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Warp wrote:
The Numberphile youtube channel just posted a video showing the extraordinary result that the sum of all natural numbers equals -1/12.
Of course many people would instinctively rather call it an outrageous claim instead, but seemingly there's some mathematical truth to the result, and there are many ways to prove it. (Note that this is not talking about what the limit of the series is, or what it converges towards, but rather the sum of an infinite amount of terms, which is different.)
Since one of the simple proofs shown in the video deals with infinities, I instinctively get the idea that "you can prove anything with infinities" (in the same way as you can prove anything with something that equals eg. 0/0.) In other words, using some other mathematical trickery you could make the sum equal to whichever value you want, like -1/25 or 50.2. However, apparently -1/12 is the only result you get, no matter which method you use to solve it.
While the video (and its extra footage) has some talk about this, I would like to hear if someone else had some other insight into understanding the rather puzzling result.
(Also, why "12" and not something else? Where is precisely 12 coming from? I mean I can see the proof in the video, but it's still puzzling why its 12. Is there some kind of odd relation with precisely that fraction and the infinite sum of all natural numbers?)
I just finished commenting on that video. Fancy that.
Some of the assumptions I noticed:
1) 1-1+1-1... = 1/2 (wild theorycrafting about the nature of infinity)
2) The power series for 1/(1-x) being valid at x=-1 (the power series is only valid for |x|<1)
3) Using analytic continuation to say the Reimann Zeta function converges for s<=1 ("Well the definition says P, so let's assume Not P and see where that gets us"). I can use analytic continuation to find factorials for negative numbers, it doesn't mean my result has any meaning.
4) That it kind-of makes sense because you go "all the way to infinity" instead of using partial sums. If you're going to make me include infinity as a number in my calculation, then you're working in the Extended Reals and all bets are off.
You can use 1-1+1-1... and creative use of parentheses to prove that 0=1. You have to be careful when doing anything with non-convergent series, and I don't see any of that in the video.
I'm okay with non-mathematicians skimping on the mathematical rigor in favor of application, and under normal circumstances I would have been fine with a non-intuitive answer if there was real world evidence backing it up. But this is an extreme case of that and I'm going to need something a little more "real world" than string theory or quantum field theory to accept it.
And honestly, if you need a non-convergent series to have a finite value for your theory to work, maybe the proper response is to take a step back and question whether or not the universe really has 26 dimensions.
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DRN wrote:
One more question, what represents "doing nothing"?
Because I'm trying to get rid of the portions where I just stop and do nothing,
but what happens is that it just messes up my TAS and do really random things.
Thanks.
A frame with no controller input looks like this:
00 00 00 00 80 80 80 80
There's a couple of them in your screenshot starting at 0x100.
If you just put a bunch of nulls then it translates to mashing down left on the joysticks, so that could be messing you up. I'm not exactly sure what you're going for though. If you're removing frames it might be causing desyncs with the RNG, or maybe you accidentally edited the header or misaligned the frames. You might also miss an input if it gets shifted into a lag frame, which would probably happen pretty often if the game has a lot of pauses.
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Oh snap a math thread. I'm in.
Warp wrote:
I understand now why the program prints "50%" (which was surprisingly difficult to predict and even understand after I saw the result.)
However, it still kind of puzzles me if and why the host's overall strategy would affect what you should choose in one single round. After all, if you are playing one single round, and the host opens a goat door, the situation is identical regardless of whether the host always opens a goat door or a door at random...
(Although it gives some insight into that question that if the host always opens the car door unless it was the one you selected, then, even though it appears to be an identical situation, it would nevertheless be disadvantageous to switch because you would lose by doing so. In this case the host's behavior does indeed affect the single round.)
The probabilities are different but the decision is the same because in neither case is it a disadvantage to switch. If the issue is random goat vs. standard Monty Hall then you're decision is always to switch.
That said, there are host strategies where it's better to stay, like you said. Suppose he had the following strategy:
1) If the player picked a goat, open the player's door (2/3 chance)
2) If the player picked the car, open a different door (1/3 chance)
In the 1/3 chance that the host didn't open the player's door, you are guaranteed to get the car if you stay. In this case it looks like the standard Monty Hall problem but you're actually better off staying. Even though the individual round looks identical to Monty Hall, knowing the host's strategy greatly affected the probabilities. Likewise in the random goat vs Monty Hall strategies, knowing that the host could have picked the player's door makes the random strategy a 50/50 instead of the Monty Hall's 33/66.
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ALAKTORN wrote:
I really don’t believe the primary goal was “time”
I’m sure that there are loads of situations where spamming the exact same move will result in faster times, but this TAS is extremely varied and clearly trying for entertainment, not speed
shouldn’t speed be the secondary goal?
I'm glad you brought this up, because this is something that bugged me when I wrote up the submission info. I considered both "time" and "entertainment" primary goals, but at the end of the day I went with "time" because I would have considered the TAS a failure if it didn't beat every existing run.
Based on other submissions and the default prefill text, I interpreted "Game Objectives" as "what kind of categories would the run fall under." If I were to write them based on what I had intended the run to be, it would look like this:
- Make a run that is entertaining to both the TAS crowd (who value creative optimization strategies) and the Smash crowd (who value style).
- The run must be acceptable for publication on this site (that is: beats all existing records, contains no obvious optimization flaws, etc).
- KOs should involve a high degree of variety. Use items and stage hazards whenever possible.
- Entertainment tradeoffs are only acceptable if the optimum solution is boring. Nobody wants to watch Samus down tilt everything in a team battle.
- Optimal play. That is, if I'm going to perform a certain combo, it should not be possible to do it faster.
I'd rather have this as the Game Objectives section because I think it's more descriptive of my actual objectives. I wouldn't be surprised if it was too late in the submission process to change it though. I'd also like to change the branch to "Adventure Mode", but again, probably too late.
Unrelated: I noticed the time was changed to 10:23.48. I really don't think the post-Giga Bowser content should count towards the official time. None of it is necessary to beat the game (the game finishes by itself after you press Start after beating him), and since I didn't skip anything there would be a solid 2 minutes of wasted time at the end. If you need a trimmed version of the input file for official timing purposes I can get you one.
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SmashManiac wrote:
First thing I did was to compare this movie to SuperDoodleMan's Adventure video back from 2006, which at the time used Action Replay to have a frame advance ability and make a segmented run of Melee's Adventure mode on Very Hard:
http://www.youtube.com/watch?v=zJr8nCgm1H8
Much to my surprise, while the in-game times are overall faster here, the times of many individual stages are significantly slower:
- Stage 1-2: +3.68s
- Stage 4-1: +5.41s
- Stage 4-2: +3.99s
- Stage 5-1: +3.53s
- Stage 6.1: +2.83s
- Stage 6.2: +5.24s
- Stage 8-2: +1.89s
- Stage 9: +9.19s
So that's a total of 35.76 seconds lost when combining the bad stages. Sure, a different character was used which may balance things out overall, and there's the possibility of worse luck due to the lack of segments, but I have a hard time believing that this movie is optimal considering this. I may be completely wrong, but that's my impression. In any case, I think that's enough evidence to verify the optimization of this movie.
Also, why play during the credits? Pretty sure any% movies are supposed to end as soon as possible according to the rules. I believe useless input it should be cut from the end of the submission.
Dont forget the stages where I outspeed him:
Stage 1-1: -3.4
Stage 2-2: -3.55
Stage 3-1: -4.41
Stage 3-2: -1.65
Stage 5-2: -5.56
Stage 5-3: -7.39
Stage 7: -23.52
Stage 8-1: -11.38
Stage 10: -6.5
Stage 11-1: -9.5
Stage 12-1: -1.03
Stage 12-2: -23.77
Even if we leave out the huge differences to account for the fact that he wasn't going for time, that's still -42.99, which comes out to -7.23 overall.
Samus and Fox are different characters with different movesets who excel in different situations. Recent TASes have gotten the Escape from Brinstar stage down to 6 seconds for Fox. Perfectly optimized Samus will never be able to beat that. Likewise Fox will never be able to beat Samus' time against the cars on the Grand Prix stage. It's all about the tradeoffs from using one character over the other. Very rarely will you ever see a character outperform another in every aspect of the game. Even Pichu has some advantages over Pikachu.
As for the credits, I see where you're coming from, but I disagree with removing it from publication. The audience is going to have to sit through the credits anyway, might as well entertain them. It doesn't affect the time of the run in any way. And I'm not sure, but I believe this exception has been made before. Correct me if I'm wrong.
On that note, if the admins want a trimmed version of the input file to prove that the game does complete by itself after frame 31,364, I can provide that.
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henke37 wrote:
I am not sure if it really is the hardest mode if the unlocked characters affect the gameplay.
But this is quite clearly great.
Most of the changes you get from having everything unlocked are trivial. The only thing I could see affecting the difficulty would be having to fight Metal Luigi along with Metal Mario. Everything else is just "substitute this character for this other character you've unlocked," and even that doesn't happen very often. You also get a couple more names in the credits, but that shouldn't matter for the overall run.
jlun2 wrote:
Er, the hell... That commit does not appear to be on any branch.
Fortunately, googlecode has snapshot links (at least if one does some URL editing).
I don't have the game iso atm, but does the run at least sync on the official.latest releases? I've read that GC emulation has stabilized quite a bit, so maybe it would work?
I wouldn't be surprised if they came out with a version that played this game correctly since I started the run, but I doubt it would play the input file correctly. I remember reading somewhere that the core changed often between releases for the earlier versions.
Fishaman P wrote:
TASVideoAgent wrote:
At one point I gave up and just did a version that got all the names on high speed
I would definitely like to see this!
Okay, so the TAS you linked for the Metal Mario strat showed me. I thought he hit 182, and you hit 172. What's up with that?
Anyway, as a competitive Melee player, I found this very entertaining. I'm not sure if it's applicable for the Vault, though.
Also, you said you aimed for time first. What's up with the Ice Climbers stage then?
Some names don't show up until you've unlocked content, i.e. you wont get Marth's voice actor until you've unlocked Marth. I can guarantee you I hit all of them, I spent more time on the credits than on any other segment.
The ICs spawn on a timer. No matter what you do they're always going to appear at the 3:08 mark. The best you can do is be prepared to hit them on frame 1.
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Thanks for the positive responses!
thatguy wrote:
Just one question, since it's years since I played this: given how you muck about on icicle mountain, i presume it's an autoscroller; however, when you move to the top of the screen it definitely scrolls upwards faster. How does this stage work again? I'm sure I'll kick myself when I find out the answer.
Derakon wrote:
I'd always assumed that the fight with Nana and Popo started after you'd climbed a certain amount of the mountain, but given that numeric faffs about instead, I'm guessing it's just on a timer.
Yup, they spawn on a timer. When I first started this run I thought it was determined by distance covered too, but after I told a friend, he said otherwise and went home and confirmed it was a timer. And then I went home and confirmed it was a timer. There's a couple other TAS's on youtube, and the Popos always spawn at around the 3:08 mark.
Derakon wrote:
I could have hoped for more playing about with items, but I imagine they're really hard to manipulate, especially on Very Hard.
I went into this with every intention of using items all the time, but they just don't spawn often on Very Hard. The timing is scripted (although the actual item and its location are manipulable), so there was nothing that could be done to get more.
Ilari wrote:
- I can't find the source code for Dolphin r7719 (obtaining the sources would be needed to dump the video properly).
This particular game desyncs constantly in the most recent versions of the emulator. My research told me that r7719 was the latest stable version that worked for this game, which is why I went with it. I ended up writing a program to brute force the a/v sync, manually adjusting the frame dumps after the fact. Worst case scenario, I can get a high quality dump and send that to somebody for encoding. I assume it would be preferred for the emulator to do it, though. I don't know if the source code for this particular version is publicly available.
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Hey guys, I'm new to this so let me know if I missed anything. I'm sure there's something.
Also I'm not sure where it got the 08:38.3 number from. The input file is 10:23.48, and the last frame of input necessary to beat the game is 08:42.73.
Thanks!
