Fun Fact: The man who solved the Poincaré Conjecture, Grigori Jakowlewitsch Perelman, rejected the prize money and told the congratulatory committee to get lost because he just wanted to be left alone.
Nobody has ever survived that many before... are you sure you’re ready for this? To sacrifice your everyday life to fall into a flintstone gummy spiral? Sacrificing your life for a math problem, don’t get me wrong it’s brave, but your life will never be the same. Are you prepared for this?
Yay, I'm about to become one million $ richer. I remember something about Paul having 10 apples and giving three of them to Mike, so I'll just take it from there and start working on this. Easy peasy!
That's not a math textbook problem. Paul needs at least 4 carts of 20 pineapples. Now Mike wants to trade 6 peaches for 1/4 pineapple each and 9 of his pubic hairs for 1 pineapple each. How many Pineapples has Paul eaten in the meantime?
The thing is: If you had the formula you would earn way more money by solving the problems selling your services to solve rather than selling the formula for a million USD.
You would have to play pretty stupid though, otherwise you would very likely give away how you proved it (or enough to deduce how to do so) and so you’d lose the advantage you have. If you’re smart enough to know how to solve one of those problems, you would very likely get a lot more money from others as a result, either through jobs or otherwise.
@@TheStrongestBaka There's also the case where the formula exists, but just proving its existence (without finding the formula itself) will also give 1 million.
Yes, comp theory was my least favorite course in computer science. My running joke in the course was to ask if the person the theorem or proof was named after had gone insane (they pretty much all had). For example, Alan Turing, Kurt Gödel, etc.
@@2011blueman I actually really enjoyed it but holy crap, those algorithm classes were some of the most difficult I took. I was always amazed by the solutions these people came up with and then I remember they did it decades ago... and now its taught in undergrad CS classes LOL. I was very fascinated by it all, but I could probably never come up with that stuff. the comp sci people of long ago were straight up geniuses and im here like...
@@Mathguy363 lol my analysis of algorithms course was basically straight up math. we didnt write a single line of code for that... unless you wanted to for fun, which I did because im a nerd lol
To be clear, even if we have a polynomial time algorithm which solves NP problems, it could still in practice be unhelpful, e.g. it could have constants greater than a googleplex or whatever and only be efficient for inconveniently large inputs.
" e.g. it could have constants greater than a googleplex or whatever and only be efficient for inconveniently large inputs." *Googologists have entered the chat*
heh. years ago I remember working on a problem like that. we had two possible algorithms. I was working on one that starts off really well but got exponentially worse as the dataset got larger. there was another that had a high setup cost (and expletive ton of RAM), but once you got everything cooked (constant+linear) put into memory (linear), the solution also became linear.
Also would the P=NP proof necessarily be constructive? If one could show that its possible to solve things in P time would it necessarily make it easier to find the algorithms? I understand that knowing that it's possible would be helpful but it wouldn't immediately resolve that issue.
> One of them has gotten significantly more attention and more failed attempts than the rest. *The Riemann Hypothesis would like to know your location*
@@RyanLucroy Oh it is, you'd have to talk about convergence, complex numbers, applications to number theory and a whole bunch of stuff that's difficult for someone without a maths or physical sciences background.
@@stardestroyer19 Yeah. You can't really describe the "core problem" without explaining a lot of background stuff. As somebody who wrote his bachelor thesis about elliptic curves, I am very interested in the Birch Swinnerton-Dyer hypothesis, but man, if I had to explain it with simple words, I would fail miserably 😄
@@RyanLucroy I know how it be man! I'm a PhD student in theoretical physics and somethings could take a long time to explain if you want to make sure people get the core idea of something without watering it down so much it becomes too simplified.
Fun fact: The man who solved the first Millennium Prize Problem turned down the $1,000,000 as well as the award, and later a Fields Medal. He then quit his job and went into seclusion. When approached by a writer in 2012, he stated “You are disturbing me. I am picking mushrooms.”
You got me, I'm just here to point out all the mistekas 2:10. Well, more of a clarification. The P?=NP problem refers to a specific type of problem, decision problems. That is any problem that can be answered with a yes or no. So, using the traveling salesperson problem as an example the decision problem would be: does there exist a path less than or equal to some given value? For sudoku, the decision problem would be: is a given sudoku layout solvable? Solving these doesn't necessarily mean we would get the solution, just the yes or no answer. As mentioned in the video these are both in NP as they are pretty easy to verify by giving the solution. A lot of problems can be expressed as a decision problem, but a lot can not be. The problems that can't be, like for example playing/winning a chess game would not become easily solvable if P was equal to NP. Well actually the problems we care about are: For a map with n stops, does there exist a TSP path less than or equal to some given value? and: given an nxn sudoku problem, is it solvable? This is because, as you mentioned at the begging, we care about how much the time increase relative to the problem size. Other than that, this was a pretty decent description of the P?=NP problem for only taking like 5 mins. It's funny, this problem is the easiest of the 7 problems to understand what the problem is saying, but that doesn't mean it isn't complicated. I mean, after all, it hasn't been solved while being one of the most attempted problems. The main difference between this problem and other complicated math problems is that for theoretical computer science, which is what field the problem is in, we just abstract out all of the complicated symbals.
A $3 million muffin is the exact opposite of a "very convoluted money laundering scheme". In fact it might be the least convoluted money laundering scheme of all time.
oh so what if I create a company which concept is to give money to people which i obtain from other companies and the people that watch me launder money. why does i have the impression someone already did that before
@@vojtechstrnad1 it's called buying and selling art, they already do it. You ever hear of shitty paintings getting sold for millions of dollars? yeah...
0:56 Oh boy, I'm a math student and I seriously tought that when you said one has gotten more attention than the rest, that you were going to talk about the Riemann Hypothesis. Would have loved an episode about that but P vs. NP is also a good one.
@@ethanl.1699 University professors get grants for millions all the time and it lasts them hardly any time at all. No idea where the money actually goes, just that it doesn't seem to last long.
Well arguably P vs NP is the most accessible of the problems, unlike RH which uses complex analysis and the other five which I barely know anything about. But yeah, if he ever makes a video on another one of the problems, it will be the Riemann hypothesis.
The proofs that P=NP (or P=/=NP) are appearing several times per month nowadays. It might be not studied more, but it certainly attracts a lot of attention.
The whole calculator part is exactly what I have to go through when I tell my students about these problems. One of my students was convinced that they found a counterexample to Goldbach's conjecture (not one of these, but still an open problen) when I couldn't immediately tell them 2 primes that sum to 1,000,000.
Short addition: a problem in P does not have to be "easy" or solvable in a fast way. Let's say I would find an algorithm for the TSP problem with a constant runtime of hundred years. That would be O(1) and in P, but probably wouldn't help me to hack any bank account.
That is a very good point. I also find it strange that most videos on P=NP seem to equate proof that P=NP with breaking encryption. Proving that a solution exists does not necessarily lead you to that solution, or does it? So knowing that a polynomial solution exists may put the encryption on shaky grounds but it will not magically make it not work overnight. Someone still has to find that solution.
Also, one could have a proof that P = NP that is not constructive. So, contrary to what is said at 1:50, just proving equivalence does not necessarily lead to new algorithms. And, even if it did, it's entirely possible that an algorithm in P has such a huge overhead that it's slower than a corresponding algorithm in NP for any input we might be interested in. This entire video is riddled with mistakes.
Yes, but the thing is: an actual algorithm would to some degree only be a side note. Breaking this exponential barrier is much more significant in itself. Any polynomial algorithm (even the one with a enormous exponent or huge constant inside the big-O) would still exploit some kind of non-trivial structure and would most likely mean there would be some insight into this class of problems. So very likely, even if that's the case, it's the crack in the problem that opens up a whole range of new research that will in all likelihood bring down the polynomial's constants and exponents. But I also don't think you can find lots of experts that believe this would be the case. As far as I know, it is believed that it's much more likely that P != NP.
@@JanStrojil "Proving that a solution exists does not necessarily lead you to that solution, or does it?" It doesn't, your understanding is certainly correct. "So knowing that a polynomial solution exists may put the encryption on shaky grounds but it will not magically make it not work overnight" Very well phrased. More to the point, simply knowing the answer(yes or no) to P=NP is not much better than pretending you know the answer, it would only tell you whether or not your attempts at a proof for or against are futile or not.
@@frankkobold I (a CS student) can confirm I've tried to proof that P=NP and P!=NP and failed at both. I don't even know what the Riemann hypothesis is though :D
You get the reward for settling the P =?= NP problem; winning it doesn't have to mean that P = NP. Furthermore, even if you prove that P = NP doesn't imply you have a non-exponential algorithm for NP problem. (Having such an algorithm of course means P = NP, but the reverse doesn't). (Also, an algorithm which takes n^1000000 steps technically is in P, but in practice, that won't give us efficient algorithms)
How again does P=NP not imply a polynomial algorithm for all NP problems? I mean of course there may be non-constructive proofs but in theory there should be algorithms.
Actually, I see we're all latched onto that idea of non-constructive proofs being a thing. I think @Caesim9 is saying, regarding the statement: "Having such an algorithm of course means P = NP, but the reverse doesn't" he interpreted it to imply "even if P=NP that doesn't mean a non-exponential algorithm exists", which would be wrong, its definitely the case that if P=NP then such an algorithm does indeed exist.
3:20 Unfortunately the problem described here is not equivalent to the traveling salesman problem, and actually could be solved with a greedy algorithm within polynomial time. The mistake in the video is that the points are labeled in a particular order and the difficulty is being described as finding the streets to take to traverse them in that order. This is equivalent to graph traversal and can be solved in linear time using the A* search algorithm. The key point that makes traveling salesman an NP-hard problem is that you're not given a particular order to traverse the nodes. Checking every possible permutation of the nodes is what makes it explode into factorial time.
Finally a topic I knew about before an HAI video! Also, in case anyone is wondering why the problems are so difficult to discuss - I'm a senior in a Math/CS double major, and I can only fully understand what 2 of the 7 problems are even asking. I'd bet a lot that the majority of math *professors* can't understand more than 4 of the statements.
The interesting thing is that the opposite hasn't been proved (although is the same question tho). We can't prove they're the same, but we can't prove they're different things either.
Yep. I'm pretty sure P != NP but don't know how to prove it. Maybe going over the axioms of the system questioning its provability incompleteness theorem style or something in that ballpark.
I really like that right after you note about people catching all the mistakes you are about to make, you say that P is things which can be solved in time which is not exponential. But this isn't the same as being polynomial. There are things which have time complexity which is worse than polynomial but still not exponential. For example, the best known algorithm for solving graph isomorphism has this level of intermediate complexity time. But well done video anyways! (Also we do have algorithms for traveling salesperson problem that are better than brute force checking everything. But the savings for it aren't that great.)
2:18 It's Actually all the problems that can be solved in polynomial time, not those whose solution time is not exponential. If the time complexity of a problem were say 2^sqrt(n), then it would satisfy your definition because it grows slower than all exponentials. But it would still not be in P since it grows faster than all polynomials.
My brother was working on this problem back in high school (~2005). He had his work copy written so as to date it. I have no idea how close he came to solving, because none of us knew what tf he was talking about lol. Will have to bring it up with him the next time we’re together
Spoiler alert: not close at all. Still, it’s nice to have a crack at problems even if they are famously unsolved by the greatest minds in the field of maths, it can still be an interesting experience and you’ll learn something probably
@@henryginn7490 We don't really know, I mean the great minds were not able to solve Poincare conjecture as well, but now it is solved. But yeah, the chances that he wasn't able to solve it are higher.
Nice video, but actually, the TSP is NP-hard but not NP-complete. It only becomes NP-complete when you turn it into a decision problem, eg "Is there a route that costs less than X?", which is easily verifiable inP time (in the original problem, the only way to verify if the route that you have is the shortest one is by computing every other route) Correction: I messed up, I somehow missed that the way you phrased it was the decision problem, so It's all good
I don't fail to appreciate the smug humour that this has generated, but it's also a quiet scandal and merits some sort of response. Presumably Sam can be bothered to give a shit when his promo codes don't work, but ... it would sure be nice to know for sure.
Man, I'd be shocked if HAI could figure out the question. Not because I think he's stupid - I'm a senior in a Math/CS double major, and I can't figure out the question.
One of the best moments in my cs class was when our lecturer showed how you can change one NP problem into another (reduction). it's basically saying you show that problem X is at least as hard as problem Y (which you know is NP) so X is at least NP. Iirc he showed 3 SAT (NP problem) and reduced to Traveling salesman
I'm sorry Sam but those lifetime warranties are always a scam. Either A) the life time warranty covers the first knife but not the replacement so when the second one goes bad you have no recourse. (I've also know companies who send products that don't meet quality standards as the replacement product.) B) They do what Eddie Bauer did. Have a life time warranty for 5-10 years and then revoke the warranty saying that you have 1 year to get your replacements before it is now invalid. or C) the most common, after 2-5 years close the company and reopen it with a new name and no outstanding warranties. In this age of the internet that is a really easy thing to do.
These are *not* the most difficult maths problems, as stated at 0:40. They're amongst the most important problems though, because of how big an impact on the rest of the mathematics they would have. There are plenty of probably more difficult problems out there, but just fewer people (or rather, mathematicians) care about them.
It's really difficult to determine how difficult a maths problem is, before it gets solved. For example the Collatz Conjecture sounds easy at a first glance but Erdős said we'd probably need an entire new proof framework for it. Wasn't there a problem in knot theory that kept many mathematicians busy that got recently solved and had a simple solution?
@@Caesim9 It's hard to even _define_ difficulty in mathematics except to say that many people have tried and failed. But the clay prizes were not chosen for their difficulty but because they were important to the rest of maths. It just so happens that important problems get more attention, and so the only unsolved important problems that stick around for decades are the ones that are difficult.
The prize isn’t for solving that P=NP it’s for solving P=NP. Slight difference, the first is asking to show it to be true, the last is asking to show if it is true or false. P=NP probably isn’t true so my point is that if you get the million dollars you’ll be able to put it in your bank.
Okay okay I already know all of this and I am just here to point out all the mistakes. Having said that: 1. Technically P and NP contain decision problem. Decision problems are problems that only admit yes or no as an answer. So the problem of finding a shortest path from A to B is not in P while the problem asking "is there a path from A to B with length less than 10km?" is in P. 2. The normal 3x3 Soduko problem (Is there a solution extending the already given numbers) is actually in P as there are only finitely many ways to fill out the Soduko and it is pretty easy to check whether this is a solution extending the given numbers. However If you vary the size (n*n) of the Soduko, you are correct. 3. 4:15 All problems in NP are solvable, just not necessarily efficiently. There are however problems outside of NP that are proven to be unsolvable/undecidable.
Sam: Are you in crushing student debt due to a predatory poverty cycle brought on by late-stage capitalism? Also Sam: Use this code to get 15% off expensive cookware Me, a millennial: He gets us 🥰 take my money
@@youngrex7694 The joke is a reference to the fact that Hitler wanted to become a professional artist but he failed the entrance exams to some art institution.
@@1vader "Pretty much". That's right. In another words, "very nearly". The P vs NP problem is not an entirely mathematical problem. It's close, but it doesn't cut it. That honor belongs to the Riemann Hypothesis.
So these problems are like encryption functions. SHA256(data, encryption key) outputs encrypted data It's easily verifiable when you have the encryption key, but requires an infeasible amount of computing power to derive the original data or the encryption key just from the output, even though the SHA256 algorithm is public information. There's no actual proof that you can't derive the input just from the output, but SHA256 is used everywhere for online encryption because no one has ever been able to derive an input just from a given output. Once super processors become mainstream, SHA256 might be broken, but just by increasing the number of bits of the encryption keys, the problem requires exponentially more computing power to crack, something that even super processors can't keep up with. This is also how cryptocurrencies work. Transactions use SHA256 with the input data being the history of transactions up to that point, generate a random output and mining computers guess and check millions of encryption keys with the SHA256 algorithm until they find an encryption key that gives that output, at which point the transaction is verified. Because it's SHA256, it's completely infeasible to try to pull off fraudulent transactions
I'll try to explain it - HAI didn't do that well. In these examples, I'll call the number of items X. P would be "I ordered these specific X items in the menu. How much is a 15% tip?". NP would be "My bill from yesterday had the total of $123.45, but that seems high. I forgot what items I ordered, but I know the menu has X items and I ordered 5. Are there any 5 items from the menu add to $123.45?" The first problem requires adding X numbers - you can do that in X time units. The second problem doesn't have a "easy" solution - the best known solution takes 2^(XK) time units. K is a constant number you shouldn't care about here - the point is that each time you put another item on the menu, the number of units goes up a bunch more than it does in the first case.
2:53 Traveling Salesman Problem, I think you missed an obvious detail - three points is a triangle. No-matter-what it has a fourth "center point", therefore achieving that interior-position is more important than any of the exterior-positions. A simple way to thinking about it is with playing Chess, jumping with a White Knight to any of 3 Black Pawns "L-triangularly", being it's not a straight line but it is still equally reaching; *assuming the variable positions of the exterior problem is unchanging relative to its interior mechanics, simply doubling-back to and from halfway as repeatedly requiring is a method of gross efficiency to solving the defined whole at a time!* So I should argue that NP = 1/2P x3 for a triangular configuration. Upgrade it to a hexagonal configuration, and it would be (NP = 1/2P x3) x6, or, NP = 1/2P x6, and so on, so there may be a Fractal connection? If Square then x4, if Pentagram then x5, and so forth. In any case it seems that the complexity of the external defines the repetitiveness of meeting it halfway at center, infinitely expandable outwards or inwards as any scale may require. - Daniel Nicolas Martin, Windsor Ontario Canada, April 25 of 2022. Since I'm not a mathematician I don't seem to get fair representation to participate in the challenge, so using RUclips will have to suffice?
I know the scriptwriter wrote this by creating their own ELI5 for themselves, but much of the language used in this video is extremely misleading and wrought with technicalities.
actually, finding the shortest distance between two points can be solved in O(m + n log n) (faster than polynomial time). The TSP is only about the shortest so called "Hamiltonian Cycle" but your graphic implied otherwise. To your credit: Really good (entertaining and understandable) video on a not so simple topic of computer science
One major thing you got wrong in this video and that most people get wrong about P vs NP, is that a proof that P = NP may be non constructive. Meaning we could prove that there is a fast way to solve all of those problems, without knowing what that fast way is.
Finally I can fulfill my dream of correcting HAI 2:50 just because it cannot be solved in polynomial time doesn't mean the solution runs in exponential time. Integer factorisation takes subexponential (less than exponential) time, but more than polynomial.
Note that a problem with exponential complexity isn't necessarily "harder" than one with polynomial complexity, and it won't necessarily take more time to solve for realistic values. For instance, you mention that primality testing (checking if a number is prime) is now known to have polynomial time complexity. But the fastest known algorithm (the AKS primality test) takes _far_ longer than practical algorithms like the Adleman-Pomerance-Rumely primality test for any primes of a size we could realistically use anyway. It is only for stupendously large numbers far beyond what a real computer could handle that AKS's asymptotic superiority gets realized. So in the unlikely event that it were discovered that P = NP, that wouldn't necessarily mean that any _efficient_ algorithms existed in P for various hard problems like the travelling salesman or discrete logarithm. It would primarily be of theoretical interest.
Small correction for the TSP. To find an optimal solution you don’t need to check all paths. The most simple algorithm to show this is BFS, Breadth First Search, which always gives an optimal solutions (in a Euclidean space). You could theoretically design a problem where BFS takes the same time as looking through all solutions but it will always be at least as good.
BFS for traveling over every node does require checking every path. It just generates them in an organized way. And it cuts out repeats that reach the same path by starting at a different node.
The Italians tried to understand this problem for logistics purposes by going to India. They understood that Dabbawallas (Delivery people for home made food - eg:to a family members work ) had a 99% on time delivery and distribution logistics (extraordinarily efficient ) day in ,day out, passing through multiple hub distribution points in a crowded city of 10 million . Minimalistic symbology (as a lot of Dabbawallas can’t read) yet everything got there and everyone gets paid per delivery every single day at the end of the shift. Couldn’t work it out after computing the algorithms and applying want they “learnt”.The issues just became exponentially harder.
You know, I went to the website to checkout this thing, maybe help out our bro Sam here. And the knife that he shows here? That's $134, after discount. The while set? Nearly 4 times that! Maybe it's my country and its high inport costs? I don't know... Still, il keep my $20 knife and sharpen it when it gets dull, use it for a few years and get a new one. Oh....good video by the way. Hey! I've got my priorities okay?!
As a computer science student i have to complimet you! Awsome 6 minute summary of a topic, i would regard as one of my hardest during my bachlelor degree!
_Hey, psst-do you want to get rich quick? Have you exhausted all the other get-rich-quick schemes on the internet?_ _Do you have absolutely no marketable skills because you pursued a degree that became obsolete shortly after graduation due to an unstable and rapidly shifting job market, which then ironically drove you into crushing student-loan debt that compounded with the pressures of late-stage capitalism to create a predatory cycle of poverty that has ultimately forced you to desperately scrape the internet for schemes to support yourself financially?_
There is also another problem with a prize of $1,000,000. It is called Beal's conjecture. The problem asks whether or not there exists three coprime natural numbers a,b,c such that aˣ+bʸ=cᶻ, when x,y,z are all at least 3.
The ad in the end was like "and here is the base for trying to crack down this world class math problem, and here's a knife". I honestly thought the knife was for protecting yourself from criminals trying to steal your prize money in case you got it.
Fun Fact: The man who solved the Poincaré Conjecture, Grigori Jakowlewitsch Perelman, rejected the prize money and told the congratulatory committee to get lost because he just wanted to be left alone.
I saw the video Count Dankula did on that
"It's a million dollars man! Just take the money!!" 😂
count dankula made a video on him
@@sabersz Some things in life are so unbelievable. That you deny them.
He returned the money because they didn't recognize another professor that did a lot of work in solving them.
@@lool12366 Why didn't he just give the money to that other professor, then?
Just give me 3 flintstone gummies, I'll handle them.
You'll overdose.
You're a madman! You'll break yourself with that many flintstone gummies
@@Potatoinator but before they do they'll solve it nah I'm kidding they will just die
Nobody has ever survived that many before... are you sure you’re ready for this? To sacrifice your everyday life to fall into a flintstone gummy spiral? Sacrificing your life for a math problem, don’t get me wrong it’s brave, but your life will never be the same. Are you prepared for this?
Take 3 and eat all your vegetables
Yay, I'm about to become one million $ richer. I remember something about Paul having 10 apples and giving three of them to Mike, so I'll just take it from there and start working on this. Easy peasy!
That's not a math textbook problem. Paul needs at least 4 carts of 20 pineapples. Now Mike wants to trade 6 peaches for 1/4 pineapple each and 9 of his pubic hairs for 1 pineapple each.
How many Pineapples has Paul eaten in the meantime?
@@NuclearTopSpot 2 1/2 pineapples. Unless you count the one in his ass. Then 3 1/2.
I'll beat u to it
Tell us how it goes
@@notyourfriendlyneighbor2733 turns out p doesn't equal np 😤😅😭🤔🤣
The thing is: If you had the formula you would earn way more money by solving the problems selling your services to solve rather than selling the formula for a million USD.
You would have to play pretty stupid though, otherwise you would very likely give away how you proved it (or enough to deduce how to do so) and so you’d lose the advantage you have.
If you’re smart enough to know how to solve one of those problems, you would very likely get a lot more money from others as a result, either through jobs or otherwise.
But it's likely that the "formula" doesn't exist and a million dollars would be awarded to someone who proves that.
@@TheStrongestBaka There's also the case where the formula exists, but just proving its existence (without finding the formula itself) will also give 1 million.
@@TheStrongestBaka you cant prove a negative tho
@@TheStrongestBaka cirno
me a computer science graduate: ah yes, my nightmares have returned
Yes, comp theory was my least favorite course in computer science. My running joke in the course was to ask if the person the theorem or proof was named after had gone insane (they pretty much all had). For example, Alan Turing, Kurt Gödel, etc.
It was a nightmare but at the same time pretty interesting, in particular the p=np problem. I don't know why there aren't more videos about this.
Lmao I was thinking the same
@@2011blueman I actually really enjoyed it but holy crap, those algorithm classes were some of the most difficult I took. I was always amazed by the solutions these people came up with and then I remember they did it decades ago... and now its taught in undergrad CS classes LOL. I was very fascinated by it all, but I could probably never come up with that stuff. the comp sci people of long ago were straight up geniuses and im here like...
@@Mathguy363 lol my analysis of algorithms course was basically straight up math. we didnt write a single line of code for that... unless you wanted to for fun, which I did because im a nerd lol
To be clear, even if we have a polynomial time algorithm which solves NP problems, it could still in practice be unhelpful, e.g. it could have constants greater than a googleplex or whatever and only be efficient for inconveniently large inputs.
" e.g. it could have constants greater than a googleplex or whatever and only be efficient for inconveniently large inputs."
*Googologists have entered the chat*
heh. years ago I remember working on a problem like that. we had two possible algorithms. I was working on one that starts off really well but got exponentially worse as the dataset got larger. there was another that had a high setup cost (and expletive ton of RAM), but once you got everything cooked (constant+linear) put into memory (linear), the solution also became linear.
I’m sorry that it’s unrelated, but I couldn’t help but notice the lick
Also would the P=NP proof necessarily be constructive? If one could show that its possible to solve things in P time would it necessarily make it easier to find the algorithms? I understand that knowing that it's possible would be helpful but it wouldn't immediately resolve that issue.
Or it could have a complexity of O(n^100), which is polynomial but not practical.
Thanks for the callout... 0:08
hey but now you have keyboards 😦👍
lol nice
hipyo tf you doin here
> One of them has gotten significantly more attention and more failed attempts than the rest.
*The Riemann Hypothesis would like to know your location*
Thats what I thought too. Then I realized that the RH is probably not very suitable for HAI, since it might be difficult to explain in a simple way 😄
@@RyanLucroy Oh it is, you'd have to talk about convergence, complex numbers, applications to number theory and a whole bunch of stuff that's difficult for someone without a maths or physical sciences background.
@@stardestroyer19 Yeah. You can't really describe the "core problem" without explaining a lot of background stuff.
As somebody who wrote his bachelor thesis about elliptic curves, I am very interested in the Birch Swinnerton-Dyer hypothesis, but man, if I had to explain it with simple words, I would fail miserably 😄
I have been waiting for this comment
@@RyanLucroy I know how it be man! I'm a PhD student in theoretical physics and somethings could take a long time to explain if you want to make sure people get the core idea of something without watering it down so much it becomes too simplified.
Fun fact: The man who solved the first Millennium Prize Problem turned down the $1,000,000 as well as the award, and later a Fields Medal. He then quit his job and went into seclusion. When approached by a writer in 2012, he stated “You are disturbing me. I am picking mushrooms.”
Source: en.wikipedia.org/wiki/Grigori_Perelman
You got me, I'm just here to point out all the mistekas 2:10. Well, more of a clarification. The P?=NP problem refers to a specific type of problem, decision problems. That is any problem that can be answered with a yes or no. So, using the traveling salesperson problem as an example the decision problem would be: does there exist a path less than or equal to some given value? For sudoku, the decision problem would be: is a given sudoku layout solvable? Solving these doesn't necessarily mean we would get the solution, just the yes or no answer. As mentioned in the video these are both in NP as they are pretty easy to verify by giving the solution. A lot of problems can be expressed as a decision problem, but a lot can not be. The problems that can't be, like for example playing/winning a chess game would not become easily solvable if P was equal to NP.
Well actually the problems we care about are: For a map with n stops, does there exist a TSP path less than or equal to some given value? and: given an nxn sudoku problem, is it solvable? This is because, as you mentioned at the begging, we care about how much the time increase relative to the problem size.
Other than that, this was a pretty decent description of the P?=NP problem for only taking like 5 mins. It's funny, this problem is the easiest of the 7 problems to understand what the problem is saying, but that doesn't mean it isn't complicated. I mean, after all, it hasn't been solved while being one of the most attempted problems. The main difference between this problem and other complicated math problems is that for theoretical computer science, which is what field the problem is in, we just abstract out all of the complicated symbals.
I too assumed this would be about the Riemann Hypothesis. As a long-time computer nerd techie type I've followed P=NP for a long time.
Fedex: thx for solving the hardest problem in the world
Me: np
Gödelmmit
*slow claps*
This fkin guy right here.
@@imveryangryitsnotbutter Your reply is incomplete
A $3 million muffin is the exact opposite of a "very convoluted money laundering scheme".
In fact it might be the least convoluted money laundering scheme of all time.
Someone needs to try this and see if such a simple scheme can actually work.
oh so what if I create a company which concept is to give money to people which i obtain from other companies and the people that watch me launder money. why does i have the impression someone already did that before
Like a $1tn coin to avoid the debt ceiling simple.
@@vojtechstrnad1 it's called buying and selling art, they already do it. You ever hear of shitty paintings getting sold for millions of dollars? yeah...
0:56 Oh boy, I'm a math student and I seriously tought that when you said one has gotten more attention than the rest, that you were going to talk about the Riemann Hypothesis. Would have loved an episode about that but P vs. NP is also a good one.
The core of P=?=NP is way easier to explain in like 5 minutes than Riemann. :-)
Why do I feel like half as interesting son will make a video in 2069 titled "why did they 2020 Olympics happen in 2021"
...Because Covid hadn't already killed enough people by the time 2021 came around. #FIFY
Kkkkkkkk true
@@dompedroii4656 Better not use the "brazilian laugh" in other languagues.
@ KKKKKKKKKKK
@@hipato6838 Not again.
I have a feeling that Sam wants us to solve this, so he can claim the money.
$1m isn't that much. It's a lot of playstations, but it's not a lot of years of salaries for well qualified people, buildings etc.
@@tomx641 if it’s not taxed, it will still give someone who makes 100k a year 10 years of salary considering their salaries are also not taxed
@@ethanl.1699 It all depends on contractual terms, but I'm talking about Universities in general.
@@tomx641 for a university professor, it’s still a few years of work saved, but yes, it’s nothing compared to a building lol
@@ethanl.1699 University professors get grants for millions all the time and it lasts them hardly any time at all. No idea where the money actually goes, just that it doesn't seem to last long.
After the intro I thought for SURE we'd be talking about the Riemann Hypothesis. Not sure if P=NP is more studied than RH.
Well arguably P vs NP is the most accessible of the problems, unlike RH which uses complex analysis and the other five which I barely know anything about. But yeah, if he ever makes a video on another one of the problems, it will be the Riemann hypothesis.
@@vojtechstrnad1 When he does I'll click on it so fast!
Me too.
The only ones I somewhat heard of was P vs NP, Navier Stokes and the RH
The proofs that P=NP (or P=/=NP) are appearing several times per month nowadays. It might be not studied more, but it certainly attracts a lot of attention.
It would make more sense if the promo code "half" gave 50% off...
missed opportunity to say 15% as interesting, tbh
Its supposed to make money not sense dear
Next video on Wendover Productions:
The Logistics of Why You Should’ve Paid Attention in Math Class
So, basically this video on how to survive The Cube
ruclips.net/video/XkYvo6S82LE/видео.html
The whole calculator part is exactly what I have to go through when I tell my students about these problems. One of my students was convinced that they found a counterexample to Goldbach's conjecture (not one of these, but still an open problen) when I couldn't immediately tell them 2 primes that sum to 1,000,000.
Short addition: a problem in P does not have to be "easy" or solvable in a fast way. Let's say I would find an algorithm for the TSP problem with a constant runtime of hundred years. That would be O(1) and in P, but probably wouldn't help me to hack any bank account.
That is a very good point. I also find it strange that most videos on P=NP seem to equate proof that P=NP with breaking encryption. Proving that a solution exists does not necessarily lead you to that solution, or does it? So knowing that a polynomial solution exists may put the encryption on shaky grounds but it will not magically make it not work overnight. Someone still has to find that solution.
Also, one could have a proof that P = NP that is not constructive. So, contrary to what is said at 1:50, just proving equivalence does not necessarily lead to new algorithms. And, even if it did, it's entirely possible that an algorithm in P has such a huge overhead that it's slower than a corresponding algorithm in NP for any input we might be interested in.
This entire video is riddled with mistakes.
Yes, but the thing is: an actual algorithm would to some degree only be a side note. Breaking this exponential barrier is much more significant in itself. Any polynomial algorithm (even the one with a enormous exponent or huge constant inside the big-O) would still exploit some kind of non-trivial structure and would most likely mean there would be some insight into this class of problems. So very likely, even if that's the case, it's the crack in the problem that opens up a whole range of new research that will in all likelihood bring down the polynomial's constants and exponents.
But I also don't think you can find lots of experts that believe this would be the case. As far as I know, it is believed that it's much more likely that P != NP.
It might be a bit confusing to suggest there could exist a constant time algorithm for the TSP. Trivially it's at least O(n).
@@JanStrojil "Proving that a solution exists does not necessarily lead you to that solution, or does it?"
It doesn't, your understanding is certainly correct.
"So knowing that a polynomial solution exists may put the encryption on shaky grounds but it will not magically make it not work overnight"
Very well phrased. More to the point, simply knowing the answer(yes or no) to P=NP is not much better than pretending you know the answer, it would only tell you whether
or not your attempts at a proof for or against are futile or not.
“One of them has gotten significantly more attention and failed attempts to solve it than the rest - P vs NP”
*Riemann: Hold my hypothesis*
Well, I would say every math student was at one point trying to proof both, but at least p=np was also tried by some computer scientists^^
@@frankkobold "but at least p=np was also tried by some computer scientists"
guilty as charged
@@frankkobold I (a CS student) can confirm I've tried to proof that P=NP and P!=NP and failed at both. I don't even know what the Riemann hypothesis is though :D
You get the reward for settling the P =?= NP problem; winning it doesn't have to mean that P = NP. Furthermore, even if you prove that P = NP doesn't imply you have a non-exponential algorithm for NP problem. (Having such an algorithm of course means P = NP, but the reverse doesn't). (Also, an algorithm which takes n^1000000 steps technically is in P, but in practice, that won't give us efficient algorithms)
How again does P=NP not imply a polynomial algorithm for all NP problems?
I mean of course there may be non-constructive proofs but in theory there should be algorithms.
@@Caesim9 I think Abi's saying that it's possible to prove that P = NP without actually coming up with a formula for an NP problem
@@NerdTheBox indeed, a --"nonconstructive proof" would be accepted-- just noticed the first reply literally said non-constructive proof oopsie
Actually, I see we're all latched onto that idea of non-constructive proofs being a thing.
I think @Caesim9 is saying, regarding the statement: "Having such an algorithm of course means P = NP, but the reverse doesn't"
he interpreted it to imply "even if P=NP that doesn't mean a non-exponential algorithm exists", which would be wrong,
its definitely the case that if P=NP then such an algorithm does indeed exist.
Imma pretend I understand that.
3:20 Unfortunately the problem described here is not equivalent to the traveling salesman problem, and actually could be solved with a greedy algorithm within polynomial time. The mistake in the video is that the points are labeled in a particular order and the difficulty is being described as finding the streets to take to traverse them in that order. This is equivalent to graph traversal and can be solved in linear time using the A* search algorithm. The key point that makes traveling salesman an NP-hard problem is that you're not given a particular order to traverse the nodes. Checking every possible permutation of the nodes is what makes it explode into factorial time.
Finally a topic I knew about before an HAI video! Also, in case anyone is wondering why the problems are so difficult to discuss - I'm a senior in a Math/CS double major, and I can only fully understand what 2 of the 7 problems are even asking. I'd bet a lot that the majority of math *professors* can't understand more than 4 of the statements.
The interesting thing is that the opposite hasn't been proved (although is the same question tho). We can't prove they're the same, but we can't prove they're different things either.
Yep. I'm pretty sure P != NP but don't know how to prove it. Maybe going over the axioms of the system questioning its provability incompleteness theorem style or something in that ballpark.
We did it. The "Who Wants to Be A Millionaire?" in math is here.
Perelman clearly didn't want to.
@@GURken Perelman is a russian wizard. He has no need for earthly goods.
This isn't going to fix your dept greece.
(I'm greek myself so dont start ww3 here)
@@GURken "I'm gardening"
@@janno288 wow I'm not the only Greek youtuber...
I really like that right after you note about people catching all the mistakes you are about to make, you say that P is things which can be solved in time which is not exponential. But this isn't the same as being polynomial. There are things which have time complexity which is worse than polynomial but still not exponential. For example, the best known algorithm for solving graph isomorphism has this level of intermediate complexity time. But well done video anyways!
(Also we do have algorithms for traveling salesperson problem that are better than brute force checking everything. But the savings for it aren't that great.)
The math problem my mom expects me to solve after watching the 3 minute video
I thought a way to solve the P-NP conjecture.
Unfortunately, this comment bar is to small for writing it in.
Fermat, is this you?
Classic Fermat😂
Whose Fermat?
Does P = NP?
Only if P is equal to zero or if N is equal to one
Or P is 1 and N is also 1
2:51 this graph also applies to other activities
Literal utopia, we all know it's too good to be possible, but good luck proving that as an abstract
The "Hey, you want to get rich quick" with the guy standing there with his finger at the beginning sounded like an advertisement for Honey.
At 1:26 I took my TI-84 plus into my hand and silently whispered "He didn't mean it!"
2:18
It's Actually all the problems that can be solved in polynomial time, not those whose solution time is not exponential. If the time complexity of a problem were say 2^sqrt(n), then it would satisfy your definition because it grows slower than all exponentials. But it would still not be in P since it grows faster than all polynomials.
This is amazing.
This is amazing.
@@kkmac7247 you are amazing
Will you marry me ?
Please
Smells like desperation in here
0:25 I think you forgot the part about having no skills attatched to your degree
Therapist: "Stock Footage Anonymous Hacker Guy can't hurt you"
Stock Footage Anonymous Hacker Guy: 4:04
i watch these when im high and it always fades into commercial in a very sneaky way.. i kinda love it
I like the knife segment...thanks.
As well great video
Genuinely thought you were gonna talk about the Riemann Hypothesis since that problem is even more studied than P vs NP.
2:52 they definitely knew what they were doing with the labels on that graph
My brother was working on this problem back in high school (~2005). He had his work copy written so as to date it. I have no idea how close he came to solving, because none of us knew what tf he was talking about lol. Will have to bring it up with him the next time we’re together
Spoiler alert: not close at all. Still, it’s nice to have a crack at problems even if they are famously unsolved by the greatest minds in the field of maths, it can still be an interesting experience and you’ll learn something probably
@@henryginn7490 We don't really know, I mean the great minds were not able to solve Poincare conjecture as well, but now it is solved. But yeah, the chances that he wasn't able to solve it are higher.
Nice video, but actually, the TSP is NP-hard but not NP-complete. It only becomes NP-complete when you turn it into a decision problem, eg "Is there a route that costs less than X?", which is easily verifiable inP time (in the original problem, the only way to verify if the route that you have is the shortest one is by computing every other route)
Correction: I messed up, I somehow missed that the way you phrased it was the decision problem, so It's all good
They did phrase it in terms of whether or not it was shorter than a specific distance, so they are all good.
@@joshuazelinsky5213 You are right, I somehow missed the way they phrased it
Why can't math grow up so it could solve it's own problems?
Your promo code “HALF” doesn’t work. The checkout process says “The provided code is invalid.”
Incidentally it's because the encryption was longer to solve that to verify
maybe it applied automatically and now you try to apply the second time, MAYBE
Does not work for me either and no there is no discount already applied
Try using the code WHOLE
I don't fail to appreciate the smug humour that this has generated, but it's also a quiet scandal and merits some sort of response. Presumably Sam can be bothered to give a shit when his promo codes don't work, but ... it would sure be nice to know for sure.
Expected a Navier-Stokes rundown and got an N = NP instead
Still loved it nonetheless
I expected Reimann Zeta, that's the millennium problem that seems to get the most attention
Man, I'd be shocked if HAI could figure out the question. Not because I think he's stupid - I'm a senior in a Math/CS double major, and I can't figure out the question.
One of the best moments in my cs class was when our lecturer showed how you can change one NP problem into another (reduction). it's basically saying you show that problem X is at least as hard as problem Y (which you know is NP) so X is at least NP. Iirc he showed 3 SAT (NP problem) and reduced to Traveling salesman
Hardest problems to solve
When will be another bricks video
I'm sorry Sam but those lifetime warranties are always a scam.
Either A) the life time warranty covers the first knife but not the replacement so when the second one goes bad you have no recourse. (I've also know companies who send products that don't meet quality standards as the replacement product.)
B) They do what Eddie Bauer did. Have a life time warranty for 5-10 years and then revoke the warranty saying that you have 1 year to get your replacements before it is now invalid.
or C) the most common, after 2-5 years close the company and reopen it with a new name and no outstanding warranties. In this age of the internet that is a really easy thing to do.
These are *not* the most difficult maths problems, as stated at 0:40. They're amongst the most important problems though, because of how big an impact on the rest of the mathematics they would have. There are plenty of probably more difficult problems out there, but just fewer people (or rather, mathematicians) care about them.
It's really difficult to determine how difficult a maths problem is, before it gets solved.
For example the Collatz Conjecture sounds easy at a first glance but Erdős said we'd probably need an entire new proof framework for it.
Wasn't there a problem in knot theory that kept many mathematicians busy that got recently solved and had a simple solution?
@@Caesim9 It's hard to even _define_ difficulty in mathematics except to say that many people have tried and failed. But the clay prizes were not chosen for their difficulty but because they were important to the rest of maths. It just so happens that important problems get more attention, and so the only unsolved important problems that stick around for decades are the ones that are difficult.
The prize isn’t for solving that P=NP it’s for solving P=NP. Slight difference, the first is asking to show it to be true, the last is asking to show if it is true or false.
P=NP probably isn’t true so my point is that if you get the million dollars you’ll be able to put it in your bank.
Holy hell! That 4 knives set costs same as my monthly salary here in India 😂
Okay okay I already know all of this and I am just here to point out all the mistakes. Having said that:
1. Technically P and NP contain decision problem. Decision problems are problems that only admit yes or no as an answer. So the problem of finding a shortest path from A to B is not in P while the problem asking "is there a path from A to B with length less than 10km?" is in P.
2. The normal 3x3 Soduko problem (Is there a solution extending the already given numbers) is actually in P as there are only finitely many ways to fill out the Soduko and it is pretty easy to check whether this is a solution extending the given numbers.
However If you vary the size (n*n) of the Soduko, you are correct.
3. 4:15 All problems in NP are solvable, just not necessarily efficiently. There are however problems outside of NP that are proven to be unsolvable/undecidable.
Sam: Are you in crushing student debt due to a predatory poverty cycle brought on by late-stage capitalism?
Also Sam: Use this code to get 15% off expensive cookware
Me, a millennial: He gets us 🥰 take my money
5:19 I gotta be honest, that's the smoothest progression I've ever heard.
"Half" should be a 50% off promo code, change my mind.
"-------, Found out why the box has a band-aid,"
lol
I would like to get rich quick, this is why I chose to become a painting major. Never mind u said math.
No. Please go into German politics.
@hi there What happing in German politics, aren’t y’all rank high for the least corrupt governments
@@youngrex7694 The joke is a reference to the fact that Hitler wanted to become a professional artist but he failed the entrance exams to some art institution.
I feel called out (2:09)
But I did not notice any mistakes at all, good job on this one :)
Am I the only one who actually doesn't want to join my friends at the "P" party....? That's just nasty....
Easy
2+2 equals 4 and it is easily checkable and solvable
Now give me my 1 million dollars so I can buy more legos
Even this knife you will replace after a year. Learn how to sharpen one, and your 20 dollar knife will last a long time
Wasn't expecting to see Barcelona appear on minute 3:10. Especially just the place where I used to live. Thanks for the surprise, HaI!
The only video without shilling in the first 10 seconds has a 20 second long rant written by a redditor
Now I have context for that one Elementary episode.
Sam sounds like he solved this equation.
P does not equal NP because one is a Problem while the other is No Problem.
Now where's my money?
P loses because he's alone and NP is two so basically its a 1v2.
It's been a while where my 1 million?
But what if P is Dream?
@@NoodleProductions He wins for a while then people finds out he used pvp cheats so he loses. Wins at the start but at the long run he losses.
Alright. I need a lot and I mean A *LOT* of pens and paper. Oh, and also like maybe 4000 pieces of mango flavored jelly?
To clarify, the P versus NP problem is a computer science problem, not a mathematical problem.
Well, it's theoretical computer science which is pretty much just a branch of mathematics.
P vs NP is a problem in theoretical computer science, which can be viewed both as a subset of computer science and mathematics.
@@1vader "Pretty much". That's right. In another words, "very nearly". The P vs NP problem is not an entirely mathematical problem. It's close, but it doesn't cut it. That honor belongs to the Riemann Hypothesis.
@@ultraviolet.catastrophe A solution to P=NP would be a mathematical proof. I don't see why it wouldn't count as an entirely mathematical problem.
@@Vaaaaadim If P=NP will be a mathematical solution, what about P!=NP?
So these problems are like encryption functions.
SHA256(data, encryption key) outputs encrypted data
It's easily verifiable when you have the encryption key, but requires an infeasible amount of computing power to derive the original data or the encryption key just from the output, even though the SHA256 algorithm is public information.
There's no actual proof that you can't derive the input just from the output, but SHA256 is used everywhere for online encryption because no one has ever been able to derive an input just from a given output.
Once super processors become mainstream, SHA256 might be broken, but just by increasing the number of bits of the encryption keys, the problem requires exponentially more computing power to crack, something that even super processors can't keep up with.
This is also how cryptocurrencies work. Transactions use SHA256 with the input data being the history of transactions up to that point, generate a random output and mining computers guess and check millions of encryption keys with the SHA256 algorithm until they find an encryption key that gives that output, at which point the transaction is verified. Because it's SHA256, it's completely infeasible to try to pull off fraudulent transactions
So
P would be a 20% tip
NP would be a 20% that took the tip into account of the total so it would keep increasing,?
I'll try to explain it - HAI didn't do that well. In these examples, I'll call the number of items X.
P would be "I ordered these specific X items in the menu. How much is a 15% tip?".
NP would be "My bill from yesterday had the total of $123.45, but that seems high. I forgot what items I ordered, but I know the menu has X items and I ordered 5. Are there any 5 items from the menu add to $123.45?"
The first problem requires adding X numbers - you can do that in X time units. The second problem doesn't have a "easy" solution - the best known solution takes 2^(XK) time units. K is a constant number you shouldn't care about here - the point is that each time you put another item on the menu, the number of units goes up a bunch more than it does in the first case.
You've got a better shot at winning the lottery than cracking these maths puzzles.
Crazy Fact: In 2006, a Coca-Cola employee offered to sell Coca-Cola secrets to Pepsi. Pepsi responded by notifying Coca-Cola.
2:53 Traveling Salesman Problem, I think you missed an obvious detail - three points is a triangle. No-matter-what it has a fourth "center point", therefore achieving that interior-position is more important than any of the exterior-positions. A simple way to thinking about it is with playing Chess, jumping with a White Knight to any of 3 Black Pawns "L-triangularly", being it's not a straight line but it is still equally reaching; *assuming the variable positions of the exterior problem is unchanging relative to its interior mechanics, simply doubling-back to and from halfway as repeatedly requiring is a method of gross efficiency to solving the defined whole at a time!* So I should argue that NP = 1/2P x3 for a triangular configuration. Upgrade it to a hexagonal configuration, and it would be (NP = 1/2P x3) x6, or, NP = 1/2P x6, and so on, so there may be a Fractal connection? If Square then x4, if Pentagram then x5, and so forth. In any case it seems that the complexity of the external defines the repetitiveness of meeting it halfway at center, infinitely expandable outwards or inwards as any scale may require. - Daniel Nicolas Martin, Windsor Ontario Canada, April 25 of 2022. Since I'm not a mathematician I don't seem to get fair representation to participate in the challenge, so using RUclips will have to suffice?
I know the scriptwriter wrote this by creating their own ELI5 for themselves, but much of the language used in this video is extremely misleading and wrought with technicalities.
2:08, they knew we'd come and point out mistakes LOLOL
actually, finding the shortest distance between two points can be solved in O(m + n log n) (faster than polynomial time). The TSP is only about the shortest so called "Hamiltonian Cycle" but your graphic implied otherwise.
To your credit: Really good (entertaining and understandable) video on a not so simple topic of computer science
Your traveling salesman example is incorrect. That's an NP-Hard problem, not NP.
I was going to say that the explanation didn’t make sense to me!
His formulation was a decisional problem so it's NP. Also, "your"
One major thing you got wrong in this video and that most people get wrong about P vs NP, is that a proof that P = NP may be non constructive. Meaning we could prove that there is a fast way to solve all of those problems, without knowing what that fast way is.
Today I learned: The probability of a blue lobster existing is widely touted as being one in two million.
Me who fell asleep in the middle of the video then woke up in 5:00 : how tf did math problem became cooking problem
Finally I can fulfill my dream of correcting HAI
2:50 just because it cannot be solved in polynomial time doesn't mean the solution runs in exponential time. Integer factorisation takes subexponential (less than exponential) time, but more than polynomial.
Note that a problem with exponential complexity isn't necessarily "harder" than one with polynomial complexity, and it won't necessarily take more time to solve for realistic values. For instance, you mention that primality testing (checking if a number is prime) is now known to have polynomial time complexity. But the fastest known algorithm (the AKS primality test) takes _far_ longer than practical algorithms like the Adleman-Pomerance-Rumely primality test for any primes of a size we could realistically use anyway. It is only for stupendously large numbers far beyond what a real computer could handle that AKS's asymptotic superiority gets realized.
So in the unlikely event that it were discovered that P = NP, that wouldn't necessarily mean that any _efficient_ algorithms existed in P for various hard problems like the travelling salesman or discrete logarithm. It would primarily be of theoretical interest.
I knew what the video was going to be about just from the title, and I STILL don't fully understand it... but you certainly helped make it easier.
Small correction for the TSP. To find an optimal solution you don’t need to check all paths. The most simple algorithm to show this is BFS, Breadth First Search, which always gives an optimal solutions (in a Euclidean space). You could theoretically design a problem where BFS takes the same time as looking through all solutions but it will always be at least as good.
BFS for traveling over every node does require checking every path. It just generates them in an organized way. And it cuts out repeats that reach the same path by starting at a different node.
The Italians tried to understand this problem for logistics purposes by going to India.
They understood that Dabbawallas (Delivery people for home made food - eg:to a family members work ) had a 99% on time delivery and distribution logistics (extraordinarily efficient ) day in ,day out, passing through multiple hub distribution points in a crowded city of 10 million .
Minimalistic symbology (as a lot of Dabbawallas can’t read) yet everything got there and everyone gets paid per delivery every single day at the end of the shift.
Couldn’t work it out after computing the algorithms and applying want they “learnt”.The issues just became exponentially harder.
You know, I went to the website to checkout this thing, maybe help out our bro Sam here. And the knife that he shows here? That's $134, after discount. The while set? Nearly 4 times that!
Maybe it's my country and its high inport costs? I don't know...
Still, il keep my $20 knife and sharpen it when it gets dull, use it for a few years and get a new one.
Oh....good video by the way. Hey! I've got my priorities okay?!
As a computer science student i have to complimet you! Awsome 6 minute summary of a topic, i would regard as one of my hardest during my bachlelor degree!
I was gonna quote that super long sentence at the beginning of the video and then make fun of it.
But it’s literally so long to quote. I just can’t. 😂
_Hey, psst-do you want to get rich quick? Have you exhausted all the other get-rich-quick schemes on the internet?_
_Do you have absolutely no marketable skills because you pursued a degree that became obsolete shortly after graduation due to an unstable and rapidly shifting job market, which then ironically drove you into crushing student-loan debt that compounded with the pressures of late-stage capitalism to create a predatory cycle of poverty that has ultimately forced you to desperately scrape the internet for schemes to support yourself financially?_
2:06 Called me out lol
5:30 swiss army pocket knife : amateurs
Teacher: the test will be easy
The test:
There is also another problem with a prize of $1,000,000. It is called Beal's conjecture. The problem asks whether or not there exists three coprime natural numbers a,b,c such that aˣ+bʸ=cᶻ, when x,y,z are all at least 3.
Wasn't that solved? I thought numberphile did a video on it.
Sudoku is a great example of a problem that’s easy to check but hard to solve.
Wasn't expecting that opening, but y'know I'll take it
His sponsor, “made-in”, is no joke. I have some made-in products and they’re absolutely amazing.
Ok let's see how well he really does when trying to describe this
Sam, I don't have any expectations in you
The ad in the end was like "and here is the base for trying to crack down this world class math problem, and here's a knife". I honestly thought the knife was for protecting yourself from criminals trying to steal your prize money in case you got it.