The Sudoku Proof That P=NP

[@damasoscibetta]

Beautiful solve, Simon! damasosos92 here: as dumediat already said in another comment, this one was set for a speed setting competition in 90 minutes, but it turned out so well in terms of pacing that I decided to polish it a little bit and publish it! I'm so glad to see that featured! And thank you to everyone who recommended P=NP to this channel!

[@arcanelizard4646]

You did an absolutely lovely job with this. It's very rare for me to spot a break-in in the first 5 minutes and still stop and think to myself "what a beautiful first step." Using two given digits to immediately figure out which pentominoes house which constraint and then immediately being able to place a digit based on that (the 2 in box 1, which can be found very quickly when thinking about the geometry of its pentomino's constraint)... just sublime. Normally if I spend 30 minutes on a puzzle just to find out I made a fatally wrong assumption very early on, I'll give up and watch Simon solve it, but even just from doing the break-in correctly I knew I would regret not solving it myself.

[@trace_tomorrow]

Quite enjoyed this puzzle! Multiple satisfying aha moments when I was solving. Great work!

[@katiekawaii]

That's amazing. Excellent puzzle.

[@StephanSpelde]

It hurt my head, but absolutely loved it!

[@dumediat_]

Would it blow your mind to learn that damasosos92 set this puzzle in 90 minutes for a speed setting contest? Because it certainly blew mine when I solved it during judging!

[@lorenzkarbach1352]

That is absolutely insane.

[@ArgumentumAdHominem]

What?? How is that possible? These people must live in a different universe in terms of cognitive ability

[@hom-sha-bom]

This sudoku had me saying ridiculous nonsense like "if this cage has a 1 in it, then it can't have a 1 in it"

[@epeo22]

A reference to the halting problem proof? If it is, I LOVE IT!

[@eninn]

LOL! I'm genuinely curious, which scenario lead you to say that?

[@saschabaer3327]

The P=NP thing is exactly the other way around. In simplified terms, a problem in P is one that can be solved quickly. A problem in NP is one where if you guess the solution, verifying that it actually works is quick. Like with sudoku, if I give you a classic sudoku it can be quite challenging to solve, but if I claim that I have a solution, all it takes is for you to go through every row, box and column and check for duplicates. It is easy to verify sudoku, but unknown whether there are fast algorithms for solving ones (specifically, ones that stay fast if you make the set of numbers bigger, like go to 16x16 or 25x25 and so on)

[@AaronTheGerman]

Just to expand on this a bit, if P = NP was true, that would mean all the problems that are easy to verify were also easy to solve. If someone could prove P = NP, that would make a lot of scientists very happy, because it means all those really hard problems that need huge computers to solve would suddenly become much easier. But it would also break the internet, because finding out someone else's password would suddenly be as easy as checking if it's correct (I'm simplifying a bit here, but you get the idea).

[@darci4434]

I still don't understand why P=NP can you clarify it for me

[@AudreyAzura]

@@darci4434That's the thing: we don't know if it is true or not. It is something researcher hope is true, but we don't know if it is yet.

[@darci4434]

ok@@AudreyAzura

[@fakjbf3129]

@@AaronTheGerman Proving P=NP wouldn’t break things right away because you would still need to actually find the individual algorithms. Proving P=NP just means that those algorithms definitely exist somewhere if we search hard enough.

[@floydmaseda]

It really bugged me that the thermometer cages were renban color and the renban cages were german whispers color lol

[@Anne_Mahoney]

yeah, I noticed that too! 😺

[@emilywilliams3237]

Me too! I didn't really care what the other color was, but it confused me a bit to have purple used in a puzzle that has renbans - and not being used to indicate those cages!!

[@TheRedReid]

I won't go over P=NP here as others have already done so far better than I could. However, I wanted to add that one of the reasons that the P=NP problem is so important is because all of our cryptography (I.e., encryption/security) is built upon the assumption that P does *NOT* equal NP. If it was proven that P *does* equal NP, it would mean that there exists some algorithm that would make it trivial to bypass all of our security. One can imagine the ramifications...

[@praematura]

I initially was just going to watch Simon's solve, but once he started talking about how to determine which pentominoes were which type of constraint, I immediately saw the break-in and went to try the puzzle... finished in 27:33 (conflict checker off), so I think I did okay! 😄 Wonderful puzzle from damasosos92!

[@jakemaranzatto6514]

I am once again counting this as my daily reading

[@longwaytotipperary]

😄

[@inspiringsand123]

Rules: 05:26 Let's Get Cracking: 09:01 What about this video's Top Tier Simarkisms?! Bobbins: 3x (08:39, 21:25, 43:47) Three In the Corner: 2x (44:58, 45:03) Phistomefel: 1x (00:23) And how about this video's Simarkisms?! Pencil Mark/mark: 17x (16:53, 18:51, 20:38, 20:42, 26:44, 32:14, 32:59, 33:05, 34:32, 35:23, 35:39, 36:11, 36:14, 37:58, 46:12, 48:14) Ah: 13x (11:31, 12:45, 15:41, 18:06, 18:10, 22:30, 26:16, 30:51, 30:56, 33:05, 37:47, 48:53, 50:10) Sorry: 6x (17:38, 19:40, 22:34, 42:51, 44:12, 49:38) Goodness: 4x (20:56, 21:32, 28:53, 50:45) By Sudoku: 4x (13:08, 27:52, 40:34, 48:50) Cake!: 4x (04:07, 04:07, 05:15, 05:19) What on Earth: 3x (07:00, 09:09, 27:17) Fascinating: 3x (08:54, 51:48, 52:21) Wow: 3x (22:18, 46:46, 51:39) Useless: 2x (26:27, 26:42) Naked Single: 2x (43:16, 45:05) Beautiful: 2x (24:39, 35:48) In Fact: 2x (05:32, 24:11) Whoopsie: 2x (24:58, 24:58) What Does This Mean?: 2x (14:49, 24:50) Weird: 2x (31:17, 41:16) Good Grief: 1x (46:52) Bother: 1x (48:53) Clever: 1x (41:21) Bingo: 1x (18:10) In the Spotlight: 1x (45:00) Stuck: 1x (14:40) Horrible Feeling: 1x (43:32) Lovely: 1x (49:22) Brilliant: 1x (51:42) Gorgeous: 1x (33:49) Hang On: 1x (38:32) Magnificent: 1x (52:13) Stunning: 1x (35:48) Obviously: 1x (33:35) Most popular number(>9), digit and colour this video: Ten, Fifty Eight, Seventy Eight (2 mentions) Five (77 mentions) Green (10 mentions) Antithesis Battles: Low (9) - High (1) Even (4) - Odd (0) Lower (4) - Higher (0) Row (17) - Column (9) FAQ: Q1: You missed something! A1: That could very well be the case! Human speech can be hard to understand for computers like me! Point out the ones that I missed and maybe I'll learn! Q2: Can you do this for another channel? A2: I've been thinking about that and wrote some code to make that possible. Let me know which channel you think would be a good fit!

[@honeybee8194]

What a fun little bot! I’ve been studying NLP recently and you’re absolutely right. Human speech is very difficult for a computer to understand! If you were to introduce this bot to other channels would you separate the training sets for maximum Simon accuracy, or would you have them together so it can learn as much as possible?

[@Petrolhead99999]

It appears that it picked up a duplicate "Cake" at 4:07. Great bot, cheers!

[@Kurgosh1]

That felt way more difficult than Simon's 40 minute solve would suggest.

[@dinane]

As a computer scientist I could not resist this one as soon as I saw it! Fun puzzle, but still we do not know if p=np just yet ;-)

[@longwaytotipperary]

Lots of fun stuff to watch on the channel and on patreon!! Sudokus, crosswords. Luckily, I just bought a 12 pack of microwave butter popcorn. Let the fun begin!! 🍿

[@emilywilliams3237]

Hilarious!

[@LiquorStoreJon]

9 given digits?? What a treat on a Friday!

[@HunterJE]

16:18 A nice clear way to visualize the point about where end points are allowable is using checkerboard/bishop color logic-an orthogonal path covering an odd number of cells will always end on the same color it started on, and since the 2 is not the same "color" as the leg of the P it cannot be the end

[@HunterJE]

(This kind of "parity vs checker color" logic is used a lot in the pencil puzzle type _rassi silai_, which involves placement of lines and endpoints filling regions in a grid)

[@Paolo_De_Leva]

@@HunterJE Thanks for reminding me about that property of "non-diagonal" paths.

[@HunterJE]

39:59 "This is the bulb, and that's going to take all the pressure off"-if you can consider can-narrow-down-to-two-options "all the pressure off" (since it's the bulb and 3 steps away from the 5 it can only be a 12, and as a bonus the cell to the left can only be 23 for similar reasons)

[@CorrectHorseBatteryStaple472]

Proud of myself for seeing that one fairly early on. That fact also makes a 123 triple.

[@emilywilliams3237]

So interesting!! I think that this is a hard puzzle, but I actually think I probably can solve it. It is clearly worth the effort and time - it is a beautiful puzzle with an interesting logical method. (But Simon, to use purple to color cages that are NOT renbans when there ARE renbans in the puzzle - you are breaking my brain!) Thanks for this wonderful video.

[@LaytonBehelit]

As a by the way Simon, P=NP is the other way around, solving a sudoku easily means the solution can be verified easily, but if I give you a filled sudoku grid that fulfills all the rules the solution path is not necessarily easy (this is not mathematically precise but an allegory).

[@jonh6585]

I thought it was more a completed sudoku can be proved correct instantly...but the start can't be easily proved solvable without solving it (but actually sudoku is not truly NP as you can use methods to check it will resolve)

[@LaytonBehelit]

@@jonh6585I'm not certain whether proving that a solution exists is in NP. Intuitively it is certainly easier than trying to solve the sudoku, but they might be equivalent problems. I'm talking about solving though. Sudoku with nxn grid size has been proven NP-complete and solving a 9x9 grid is in P because fixing n is just making the complexity function a constant.

[@abcadef6171]

@@jonh6585Sort of: it's trivially easy, given a completed grid of a sudoku puzzle, to check if that's actually a solution to the sudoku grid that you started with (you just check that each row, column, and box has the digits 1-9 once each in, and that it contains all of the given digits from the puzzle), but it can be relatively hard to solve it given just the original constraints. P=NP essentially says that if you've got a problem whose solutions can be checked quickly, then the actual solving problem can only be polynomially slower to solve (this is still not perfect, as P=NP is about decision problems, but it's closer).

[@woodchuk1]

@@abcadef6171So basically, if we can prove that P=NP, does that essentially mean that any problem with a solution that can be quickly verified as correct also has a solution path that allows us to find that correct answer in roughly the same amount of time?

[@saschabaer3327]

​@@woodchuk1Pretty much, yeah, though there’s a big caveat in terms of practicality: all this “polynomial time” stuff doesn’t really mean fast per se, it means that the problem scales up not too badly as you make the inputs larger. Like for sudoku, our usual sudoku uses 9 digits, so we could say the input size is 9. The question is then, how much longer does a solving algorithm take as you scale that up and go to, say, 16x16 sudoku, or 49x49 etc. If the time required, say, doubled for each additional digit, then that would be exponential growth, which is not in P. But we know verification is in P: you only need to scan each row, each column and each box, so you basically need to go through the puzzle three times, which takes time proportional to n² if the puzzle uses digits 1 to n. That’s quadratic time, which is pretty fast (and polynomial). Since verification is in P, determining solvability of sudoku is in NP. But even if P=NP, a solver would probably be MUCH slower than linear time, just not as bad as the previous example of exponential time.

[@TurquoizeGoldscraper]

Tried it and figured out which were thermos and renbans, but got stuck shortly after. Good solve, Simon.

[@fredgoodyer4907]

Me at 48:52 : I thought he would solve this by thermo logic, but instead he will use sudoku... Me at 49:20 : Here we gooooh wait, he's missed the sudoku, and uses the thermo logic, classic! I love this channel, the puzzles, and your solves, Simon; you are a gem 💎

[@anaayoung9142]

This was a great puzzle, I would not be able to solve it without your break-in at the start. Thank you! 😀

[@khayman9574]

Once I figured out that the Ns were renban lines, I used the 5 trick (there must be a 5 in every 5-cell long renban) and the puzzle just unraveled.

[@HunterJE]

28:36 One small thing available here - since you can rule out a 9 from the two selected renbans, you can actually pencil mark 9 to two cells in box 5, since if it's in the thermo cage it would have to be the tip

[@jeremypalmer5695]

I just had an idea for the puzzle masters, what if we flipped the meta and made a sudoku puzzle with the most given numbers. I.e. give Simon a sudoku puzzle without Sudoku. Have a grid with only a chocolate teapot of deadly pattern left, and then have a list of hidden rules that are needed to disambiguate it.

[@psiphiorg]

The break-in was quite nice, although it took a bit of hunting for me to find it. My time today was 33:22, solver number 145! (All digits combined would make a renban line, if duplicates were allowed.)

[@MattYDdraig]

43:31 Intricate and clever. The restrictions on paths for the thermometers proved particularly useful.

[@Raven-Creations]

What a fascinating puzzle. Not easy to get started, but it seemed to accelerate throughout. Although our solves were similar, I think you found a slightly smoother path than me.

[@jussikuusela7345]

I found the same break-in as Simon. I then went to track the 5's but made a mistake and got stuck... then watched the video to 24:54, saw my error, and eventually finished the puzzle.

[@selenasilverstep7981]

col 6 and col 7 with two cages stacking on top of each other like that had my spidey sense tingling like crazy

[@chocolateboy300]

I got 54 minutes. This was a really nice puzzle and very satisfying to see how each deduction gave a treasure trove of information around the grid.

[@Tyranisaur]

Out of the 5 cells that make up a P shape, both the tip and the bulb end of a thermometer can only go in 3 of those cells.

[@VeritasUnae]

So this one was an ordeal for me. I spent 60 minutes going down frustrating rabbit holes and not really getting anywhere. So I took stock, cleared the grid and started again. I wish I had realised that 4 couldn’t be in the 2 pentomino - instead, I realised that either 2 or 8 had to be in the middle position of the left-most N pentomino from the starting placement of digits - they couldn’t both go into the top P pentomino because in both cases it would break either renban or thermo logic. And in either case for the N pentomino, you couldn’t make thermos with a 2 or an 8 surrounded by two squares in either direction. So Ns had to be renbans. But, like yourself Simon, I didn’t want to assume that immediately meant that P pentominos were thermos - I wanted to make sure that was something that could be deduced. In the end, after slightly over 103 minutes (so 163 all up) I managed a solution following that path of logic. Very satisfying once I got those deductions tumbling!! What a wonderfully set puzzle, and it was great to see your solution c: I still have much to learn.

[@MasterHigure]

Indeed, P=NP is commonly explained in terms of whether a puzzle has easy-to-check solutions (i e. the problem is an NP-problem) means the puzzle is easy to solve (it is a P-problem). Of course, the formal definition is more specific than this. There are problems/ puzzles in this space that have the property that if they are indeed easy to solve, then they are ALL easy to solve. And if I recall correctly, someone made a general way to translate these universal puzzles into sudokus. Or rather, many separate sudokus with interlocking boxes. Which is to say, if you can solve such sudokus with a general algorithm that's significantly better than brute force, then you can solve all of these problems.

[@iabervon]

P is how hard it is to check whether someone's solution to a puzzle follows the rules. NP is how hard it could be to find a solution yourself, if the setter hasn't left any clues. The question is whether NP is exponentially harder, which is to say that there would be some sort of puzzle where the only technique for solving it would be total bifurcation on a portion of the cells. P=NP would mean that there's a solve path for every puzzle that's faster than just guessing, even if the setter didn't plan one. This question applies more to puzzles like nurikabe with arbitrary size grids than sudoku, since part of it is being able to increase the grid by a fixed amount and double the difficulty by adding another spot that needs to be guessed.

[@gi0nbecell]

I can‘t believe that I was more efficient than Simon this time. I never managed this before, but I solved this puzzle in 34 minutes - which for me is incredibly fast.

[@thenatundi9009]

I'm glad that assuming that the Ns were renbans after figuring out the Ps didn't ruin me in my solve! Not sure between me and Simon who is reading the rules long to be frank

[@tomhoward4906]

I must be missing something in the rules. Since each pentomino is filled with either a renban line or thermometer, it means no pentomino can have a repeated digit. From the start then, we see R4C1 must be a 4, since it’s the only cell in box four that isn’t in a pentomino with R6C4. But then by a similar argument we can conclude that R1C3 must be a 4, yet it contains a given 7.

[@travisporco]

the pencil marking needs some greater flexibility, because you have one type of pencil mark that is box-centric and one that is digit-centric. You need to be able to say "the digit in this box is in this set" and also to be able to say "the box for this digit is in this set". But for the latter type of pencil mark you have this restriction based on box boundaries. This is not logical.

[@MaierFlorian]

The title triggered me juuuuust SLIGHTLY :'D Maths/Informatics Input for all interested: P stands for (deterministic) Polynomial and is the set of all problems that can be solved (by computers) whithin time t = any sort of polinomial (but not exponential) function. NP stands for Non-deterministic Polynomial and means "If I would have an oracle that can can tell me the next part of the solution whithin P time", than the whole function is polinomial. We know P is a subset of NP, but we don't know if P = NP (this is the part where Simon stated to make a lot of money, if someone could prove or disprove P = NP).

Also, if you're going to get into what polynomial time means, polynomial functions and such, then you should probably also say what they are functions of, otherwise there's no point getting into it. They are polynomial functions of the problem size (as in number of elements, e.g. number of cells to fill).

[@MaierFlorian]

@sorry, thanks, I guess that was a "I'm so used to it that I thought it would be clear to everyone" x)

[@jkdewar]

Simon, I think you sound like Neil Tennant from the Pet Shop Boys when you sing "that's 3 in the corner". 🎵🎶

[@Sev92]

Funny how the wording of the rules gives away that the P's are the thermos

[@piarittersporn]

Absolutely wonderful and pretty puzzle.

[@wrightn9]

In sudoku, it is very easy to show that a solution is correct once you have the solution. You just have to check that each row, column, and box has the digits from 1 to 9. However, it is always much more difficult to solve a puzzle from scratch. Checking takes seconds. Solving takes minutes or more. P versus NP asks whether the set of questions that are "easy" to answer is different from the set of questions that are "hard" to answer.

[@RichSmith77]

Eh? By definition, a question that is easy to answer isn't also going to be a question that's hard to answer. So I don't understand how the two sets could possibly be the same?

[@BryanLu0]

Simon, answer check for Sudoku is very easy. Just scan the rows, columns, and boxes. I know you don't like to do Sudoku in Sudokus, but the rules of Sudoku tell you how to verify the solution. On the other hand, you've featured a few puzzles which computer solvers considered very hard

[@BozoTheBear]

Fantastic puzzle!

[@MyriamTT]

Excellent puzzle !!!!!

[@spudd86]

Not quite, PvsNP is about the difficulty of finding a solution vs checking if a potential solution is correct. Basically if P=NP anything that it is fast to check that a solution is correct also means you could find the solution quickly. The existance of puzzles sort of depends on P and NP not being the same set.

[@leefisher6366]

With a solve time of 42:42, you definitely found the answer to this one, Simon.

[@IgorPellinen]

Simon messed with the colours. Renbans should be purple, thermos are usually grey…

[@srwapo]

55:23, took like 15 of those minutes not seeing what the 4 in Box 5 was doing the whole time. The rest of my time was struggling with renban lines, like usual.

[@angelowentzler9961]

P and NP are terms used in the field of 'time complexity', ie answering the question: how much longer would an algorithm take to find a solution if the input increases in size? An algorithm's time complexity is expressed as a function of input length N. For P (polynomial) algorithms, this function would be a polynomial over N, so N, N^2, N^3, etc For NP (polynomial) algorithms, this function would be an exponential, ie 2^N,3^N, etc. In general an NP problem will result in a form of search with backtracking ('bifurcation'), while P problems do not. The as-of-yet-unsolved question is, can any NP solution be changed in such a way that it becomes a P solution? We can't prove it, but we strongly suspect the answer is 'no': some problems are intrinsically unsolvable in P time. (Interestingly, while it may take NP time to find a solution, verifying that solution usually DOES take P time!)

[@chaosme1ster]

Was looking for this reply, hoping it'd be there, so I wouldn't have to write it :).

[@_-_-Sipita-_-_]

34:43. i tripped over and over like shit

[@guruone]

P vs NP asks whether every problem whose solution can be quickly verified can also be quickly solved by a computer. ...... AI told me so :PPPPPPPPPPPPP .....ps don't 4get Cyadonia for the next game stream. U both will LUV IT

[@Gonzalo_Garcia_]

14:42 for me. Nice puzzle!!

[@TheSane42]

Thermo in box 4 getting completely ignored despite knowing where the bulb was and that it had a 5 in the 4th position. Missed out on a very easy and quite useful 123 triple.

[@frankjiang1857]

Finished in 36:22. I actually went at least a couple of minutes into wondering if the N pentaminos were thermometers and after looking at some possibilities in the N pentaminos looked at what happened if P pentominos were renbans and realized that the P pentomino mainly in box 5 with the 2 in couldn't be a renban because the 4 was looking at all the possible spots for it. What a waste of time where if I had just looked at renban possibilities on 1 pentomino I'd have easily figured out which was which. Also, shouldn't this puzzle be titled P != NP? Since Ps and Ns are not the same thing? Fun puzzle, either way!

[@persephonetsebelis8948]

This was such a nice solve. 14:56 mins for me!

[@parkerbond9400]

I think the rules are a little vague, there are technically 2 different shapes of N and 2 different shapes of P pentominoes, but the rules just specify different shapes, not different types of pentominoes...

[@Squishy3757]

So since you know P is thermo then ask yourself where 8 and 9 go in box 1. 8 can’t go in row 3 and it can’t go into the P so it must go in cell 6 which is the green cage. Because of where the 8 is placed in that cage it must not be a thermo and green must be renban.

[@damsonrhea]

"I probably can do more with the thermometers." *Seven exist as the tip in the upper left corner, strongly restricting the cage.* Goes back to renban.

[@Timlagor]

Weird that anyone thought this was 3* difficulty

[@karsaanita]

It bothered me so much that the thermometers were purple and the renbans were green 😅

[@sebastiancruxian6518]

Anyone here learned about P vs NP by watching the series Elementary?

[@MrBluemoon74]

Maybe you should use a different color for the dodgy pencilmarks?

[@alienrenders]

42:54 is my time. This was a tough one.

[@themuffin437]

34:52 for me, very satisfying solve

[@pranavsetpal]

I've not seem the whole video yet, but P=NP argh!!! I can already see the P-pentominos having the same constraint as the not P-pentominos! Edit: I was wrong. Missed opportunity though

[@xerodeus2337]

imagine if it actually was all thermometers... would have been fitting for the name... that P=NP, cause the Ps being thermos, is what the Ns and Ps are. Despite saying Ns and Ps are different lol.

[@butsukete1806]

Yey! computer science is over, good thing because it wasn't about computers and it was never a science.

[@LednacekZ]

33:22 for me. very confusing solve.

[@jaron5843]

took 19:55

[@robalanzo4519]

This is a millennium problem. Get your million dollars

[@HolySerega]

SAME SHAPE HIDE THE SAME CONSTRAINT reading the rules explain the rules

[@bobblebardsley]

"All of the P pentominoes are renbans and all of the N pentominoes are also renbans" would satisfy that, it's not a question of whether all cages of the same shape hide the same constraint, it's a question of whether the two different shapes of cages _necessarily_ hide different constraints.

[@mxpxorsist]

24:34 for me

[@robertcousins2274]

32:52 for me

[@theredstoneengineer6934]

21:24 for me

[@bobfish7699]

Nice puzzle - I thought the solve was poor. Many opportunities for Thermo logic missed - especially in Box 4.

[@Daymickey]

Hmm interesting. Can you give an example?

[@bobfish7699]

@@Daymickey Yes I could. But i'm not going to - because this is my feedback and I didn't leave it to once again get involved in a debate.

[@Daymickey]

Oh ok no problem. I was just interested in seeing that thermo logic because I was confused why he didn’t use it more throughout the video.

[@terracottapie]

Reading out Joya's birthday reminded me that there was a kids' show, about 40 years ago, where a beautiful woman named Joya Sherill, a former singer with Duke Ellington's orchestra, used to sometimes open the show by reading out birthdays sent in by the parents every morning. I had my name read out on there when I was pre-kindergarten. Cool coincidence! kzbin.info/www/bejne/nmTOqnykjq5giac

[@longwaytotipperary]

I watched that link. Joys is beautiful and the lamb is adorable!

[@romsegal7400]

first