comment that it saves time to approach trainers by up to 3 steps (7 if bicycling). Does this include the additional time it takes to step around them afterwards? I know it's standard procedure to approach trainers, but it just seems fishy to me that you'd want to walk up and back rather than wait for the exclamation mark to pop up, especially if they're already next to you.
Click to expand.