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.

me a computer science graduate: ah yes, my nightmares have returned

Just give me 3 flintstone gummies, I'll handle them.

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!

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.

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.

> One of them has gotten significantly more attention and more failed attempts than the rest. *The Riemann Hypothesis would like to know your location*

Thanks for the callout... 0:08

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.

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.”

Why do I feel like half as interesting son will make a video in 2069 titled "why did they 2020 Olympics happen in 2021"

“One of them has gotten significantly more attention and failed attempts to solve it than the rest - P vs NP” *Riemann: Hold my hypothesis*

Fedex: thx for solving the hardest problem in the world Me: np

I have a feeling that Sam wants us to solve this, so he can claim the money.

Next video on Wendover Productions: The Logistics of Why You Should’ve Paid Attention in Math Class

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.

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.

It would make more sense if the promo code "half" gave 50% off...

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.

The math problem my mom expects me to solve after watching the 3 minute video

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 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.

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.

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.

We did it. The "Who Wants to Be A Millionaire?" in math is here.

I thought a way to solve the P-NP conjecture. Unfortunately, this comment bar is to small for writing it in.

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)

2:51 this graph also applies to other activities

Does P = NP? Only if P is equal to zero or if N is equal to one

0:25 I think you forgot the part about having no skills attatched to your degree

Your promo code “HALF” doesn’t work. The checkout process says “The provided code is invalid.”

Literal utopia, we all know it's too good to be possible, but good luck proving that as an abstract

Why can't math grow up so it could solve it's own problems?

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.

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.

Genuinely thought you were gonna talk about the Riemann Hypothesis since that problem is even more studied than P vs NP.

The ad that played before the video just fit perfect

"-------, Found out why the box has a band-aid," lol

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

This is amazing.

Therapist: "Stock Footage Anonymous Hacker Guy can't hurt you" Stock Footage Anonymous Hacker Guy: 4:04

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.)

i watch these when im high and it always fades into commercial in a very sneaky way.. i kinda love it

Expected a Navier-Stokes rundown and got an N = NP instead Still loved it nonetheless

2:52 they definitely knew what they were doing with the labels on that graph

2:06 Called me out lol

Hardest problems to solve When will be another bricks video

"Half" should be a 50% off promo code, change my mind.

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.

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

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

Am I the only one who actually doesn't want to join my friends at the "P" party....? That's just nasty....

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. 😂

I would like to get rich quick, this is why I chose to become a painting major. Never mind u said math.

The only video without shilling in the first 10 seconds has a 20 second long rant written by a redditor

5:19 I gotta be honest, that's the smoothest progression I've ever heard.

0:01 man went rapping 😂

Even this knife you will replace after a year. Learn how to sharpen one, and your 20 dollar knife will last a long time

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,?

5:30 swiss army pocket knife : amateurs

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 😂

Sam sounds like he solved this equation.

Now I have context for that one Elementary episode.

Teacher: the test will be easy The test:

P loses because he's alone and NP is two so basically its a 1v2. It's been a while where my 1 million?

Your traveling salesman example is incorrect. That's an NP-Hard problem, not NP.

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?

Me who fell asleep in the middle of the video then woke up in 5:00 : how tf did math problem became cooking problem

To clarify, the P versus NP problem is a computer science problem, not a mathematical problem.

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.

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 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.

Today I learned: The probability of a blue lobster existing is widely touted as being one in two million.

Crazy Fact: In 2006, a Coca-Cola employee offered to sell Coca-Cola secrets to Pepsi. Pepsi responded by notifying Coca-Cola.

I feel called out (2:09) But I did not notice any mistakes at all, good job on this one :)

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 like the knife segment...thanks. As well great video

"Thanks for solving the hardest problem in the world." Me : *No need to Thank me*

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!

You've got a better shot at winning the lottery than cracking these maths puzzles.

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

This is mentioned in cyberpunk settings if I remember correctly, shadowrun had something like this

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 UKposts will have to suffice?

Woah, just a few days ago I was browsing Wikipedia and got to the article about this. I didn’t totally understand it so, thanks for this video

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.

0:20 Yes!

Sounds interesting and i would want to try solving it but i didn't even understand your simplified version of the question so i dont think i will

5:22 "Found out why the box includes a bandaid"

I feel a tweaker is likely to solve this. They stay awake for a long time and are laser focused on something as long as they are not around flash lights or power cords😂

Wasn't expecting that opening, but y'know I'll take it

I didn't think I'd be listening/looking at P problems for about another 30 years.

It's nice you used a map of Barcelona and marked the route from my home to the comic shop

1:26 Humorously enough, I actually have my TI-84 Plus sitting next to me right now.

You should talk about the Collatz Conjecture.

Your visualization of the travelling salesman problem actually shows the problem of finding the shortest distance between two vertices in a graph, which is solvable in polynomial time. The distance between each pair of points is given. The problem is to find the order in which the points shall be visited.

You know full well that the HAI team saw *that* meme and didn't get it, resulting in some research and the subsequent creation of this video.

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.