For some reason there seem to exist two slightly different versions of the Fundamental Theorem of Arithmetic. For example, this numberphile video gives the following definition:
"Every positive whole number can be written as a unique product of primes."
This website gives a similar definition:
"Every natural number is either prime or can be factored as a product of primes in a unique way."
On the other hand, most other websites have a slightly altered definition. For example:
"For every integer n≥2, it can be expressed as a product of prime numbers."
Or:
"Every integer n≥2 can be factored into a product of primes in exactly one way (aside from rearranging the factors)."
Some definitions include the requirement that the number be greater than 1, others don't. Which one is correct?
The host of the numberphile video points out that the definition of the theorem seems to imply that 1 isn't a whole number (because surely it can't be written as a product of primes).
The answer given to this is that 1 is the result of the empty product. I suppose this makes sense when you consider that every natural number can be expressed as:
n = 2p1⋅3p2⋅5p3⋅7p4⋅11p5⋅13p6⋅...
as a combination of non-negative integer values of p1, p2, p3, etc. Nothing stops all of the powers from being zero (which is, thus, the unique prime factorization of 1).
but in the case of a game like Celeste, where the source code of the tools is openly available, there should be a way to get it on the site.
Note that the source code being available isn't some kind of magic bullet that will automatically erase all possibility of invalid runs.
Even small programs can become quickly quite complex, and software bugs are infamously common even in small open source projects. Checking even a small program for bugs is a rather large endeavor, even for a team of people.
Does this mean that all the officially accepted emulators have been 100% checked for bugs, and are guaranteed to be completely bug-free, and emulate the hardware with 100% accuracy in every possible situation? Of course not. But the line has to be drawn somewhere, if we want TASes at all. The advantage of generic console emulators is that they are being used to TAS literally hundreds of games, and many of these TASes are being verified with the actual original hardware, so this gives a very good degree of confidence that the emulators are as accurate as can be reasonably expected. The longer the emulator has been in use, and the more games have been TASed and verified with them, the more confidence it gives.
However, if we have a TASing environment that's specific to one single game, the situation is much more difficult. After all, we only have one specific game as a "test" of the "accuracy" of these tools. Sure, some people could go through the source code of these tools and try to find bugs and estimate how legit the data it produces is, but with so little data to test (usually only one run for one game), things can be easily missed. It's a bit comparable to a scientific test with a ridiculously small sample size.
And this would have to be repeated for any future single-game-specific TAS tools.
Let's talk about Celeste itself for a moment - we have full code we can look at, a patch file, and even support from the developers.
Does "patch file" in this case mean that the game itself is being modified?
One of the core principles of TASing is that the game itself ought to be unmodified. Once you start modifying the game itself, it opens yet another huge can of worms with regards to the validity of runs.
A modified game could be used to create the keypress input data for the game, but said input data ought to be, at least in principle, usable with the original unmodified game to replicate the run via controller input alone.
Somebody mentioned Doom as a huge exception to this whole principle of no game-specific TAS environments. And it is indeed quite unusual.
It is my understanding (and please correct me if I'm wrong), that Doom's own replay files do not store keypresses and replicate the run by feeding these keypresses to the game. Instead, they contain, essentially, data of the type "object X is at place Y at time Z" which (if that's indeed the case) could perfectly well be abused to create "runs" that are impossible to replicate when playing normally (eg. the player just jumping from one end of the level to the other in one frame, just because the replay file says that at the next frame the player's position is at that other end of the level).
I think the idea was to unpublish the few TASes that might still exist that are clearly and blatantly against the rules (such as not completing the game, or is known to be using faulty emulation). I have no idea if there exist any such TASes that are in a published (non-obsolete) state.
Sometimes rerecord counts get lost (there are published runs with 1 and 2 rerecords, which obviously can't be the case). Sometimes the rerecord count gets hyperinflated due to the use of scripts. In a few cases the rerecord count has been manually changed and doesn't reflect anything.
I have struggled greatly with attempting to get content from GDQ's submitted on TASVideos. There is a general resistance amongst site staff to allowing anything "new" and Celeste falls in this category.
Consider how strict of a stance sites like speedrun.com take on cheating, and justly so. In unassisted speedrunning cheating can take many forms, such as submitting a segmented run to an unsegmented category, using a spliced video (ie. snippets from different runs have been deceptively edited together to give the impression of one single unsegmented run), surreptitiously using tool-assistance, and so on. Infamous examples of such things exist plentiful (some even having been published for years before having been discovered as having been cheated), and sites like speedrun.com go to great lengths to catch and deter such things from happening.
TASvideos also has very strict standards of legitimacy for runs. Runs cannot, for instance, abuse emulation errors to do things not possible with the original game running on the original console, the game's code must not be modified using emulator features, and so on and so forth.
When a TASing toolset is game-specific, created specifically for one single game, and sometimes perhaps even requiring modification of said game in order to work, it opens a can of worms. How can we know that the end result is legit, using the above criteria? How can we know that what's happening in the run is indeed achievable with the unmodified game, running on unmodified hardware, using normal input?
Of course the runs shown at GDQ are always legit (I think it's pretty safe to make this assumption, given the reputation of the people involved), but accepting a game-specific TASing environment may in some cases open the possibility of more unscrupulous people to submit cheated runs for it.
Perhaps the toolset could be verified as making cheating impossible (assuming that's the case), but that would be quite a lot of work for the site administrators. And this work would have to be repeated for each such toolset that may be created. Just for one single game.
One could argue that perhaps just publish the run showcased at GDQ, but don't accept any further submissions for it. However, I would say that accepting a run made with a game-specific toolset, on the basis of the reputation of the people who made it, and then closing further submissions for that game, with perhaps the exception of a small group of "trusted" people, would sound strange, unusual, inconsistent and perhaps even a bit elitistic.
You nay not know something is an emulator bug until after you made it for instance. Not a fan of outright removal.
It would be a bit embarrassing to have runs that rely on faulty emulation. But I suppose these cases can be dealt with a big warning in the description of the TAS.
For one reason or another, the site has an extremely hard "we never unpublish a published TAS" rule. I'm not aware of any exception ever having been made to this (except, perhaps, some case where the author himself wanted the TAS to be removed. I remember cases where an author didn't want a submission published, and thus it wasn't, but I don't remember now if a publication has ever been removed afterwards due to the author asking.)
Perhaps this rule could be made just a tad bit less strict, and allow for clear past mistakes to be corrected. In other words, if a TAS was published in the distant past that would never be accepted today because it blatantly doesn't fulfill the requirements (eg. not actually completing the game, or abusing emulator-specific errors that make it work only in the emulator), and the case is crystal-clear, it could be retroactively unpublished (and perhaps its status to be restored as a submission, and marked as rejected).
(Of course hopefully this wouldn't create a precedent where a future change in rules would cause half of currently-published TASes to become unpublished.)
So if your movie is obviously sloppy, which means, it's easily improvable in a lot of places, it'd be rejected.
So we have a rather curious situation where a TAS that breaks all records would be rejected, even though there is no better TAS, nor anybody willing to make a better one (at least as of writing this).
Of course the difficult question is what level of optimization will be deemed enough for publication.
But by all means such a movie would be nice to have as a test run for future work!
I probably would, but given the enormous amount of work required (even if I'd just mostly replicate the unassisted route and tricks, making a moderately faster version of it), and the very high likelihood that it would just be rejected, it's not very motivating.
The run you linked depends on emulation bugs if I understand it correctly, so it's invalid by our rules,
I think I stated that quite clearly.
which means it won't be even compared to when a properly emulated submission is made.
Perhaps it won't be used for comparison (as in, in a way, to see if the new submitted TAS would "obsolete" it), but it's a clear demonstration of what kind of times can be achieved in the any% category (give or take some seconds). It is my understanding that all of those tricks showcased in that run can be performed on the Wii VC version (if not identically, almost identically.)
So the answer is, if your movie is optimal in itself, it will be accepted.
That's the critical question. If I were to make, let's say, a 17-minute (in terms of RTA timing) TAS, it would contain suboptimalities with absolute certainty. It would be 7 seconds faster than the current unassisted WR, but it's well known that with absolutely perfect play a sub-16-minute time is perfectly possible. (This is demonstrated by the N64 TAS. Even though it relies on faulty emulation on one part of the run, all of the tricks shown in it can be replicated exactly or almost exactly in the Wii VC version.)
So the question is: Would it be enough for the TAS to beat all existing official records (tool-assisted and unassisted) for it to be admissible for publication, or would the fact that it contains known suboptimalities disqualify it?
(Someone may justly ask: "Well, why would you even make a suboptimal run? You have the routing and all the tricks right there in the unofficial N64 TAS. Just replicate it, and make a sub-16-minute TAS on the Wii VC version." That would be a fair question. The thing is, that TAS is amazingly polished and optimized, and replicating everything it does, without all that knowledge and experience, would require an extraordinary amount of research and work. I have no idea how even half of the glitches are triggered. I have seen them many, many times, of course, but personally I don't know how they are done. And even then, the amount of micro-optimization in that run is staggering, and replicating all of it would be a humongous endeavor, especially for someone with little knowledge of the inner workings of the game and its speedrunning, and almost no experience on using the emulator.
Of course one could also be of the opinion that it's better to have no TAS at all than have a clearly suboptimal one, even if it beats existing records. That might also be a fair point.)
The current unassisted word record for The Legend of Zelda: Ocarina of Time any% category is 17m 07s, made on the Wii VC version of the game. There's an unofficial TAS of the same category that's 15m 43s. This TAS wouldn't be accepted at tasvideos because it's done on a N64 emulator and uses a glitch (the so-called GIM glitch) that cannot be used in an actual N64 console without the game crashing (and the sole reason why it does not crash with the emulator is because of faulty emulation due to a faulty graphics plugin).
This TAS, using the current any% route (ie. the same one used in that unofficial TAS), could be done on the Wii VC version of the game using the Dolphin emulator and, as I understand, this would be accepted at tasvideos.
My question is: What if I were to create a TAS on the Wii VC version of the game that's faster than the unassisted WR, but slower than that unofficial TAS? Let's say, for example, that the TAS would be 17 minutes flat. Would it be accepted?
It would break all existing official records (both unassisted and tool-assisted), but it would have known inefficiencies. What are the rules regarding this?
I wonder that if I made a TAS of the Wii VC version of the game that's not optimal but still beats the real-time record, if it would be accepted. After all, one of the requirements for publication is that it must break all existing records (it would, since no better run has been officially published here or in the real-time community), but it's highly unclear to me that if there are known improvements whether that would constitute grounds for rejection.
Assuming nobody else would be willing to make an any% TAS of the Wii VC version (which at this point appears to be the case), we would be in a strange situation where the fastest completion time that anybody has made would be rejected, but nobody would be willing to make a better one.
In fact, I went ahead an out of curiosity tried it, and TASed a bit of the beginning of the game with dolphin, and it seemed to work fine. (I got one desync at one point, but it went away when I redid that part. I don't know why it desynced there.)
Replicating that run in that video, however, would be quite a big endeavour, because it's so highly polished and uses many techniques I have no idea how they work...
Why not make the same TAS using Dolphin and the Wii VC version of the game? It's my understanding that it works fine there, and can be TASed just fine, and will be accepted at tasvideos.
I would love to see an any% TAS officially published. Heck, I would do one myself, but the level of optimization and work, and required knowledge, is just enormous. Someone who knows the game inside out, and has tons of experience in TASing it, would be much more suited for the job.
Are you sure it's the squarefree numbers?
12 is not squarefree (because it's divisible by 4), but I can't find any whole numbers m and n>1 for which
m2n = 122 = 144
I was wondering if there exists any pattern or formula for all square numbers that can not be represented in these two forms, where a, m and n are natural numbers, and n>1:
a2 = m2n
Rather obviously there are square numbers for which that formula holds (such as 162=28). There are also square numbers for which it doesn't (the smallest one being 4.)
While I understand the reason (mediocre ratings) that a bunch of runs are being recommended for demotion to vault...it's still a bit discouraging to see that so many runs aren't considered very entertaining.
I suppose what's happening is that as the number of TASes has reached the several thousands, and their novelty has faded a bit, the standards of entertainment have got stricter, and thus what was once considered brilliant might now be considered merely "good". And merely good might not be enough anymore.
Anyways, could we please stop considering putting a run in vault a "demotion"? I'm still rallying for the bad reputation of vault to be removed from the common consciousness. It shouldn't be considered a garbage dump where all the "boring" runs are thrown into. It shouldn't be a shame to get a run into vault. If you get your run published, that means that it's, essentially, the world record TAS for that game, and that shouldn't be something to be ashamed of. That should be an achievement worthy of pride.
