Posts for Warp


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If I understood correctly, what he's essentially saying is that x^x^x^... = y, and x^y = y are not exactly the same thing, in a similar way that sqrt(x) = y, and x = y^2 are not exactly the same thing. They behave the same within a certain range, but not outside of it. The "validity range" for the former is just a bit more complicated than for the latter.
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blackpenredpen once again provides us with some interesting math. If you have an infinite power tower equaling 2, it's quite easy to solve what x is: x^x^x^x^... = 2 Simply notice that since the entire thing is equal to 2, then the power tower starting from the second terms is also equal to 2, so you get: x^2 = 2 -> x = sqrt(2) So, the same should be the case with any other value, right? Like: x^x^x^x^... = 3 -> x^3 = 3 -> x = 3^(1/3) Except that for some reason that's not the case. If you start raising the cube root of 3 to itself over and over, the value does not approach 3, but something else. And, apparently x^x^x^x^... = 3 has no solution at all. There is no value of x for which the power tower is 3. It's not at all apparent why this is. Why does the "substitution trick" work with 2 but not for 3 (or any larger values either). It only works when the value on the right-hand-side is within a certain range. (If you don't know what this range is, that's a math challenge for you.) But why is this so? The only conclusion that I can draw is that even the original substitution trick was mathematically invalid, and it happened to give the correct answer by pure chance. There doesn't seem to be any obvious reason why the trick would be valid when there's a 2 on the right-hand-side but not when there's a 3. (Although I suppose there may well be a mathematical reason for it, but it's certainly not obvious.)
Post subject: Politics in 2020
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[MOD EDIT: Original topic: Thread #21939: itch.io Bundle for Racial Justice. 1700+ items for $5!] [MOD EDIT 2: Reminder to try to keep debate civil and follow the rules. Refrain from personal attacks or name-calling.] I hope this doesn't open the floodgates for making tasvideos.org a website for political activism.
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Ramzi wrote:
I'm just here to not answer Warp's question.
What do you mean to say with this?
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For the life of me I cannot remember if I have asked this before in this thread (it's 116 pages, and apparently the search functionality of this version of phpbb is broken in that you cannot search for posts containing all of the search terms; or at least I can't figure out a way to do it; the setting that supposedly selects "search for all the terms" seems to have no effect), and I'm too lazy to browse through 116 pages of posts, so I apologize if I already asked this and it was answered. Anyway, in this video blackpenredpen casually does ln(-2) = ln(-1) + ln(2) This got me thinking whether the logarithm rule is valid when the argument is negative. (After all, there are lots of rules in math that are valid only when the arguments are valid. Ignoring such rules is commonly the basis for "proofs" that 1=2, etc.)
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I freely admit that was so complex I'm not even sure it's an answer to the problem I posted...
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Prove that no partial sum of the harmonic series is an integer (other than, rather obviously, the very first term, which is just 1). (And by "partial sum" I mean 1+1/2+1/3+...+1/n for any given n.)
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feos wrote:
It's meaningful because it gives an interesting piece of information: How fast can the game be completed theoretically, assuming perfect play?
Perfect play isn't how we measure movie quality. If you easily achieve perfect time in a trivial game, it's of the same quality as playing SMB in a sloppy, lazy way. I can't consider lazy play a meaningful record that you need TAS tools and skills for.
I think you are completely missing my point there.
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CoolHandMike wrote:
Having a lot of low quality and not entertaining published runs would diminish the overall worth of the site. Is there something wrong with userfiles? Even they get a place on the front page and can act as a library for tases.
How many submissions do you expect to see of TASes for games that are not trivial to speedrun by a human but have a completely trivial TAS?
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Fine, don't answer my question then.
Post subject: Re: What defines the triviality of a game?
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feos wrote:
If something is trivial to TAS, it won't be a meaningful record
Of course it will, especially if speedrunning it in real-time cannot achieve the same time. It's meaningful because it gives an interesting piece of information: How fast can the game be completed theoretically, assuming perfect play? I think tasvideos should be considered like a library of perfect speedruns. An encyclopedia that collects best possible theoretical completion times of games. If someone would be interested in knowing what this theoretical best completion time is for game X, that information could be found here. Whether the TAS is "trivial" to make is inconsequential. What matters is the piece of information that it gives. What is the downside you see in having TASes of non-trivially-speedrunnable (by a human) games, even if making the TAS is trivial? What is the harm, loss, or inconvenience caused by this?
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p4wn3r wrote:
So, I was a bit off with my definition of a unit, a unit is actually an invertible element. My definition is for a root of unity, which implies it's a unit, but there are units which are not roots of unity, although not for the rings usually in number theory. The definition of unique factorization in terms of units is given here.
All that goes well above my head, but I get the feeling that you are trying to apply a wider field of mathematics into the narrower field of integer arithmetic.
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p4wn3r wrote:
The modern definition of unique factorization only requires that the factorization be unique up to permutations of the factors and multiplication by units.
I have never heard of such a definition.
A unit is an element that eventually becomes 1 if you multiply it by itself a finite number of times.
I have never heard of such a definition. A unit, or multiplicative identity element, is the only value that leaves the original value unchanged when multiplied by that element.
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p4wn3r wrote:
If you ask me, I do think the most elegant definition of prime numbers, even if we work entirely in elementary number theory, is to allow them to be negative.
One problem with that is the same as why 1 isn't considered a prime: It contradicts the fundamental theorem of arithmetic, which states: "Every positive whole number can be written as a unique product of primes." Allowing for negative primes would break the uniqueness requirement.
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Blackpenredpen recently tackled the question of whether any of the numbers of the form 1010101...01 or in other words 1 + 100 + 1002 + 1003 + 1004 + ... + 100n can be prime, other than 101. It would be interesting to see your approaches at this.
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moozooh wrote:
There is no definite answer to that, but astronomers have looked into it for decades and so far have seen no evidence of any boundaries or wraparounds. Not even a trace thereof. So it's either actually infinite or so big it is functionally infinite.
Given that the observable universe is smaller than the entire universe, what kind of "evidence of any boundaries or wraparounds" would you expect to see? The observable universe is smaller than the entire universe because the part of the universe that's farther away from us than a certain distance is receding from us faster than c, which makes it physically unobservable by any possible method we know of. (Although due to the rubberband phenomenon I think the horizon of obserbability is not exactly at the distance where space recedes from us at c. I'm not exactly sure how far it is, theoretically speaking.)
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p4wn3r wrote:
For example, a very popular algebra textbook states
One textbook does not a consensus make. The vast majority of definitions always talk about positive numbers. Perhaps the most succinct definition of prime number is "a positive integer that has exactly two positive integer factors". Definitions that include negative numbers don't seem to be completely non-existent, though. For an unfathomable reason it appears that the Merriam-Webster dictionary includes negative numbers in the definition as well.
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p4wn3r wrote:
OK, soft question now. Asking here because at another place I post about math there was some heated discussion about this and I want to know what people think. Are -2, -3, -5, -7, ..., prime numbers?
I believe the definition of "prime number" has always included that it's a natural number. The only thing that has changed over the centuries is whether 1 is a prime number or not. (Initially it was considered one, as it fits all the criteria. At some point the consensus was reached to exclude it for practical reasons. This makes 1 a special natural number, as it's not prime nor composite. It has its own, third category.)
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r57shell wrote:
Warp wrote:
For any given N >= 3, is the N-dimensional cube Hamiltonian?
If you refering to hamiltonian cycle then yes. Easy to prove. Proof: gray code Oh. No. It's wrong. Ok then I don't know. Oh, no, I'm right.
I don't understand that "proof" at all. But I thought of a marvelous proof myself. I'm not mathematically adept at writing a mathematically rigorous proof, so I can only express it colloquially. It would be a proof by induction: Assuming that an (N)-dimensional cube is Hamiltonian, we prove that it follows that an (N+1)-dimensional cube is also Hamiltonian. (Then, since a 2-dimensional "cube", which would be a square, or even a 1-dimensional "cube", which would be a line segment, is trivially Hamiltonian, it follows that all of them are.) In order to do this we notice that we can create an (N+1)-dimensional cube from an (N)-dimensional cube by extruding the latter into the direction of the (N+1) axis by one unit. In other words, we create a copy of the original cube at a distance of a unit in the (N+1) axis, and connect all the corresponding vertices. The simplest example of this would be going from a 2-dimensional "cube" (ie. a square) to a 3-dimensional one: We extrude the square into the 3rd dimension by a unit, ie. we create a copy of the square 1 unit apart in the 3rd dimension, and connect all the corresponding vertices, and we get a 3-dimensional cube. (The same works, in fact, from a 1-dimensional "cube", ie. a line segment, to a 2-dimensional one, ie. a square.) If we have a Hamiltonian path for an (N)-dimensional cube, when we extrude the cube into the (N+1) dimension, we can then continue the Hamiltonian path by going from the last vertex we visited along the new edge created in this manner, and then just do the reverse of the previous path in this new "copy" of the (N)-dimensional cube. Therefore, if we have a Hamiltonian path for an (N)-dimensional cube, we can form a Hamiltonian path for an (N+1)-dimensional cube in this manner. And since there exists a Hamiltonian path for the trivial case of the 1-dimensional "cube", then every N-dimensional cube has one.
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For any given N >= 3, is the N-dimensional cube Hamiltonian?
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Returning to the roots of this thread. Link to video
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I'm slightly disappointed at the lack of enthusiasm.
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Matt Parker's math puzzle of the week. Link to video
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Amaraticando wrote:
Can you arrange 4 pencils, such that any 2 distinct of them form a 90° angle? I can do it with 3 pencils.
Where does the 90-degree requirement come from? Isn't it, once again, just an arbitrary requirement? Why 90, and not 60, or 120, or pi?
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That got me thinking: Isn't even the notion of "three" dimensions arbitrary? Why is it precisely three? I started thinking like that because the notion of "two-dimensional space" is, ultimately, physically nonsensical. It's just a notion. A way of describing the abstract notion of a measurement. We say that eg. a length is a "1-dimensional" quantity, but only because we have decided on it being such. Likewise we have decided that area is a "2-dimensional" quantity pretty much by agreement. In real life there is no such a thing as a "2-dimensional" anything (because it would need to have zero thickness, which is only notionally, not physically, possible). So if "1-dimensional" and "2-dimensional" quantities are just abstract concepts, why wouldn't "3-dimensional" be such as well? We have just decided, for whatever reason, that there are 3 axes, and therefore space is "3-dimensional". But this is just a convention, an agreement. It's very handy, but still just an abstract notion. Or am I talking complete BS here?