A Maths Puzzle: Euler Trail and Solution

[@TrainTsarFun]

We wasted so much time in school trying to figure this out. I am glad KZbin didn't exist back then - or else I would have wasted my time on something else.

[@the_hippie88]

Well, good👍

[@loganshotrod4x464]

When I was about 10 or 12 years old, my science teacher showed me this puzzle & told me that Albert Einstein was the only known person in history to solve it on his first try. Looking back, he was probably bullshitting me, but he did his job as a teacher by sparking my interest in math & science. I played with this puzzle over the years & ultimately conceded that it was impossible. If you have determined that it's impossible, you have essentially solved the puzzle. Admitting that something is impossible is like admitting defeat, it's no wonder so many of us get hooked on this & keep trying. Excellent video, thanks for posting.

[@singingbanana]

Logan's Hot Rod & 4X4 This is a great comment. I think I like your teacher.

[@derpmanaveragegamer7791]

+singingbanana o-o still trying. I MUST FIND THE SOLUTION

[@TheDrB0B]

Logan's Hot Rod & 4X4 A basic principle of the scientific method is that if you cant prove that something is impossible then it must be possible. So what you say is true, but only if you have the evidence to prove that it is impossible in fact.

[@NS-cy1hh]

Logan's Hot Rod & 4X4 buts it's proven to be possible just google images- 5 room house puzzle solved

[@LiarraSniffles_X3]

+Nathan St John Shhh, let them have their stirring speeches about admitting defeat, they learned more than you or I with our google cop-out.

[@mueezadam8438]

I’m convinced that Euler is the most productive mathematician, nay, human in the entire goddamn cosmos. He’s so prolific it’s not even a joke to say his work in today’s era would’ve probably earned him more PhD’s then some universities have ever handed out.

[@dalirkosimov4623]

Underrated comment

[@JustSomeLeagueClips]

I have the solution. A marker the size of the puzzle is gonna do the trick. 1 line, all edges touched ONCE.

[@YG-yr3rg]

That is not a line though...

[@thelastdino8019]

+Matthew Sakamoto Point to point so a line ;) GOOD JOB LALUGAMING lol

[@aengusroberts2685]

+TheLastDino but a line has no end or beginning, it's not point to point what you see is just part of the line, and it is indefinitely small so it cannot be that thick.

[@xxx-lz3sy]

+LaluGaming no that means you will cross the lines infinitely many times, sit down

[@trenthaines3886]

+LaluGaming Theoretically lines dont have width

[@jdfranklin2]

I paused when he said pause to try it out and spent an hour and a half doing it before pressing play again :(

[@maheeiyer5235]

I feel your pain my friend.

[@DjNerDee]

+Justin Franklin Lol I starting cussing saying it was impossible, as soon as I pressed play he confirmed my thesis

[@smokingorca]

that's great! this is the most important thing we do, try and fail. everybody that succeed has failed and learned. only idiots ars afraid of trying

[@maocharlisme]

Ahaha 'President Trump' calling a man who ACTUALLY has spent alot of thought on a problem an idiot! The irony!!! ='D No, this even surpasses irony. It's in a whole other league! It's some sort of... paradoxal hyper-irony! x'P edit: *disclaimer*: I'm pretty sure that I wasn't talking to a figment of my imagination and that this mysterious "President Trump" has silently deleted his comment.

[@mastervolume1508]

+President Trump Nice Person

[@Jimsdogwashbaldivis]

This puzzle has plagued me for 25 years since I was shown it as a kid!! Thanks for finally putting my mind at ease

[@doomilaakso]

Haha at all these people. "Technically I did solve it I just went to another dimension and crossed the corners and went on a line so this is solvable." k guys, k.

[@Felixr2]

He said cross. Not intersect (sharing a point) but cross. In 2D, they're interchangable (kind of, cross is actually wrong but nobobdy would get confused), but in 3D, a pair of crossing lines is defined as a pair of lines that are not parallell, but also don't share a point. If they're not parallell but they do share a point, they're intersecting. So, if you consider the puzzle to be 3d, it's actually incredibly easy since all you have to do is draw a line that is not parallell to any line in the puzzle and also doesn't share a point with any of them.

[@ariesradke6193]

i teleported

[@EDeeseLex1]

I know how to do it! 1. Get a really really thick marker 2. Draw a huge line through the paper.

[@EDeeseLex1]

.

[@jacobdestroyer6485]

PivotMapping OMG ur a genius!!!!

[@rupin4004]

Caleb Stremcha fuck u

[@supergame2534]

ya 😂😂

[@toonoobie]

REEEEEEEEEEEE

[@NotQuiteFirst]

I thought about this in terms of there being two types of boxes - they either have five sides (odd) or four sides (even). With even-sided boxes if you start your line on the inside of this box it must also finish on the inside when crossing each segment, and likewise if you start on the outside you must finish on the outside. The odd-sided boxes are the opposite - if you start your line outside and odd-sided box, to cross each line you MUST finish on the inside, and vice versa. So to be able to escape from an odd-sided box and continue completing the puzzle, you MUST have to start on the inside of it, but given that there are three odd-sided boxes and you can't start your line inside more than one, it becomes impossible. If there were only two odd-sided boxes it would be ok because you could start inside one and finish inside the other, and you'd just pass through the even-sided ones on the way. When I worked this out I just gave up trying to find the clever solution and watched the rest of the video, only to find out there is no solution :P

[@GotYourWallet]

Yay, I figured it out myself before I continued watching. I drew a square and saw that if you start outside, you can only end outside. Then I drew a 5-sided object and saw that if you start outside you can only end inside. So if your line only has two ends there's no way your picture can have more than two uneven sided objects. I'm glad that there is no super smart way to solve it.

[@KingInky13]

Except there actually is a super smart way to solve it...

[@TheWolf-bf1uk]

What is it

[@jonigazeboize_ziri6737]

KingInky13 Nope, the diagramm at 2:29 contains all legal moves. It is impossible.

[@garrettday3623]

Thank you for actually explaining why it's impossible. I've been watching a lot of videos of puzzles like these and most of them are just reposted by people that don't understand the puzzle at all and it's very unsatisfying.

[@AmazingGriffin]

you sound just like the guy from numberphile...

[@singingbanana]

+AmazingGryphon Cool.

[@LLHLMHfilms]

He is one of the numberphile speakers!

[@AmazingGriffin]

LLHLMHfilms lol yeah maybe he went on numberphile once

[@Pouk3D]

+AmazingGryphon Once? He is one of the better known ones.

[@vestrocity9561]

+singingbanana Hi James!

[@bustystone]

roses are red, violets are blue, if you think you did it, you screwed a rule.

[@flunkshsucksattoribash5566]

roses are red, violets are blue, this comment is so true it woke king Neptune.

[@Deblivator48]

Roses are red Grass is greener When I think of you I play with my wiener

[@lastrequiem9245]

Roses are Red, that much is true! but Violets are Purple, so buddy Fuck you!

[@carmensandoval8650]

Roses are red, violets are blue, god made me pretty, what the hell happened to you!

[@carmensandoval8650]

Last Requiem ^^

[@monme6123]

Do you have to press the marker down so hard so it makes the ''RHHHHHHHHHHHHHHHHHHHHHHHHHHHH'' noise?

[@lilicruz1449]

ikr it gets annoying 😂💯😀

[@MrVagif1]

+Fennekchu ghahahha

[@pleasetakeasip4556]

You guys get easily annoyed, I'm sorry that you had to hear a sound that you dislike lol

[@cawfeedawg]

Love that sound!

[@beaudweiser]

yes

[@CasaErwin]

Since you used the word "cross" and not "intersect" then there is a technicality that makes this puzzle solvable, and that is that we can cross one of the lines (specifically the upper middle vertical line) from end to end just as you would cross a bridge. That, however, is the only way this puzzle can be solved.

[@screaminlordbyron7767]

Nice one.. I came up with this solution and you are the only other person who has pointed that out in the comment section. Exactly !!. Running along a line or edge is not the same as intersecting it.

[@ffggddss]

So three of the boxes are odd nodes, each having valence=5. You might notice that the outside region is another odd node, with valence=7. So there are actually 4 odd nodes, and 2 is the maximum for an Euler trail to be possible. (There must always be an even number of odd nodes, BTW.)

[@rohanmudumba2545]

My teacher gave us that problem in 3rd grade. I proved it was impossible then after trying the whole year. Wasted notebooks trying before I figured out it was impossible. I don't remember my exact proof but my teacher was really impressed. I went to a really advanced private school if you're curious about the advanced placement of math.

[@singingbanana]

Excellent. And you remember it! That's good teaching.

[@KenStentiford]

Rohan Mudumba what is a really advanced private school, and was it a really long year realising this is impossible?

[@everythingbobbywolfe]

Why are there dislikes?! He explained it and gave a proper video description. Do you people not like his accent?? Sheeese... KZbin smh

[@dirt2774]

Dislike bots

[@CairnOwO]

+Apocalypse For one, there is a wall that a line didn't go through, and two, he/she/you cheated and went through an intersection not an actual wall

[@RonWolfHowl]

He didn’t really prove anything. He just gave a blanket rule without explaining why it’s true.

[@yafi2475]

its because he said he would give solution. so it was kinda click bait

[@MsCrisKun]

He actually did. It has to be a pair number of lines since you want to go in as many times as you go out, except for the beginning point and the end point

[@2905sid]

I have found a solution but the KZbin comment section is too small to contain it

[@jamesnewman3851]

Its easy... figured it out in about 25 secs

[@Jimmyfpv_]

Classic Fermat trolling

[@superdrykidrobot]

there is an easy solution. think of this drawing as a 2 dimensionsional entity in a 3 dimensional space. you can go up and down through the paper. problem solved.

[@jacobhelbig6967]

Stop it, Pierre!

[@unjogratistheforbiddenmonkeygo]

uuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuhhhhhhhhhhhhh no

[@TheRedfire21]

It's important to note that this, as simple as it looks, is part of a revolutionary math paradigm that Euler did. I think one of the most remarkable problems is the "Königsberg bridge problem" which inspired Euler to do this kind of things (this is a cite from my geometry book, I do not really know if this is true and whether it is or isn't we can agree it makes for a cool story)

[@Jerryfan271]

3 of the rooms have exactly 5 walls. For a path to exist that crosses every wall exactly once, it must, for each of the 3 rooms, enter a times and leave b times. b > a only if your path starts inside the room. So for at least two of the rooms, a > b. There's no way to enter a room 4 times and leave once, or enter a room 5 times without leaving, so it must be that you entered both rooms 3 times and left both rooms twice. Since you entered once without leaving, your path must end in one of them, but also the other, which is impossible. So no path exists.

[@TheGreatRakatan]

Solving it involves proving that it's impossible. That is the solution.

[@nathandepater1881]

+TheGreatRakatan fold the paper you moron

[@aengusroberts2685]

+TheGreatRakatan Its not impossible it has been done

[@aengusroberts2685]

+Jalvlie sorry I was using different rules to solve the puzzle, I was using each black line as 1 wall: not multiple

[@AnotherGenericGamingChannel]

It is impossible, using some of Euler's maths and theories you can prove its impossible

[@TheGreatRakatan]

+Nathan de Pater Lol, oh this is gonna be good. Please explain oh master of planes.

[@GabrielAugendre]

To be really correct about Euler trails, you need to consider a “center point” at the outside of the design, because if you represent this as a graph you would have 6 nodes : one for each rectangles and one for the outside. Therefore you don’t have 3 but 4 nodes with an odd number of edges. Obviously the answer stays the same : you still need 0 or 2 nodes with odd number of edges to have an Euler trail in the graph.

[@MrDannyDetail]

It could be anywhere between one and four 'centre points' on the outside though I think, which is why he leaves that open on his diagram. I think it does gaurantee at least one more odd node on the outside though, given that there are nine crossings to the outside.

[@PrinceBarin77]

My grandfather showed me this and it kept me puzzled for years. Eventually I I worked out that it was impossible by virtue of the number of individual lines versus entry/exit points you need. But really it just reminds me of my grandfather, how wonderful he was, and how he always wanted to give me something to think about (or to keep me quiet 😂).

[@TheGrandexeno]

lol I immediately recognize the voice's guy being from the channel Numberphile

[@dPope9093]

People, please! I see the majority of people don't understand the term impossible. You've been given rules saying you can only cross through lines. You've then been told that this is proven to be impossible. And if it's mathematically proven to be impossible with these rules, then the only way you can think you did it is by doing it wrong. A lot of you say you can do it by using corners. Of course you can, but you can also do this by ignoring any other rule. I did it by completely ignoring half of the bricks but that clearly doesn't count. If I said to you, "This is impossible, but try to go through this maze by getting through all the doors while only going through each door once" and I saw you trying to walk through a corner of the wall into the other room because you're that stubborn, I would laugh.

[@woodfur00]

It's really funny that this came up in my recommendations today. I was reading about this exact problem last night. P.S. YES PEOPLE, IT'S THE GUY FROM NUMBERPHILE

[@heltrom716]

FuckYouGooglePlus true i was about to say that

[@icisne7315]

OMG you're that guy from numberphile :0

[@singingbanana]

Phonzo Cisne Yup.

[@ongansan]

+Phonzo Cisne OMFG

[@lalalalalalalalal567]

I knew it from the voice!! Then I scrolled down and saw this comment xD

[@nofanfelani6924]

+Phonzo Cisne its very familiar how he said "here" :D

[@mastervolume1508]

Some of these solutions I am reading are the same as saying 'I Solved the Rubic's Cube by pealing off the stickers and sticking the same color ones on the same side. Duh, it's so easy.

[@6infinity8]

Depending on how you tackle the problem, you can end up solving the Eulerian Trail problem which is polynomial, and more precisely linear in terms of the number of eges and vertices; or you could also end up with the Hamiltonian Path problem, and this problem is NP-Complete meaning that only exponential solutions are known (and it's very, very, very unlikely that we will ever find a polynomial time solution for that). Now of course this doesn't really matter here because the graph is not too big, but I thought this was an interesting fact to mention.

[@DerSim688]

I love all you guys saying "I Solved it" cause i know u did not :D He showed a proof that it is immposible. So, if u solved it, you did somthing wrong!

[@KWGTech]

Best way to check if its solved is to follow the line you drew, and highlight every edge you cross. Most "solutions" either have one edge not highlighted, which would normally not be detected.

[@MrLoftyDave]

It is possible to solve this puzzle, talk of going through corners and following on top of your own line is against the rules! To make it more specific; make a door in each of the walls, you must go through each of the doors only once without taking the pen off the paper. Hint: the answer lies in the phrasing of the rules, the solution is not mathematical. Have fun.

[@denilsonpw]

I paused the video before the answer and spent the last 3 days writing code modelling the problem and flipping tables wondering why my program wasn't finding any path bigger than 15 crossings. Hm... At least I wont forget what a Euler's Trail is. Mathematics can be a mean teacher, sometimes haha! (T_T)

[@kilodelta5562]

A kid showed this puzzle to me in the 6th grade in 1984. He moved away before I ever found out if it was solvable. Over the years, I have wondered was there an actual solution. Thank You for letting me off the hook, Lol

[@geminix365]

The second time I did it (And I though I solved it) I realized that the straight horizontal line counts as 4 lines, then the recangles have 5 lines to be crossed, odd number, that means that you only can end inside of it, but then you have 3 of them, it's like having 3 cages and you need to end in all of them at the same time.

[@iamrolfie3755]

geminix365 Thats bullshit since its impossible to solve.

[@SoraRaida]

"I solved it by making the puzzle 3D" "I solved it by crossing the intersections instead of the edges" And thanks for saying that you broke the rules of the game

[@xFlRSTx]

Each time you cross a line touching a box you either go from the inside to the outside or from outside in, so for a box with an odd number of lines you will go in and out sequentially an odd number of times and end up either outside if you started inside or inside if you started outside, so for each odd numbered box you will either start or end inside. This picture has 3 5 sided boxes and you can only start in one, and finish in one, meaning at least one of the boxes cannot be started or finished in and therefor cannot be crossed the requisite 5 times. edit: the outside of the boxes is also an area with an odd number of sides touching it(9) so there will actually always be at least 2 sections missing 1 cross over, not to mention if one section has a side not crossed then it must have a section next to it also missing a side.

[@huohization]

Doesn't the fact that the bigger squares have unequal entry/exit points mean that it's impossible? You need IN and OUT points and that's 2. Maybe if there was an equal number of the big squares connected it might work but there's 3

[@timothyjonesjr.6161]

+Kusinuppi Exactly

[@twalker1111]

This seemed like a great puzzle, and one I could surely solve...after carefully drawing the puzzle I proceeded to draw out all possible combinations of lines by creating triangles throughout the puzzle (which he basically does). Within ten minutes I realized that this puzzle must be impossible. However, after staring at this piece of paper for an additional hour I realized that the only way you could solve this puzzle was thought the intersections (corners)...I then thought to myself "Hey, maybe I'm not as stupid as I thought! ". Of course, another five minutes later I realized that this was cheating and confirmed that I am indeed stupid. Being unable to solve an impossible puzzle isn't grounds for stupidity...however, realizing a puzzle is impossible within the first 10 minutes, and then staring at the puzzle for another hour stubbornly searching for an "ah ha" moment is bang on stupidity. Sorry Dad, but I'm a dummy.

[@DaveScottAggie]

Cool demonstration. I recall reading about this. The Königsberg Bridges problem. For years nobody could find a solution or figure out for sure that the solution did not exist. It was Euler who proved that it was impossible in similar fashion that you did. I think it was something like - If you enter an island (box) then you must leave it, so there need to be an even number of bridges (except for start and finish as you stated).

[@theuncalledfor]

Wait, it took them years to figure this out? With several people trying? ...I solved this puzzle without help a few years back. My father presented me with the puzzle and I just couldn't find a solution, so I figured it must be impossible. But I wasn't satisfied with just thinking it's probably impossible, so I tried to solve it mathematically. Turns out I was right, it's impossible to solve it the traditional way, for exactly the same reasons that were explained in the video, so that's the solution. I can't believe it took them so long to figure out why this is impossible. O.O

[@Johnnyyoo1]

Nice solution fam, you really showed it in the vid

[@Johnnyyoo1]

Not a click bait at all

[@noahddd]

Nice observation skills fam, this video is 7 years old. I don't think the uploader will even see your comment. And if he does, he certainly won't care.

[@shadycharlotte9144]

BeFondOfJohn Well that's not true. Zero is the absence of things. You see more 0's in a day than basically any other number. You see 0 real murders (I hope), 0 living mountain dew bottles, 0 polar bears. Numbers themselves as in 1 the idea and 8 the idea it's also very real. 0 is just as much an abstract idea as all the other numbers.

[@shadycharlotte9144]

BeFondOfJohn Explain when a number isn't a measurment in some way. Numbers are ideas and 0 is the one we interact with most, making it the most real number.

[@shadycharlotte9144]

BeFondOfJohn But 0 does exist in measurements. That's pretty basic. I have 0 polar bears. There's 0 inches between me and myself. The ball traveled 0 feet. Now you can't get negative measurements that's true. But you can get a measurment of 0.

[@danjbundrick]

Let me be clear: I will not rest, and my administration will not rest, until we've solved this puzzle.

[@bugbyt3]

I just did it

[@HatsuneNeko01]

+Bugboy24MC Liar!!!

[@gyrox0031]

+Stefanie Neko start in the middle

[@geminix365]

+Daniel Bundrick Try to start with just a simple pentagon, exactly... YOU CAN'T FINISH OUTSIDE OF IT, the rectangles are basiclly pentagons, with 2 was already impossible with 3. It's like trying to draw a square with 3 straight lines, Impossible. Btw, I guess you were kidding, have a nice day.

[@landenclay8667]

So how are you 2 years without sleep

[@SchengW]

I am a Chinese and learned it while I was a pupil, really I remember this all my life. In China it is called Olympic Maths and a lot of children learn it in order to be smarter than others. Now I am guaduated my bacholer of science and my major is mathematic. Thank you making me remember that happy childhood

[@prathameshsawant5574]

can you make more videos on Euler and his proofs like bassal problem, amicable number and other? He was amazing and wonderful mathematician .

[@singingbanana]

I certainly will at some point!

[@lionelmessi-qr9cu]

What if somebody take it seriously ,paused the video,tried to solve it for a week having patience,came again to video for solution and he got to know that it was impossible....

[@brendennoonen8021]

If you draw a line larger than the rectangle it is considered only going though at one point as it is a single line and only travels in one direction, therefore never goes back to cross again

[@_S.H_]

If you think outside the box you will find a solution for this. The puzzle doesn't say you can't cross the lines at a 0 angle.

[@joshwist556]

But thats not an edge or line. It's angle so you still have to do the other ones also this is mathematically proven to be impossible like how it is mathematically impossible to fly in a place with gravity in a stand still position.

[@joshwist556]

But thats not an edge or line. It's angle so you still have to do the other ones also this is mathematically proven to be impossible like how it is mathematically impossible to fly in a place with gravity in a stand still position.

[@maocharlisme]

+Saleh Haddad Then you overlap it instead of crossing it.

[@1gallimaufry]

+Saleh Haddad The line must curve. Sorry!

[@manojkr9198]

Overlapping means infinite intersections, that means you are crossing the line more than once!

[@jamenneel342]

I can’t tell if the “Euler’s Trail” paper at the end was computer generated or hand drawn.

[@TheGrooseIsLoose]

I tried it like 5 times, and in between each I thought about how there were 3 boxes with 5 lines to cross, so it must be impossible, but I kept trying because obviously if you were about to give the solution, it must exist. Then finally, I was so sure it was impossible that I gave up and decided unpause to see your solution.

[@robertvermillion6816]

Quick version: it has more than two odd vertices and thus cannot be solved ;).

[@parodysam]

I assumed it was because of an odd number of odd vertices

[@robertvermillion6816]

@@parodysam It can be even too but still create an invalid situation only if it's 4 and over.

[@Ethromel]

WHY IS NO ONE NOTICING THIS? If you were to count each box as having it's own 4 sides and then add them up- the total should be 20 sides- NOT 21. So this whole diagram requisite is false. Here's why... They're technically asking you to go through the upper part of the middle brick's edge TWICE, when instead it should be considered ONE brick edge (for the top part of the middle brick). The bottom center box is being divided into an imaginary two squares or brick edges, when the lower middle box's upper brick edge should be considered one line, and the center-line that separates the upper two bricks should be considered the line necessary to cross that lower middle box. As is it defies logic and deliberately makes the task impossible, but when looked at the way I described- it's possible... and logical.

[@sonnygarner3752]

No, you misunderstand. Where an edge is bisected by a perpendicular line, as the top edge of the middle-lower brick is, it is considered two lines, and you have to go through each of them once.

[@Ethromel]

I understand that, but the premise is to go through each brick edge- not every line. If it's meant to be every line, then the boxes have nothing to do with it, and they should change the instructions. It shouldn't be both.

[@LeftFlamingo]

Ethromel The premise is to cross each line once and only once. A bisected line = 2 lines, as is explained in the video. The boxes are part of the explanation, not part of the initial objective.

[@AwesomepianoTURTLES]

There are no boxes. Just because you see boxes doesn't mean they are there. All I see are lines I need to cross only once.

[@maxkillaguy9358]

AwesomepianoTURTLES shh turtle fanatic group they are talking a language that doesnt fit inside google translate, i think its a new species

[@erwinthomas3667]

I have come up with two solutions out of about 6 attempts at this. Two solutions that actually work given the description that you and everybody else gave me: cross each line *once*, with one, continuous line. The only thing that is not mentioned, but clearly implied, is that you cannot start and end inside of the house. The two that I came up with involve starting in the top left, and ending in the bottom middle. The possibilities are not limited to those rooms, but those happen to be the solutions I came up with quickly. I even heard another guideline/rule in the scamschool video about this and another number problem: You could even cross the lines if you feel like it. The two solutions that I came up with don't even use this added aid.

[@bailey125]

I solved it!! I cheated

[@NinjaNerpaDerp]

***** You're right

[@rexdove1051]

you ain't solved shit!! if you had you'd be a million dollars richer!!

[@bailey125]

Oh hey it's me from 2 years ago

[@kieransia8045]

baileyboy125 what's up?

[@bailey125]

Heya +kieran sia how are you?

[@Jondude201]

I recognized his voice the moment I started watching the video THE NUMBERPHILE GUY! lol

[@jackismname]

James grime i think is his name

[@TroubleGummie]

So reading comments, corners count as two, so it's possible you count that (It isn't specified that you can't do that) the other way, is that you use horizontal line, and draw a line over top of all of them, but then you just have to pass through the vertical lines.

[@PeppermintSwirl]

Jesus Christ you sound too much like Numberphile.

[@MeAgain998]

+TheWilliamMaster You do realise that he is in a lot of numberphile videos, right?

[@singingbanana]

+TheWilliamMaster I am the guy from numberphile.

[@michaelkwright]

+singingbanana didn't think that you still read the comments on this video lol.

[@thomaswaller4517]

+TheWilliamMaster sounds like Dr James Grime

[@vestrocity9561]

+Thomas Waller (TiaTnT) YOU DONT SAY!

[@carolinejohanson3197]

Go through the corners that join multiple lines...

[@carolinejohanson3197]

I started in the top left line(not corner) then went down to the bottom corner and up through the inner corners of those first two rectangles on the left, next I went through the top right corner and kept going to make an arch that came back around to the bottom right corner then came up through the middle bottom and through its top right corner to complete the puzzle! :D

[@leion247]

Caroline Johanson My god... you're right. I just did it that way and it worked. Nice work thinking outside the box. =3

[@carolinejohanson3197]

Thanks! I spent awhile coming up with it though... this puzzle was really frustrating in the begining

[@haxpet]

Caroline Johanson So when you go through a corner, which line are you crossing then?

[@llsnart3554]

both, lol

[@natox7555]

It's just graph theory isn't it ? I counted the number of lines (and not sides) of each boxes and saw 2 boxes have 4 and 3 have 5. Since there can only be 2 boxes with an odd number of lines, this is impossible. (Because we have to enter and exit once every box but the one we begin with and the one we end with, all the boxes but 2 should have an even numbers of sides).

[@ImCovino]

These people commenting that they solved it LOL, please post a screenshot and I'll tell you how you got it wrong thanks :)

[@theuncalledfor]

I saw one who actually managed to solve it. Here, I still have the link: gyazo.com/930b795b6932820f615e11f088f547ef

[@ImCovino]

theuncalledfor yeah thats the only solution :))

[@Nikola1999]

prntscr.com/bgkoop

[@ImCovino]

Suknaman you went through the same line multiple times.

[@Nikola1999]

where ? :)

[@goku_jerome1732]

Are u on numberphile

[@singingbanana]

I am

[@sidneydamsma]

i recognised your voice immediately

[@vsause4science620]

singingbanana I knew it

[@brittanykearby5570]

I have been trying to figure this out for 20 years!!!! I just decided to Google it.... only to find out I was wasting my time doodling this in my notebooks all throughout school lol😂

[@mikepratali5582]

OK. check it... when I was in the 6 grade we had a sub teacher that gave us this problem and he swore it had a solution. I believed him, why shouldn't I he was a teacher. I spent the next 40 years working this problem and it wasn't until 3 years ago that I mathematically worked it out that it couldn't be solved. I"M STILL PISSED. Tonight, this night I saw this on the tube and my heart raced... I clicked over, never seeing this problem anywhere during my travels over the 40 years.. I thought, what if?? Then I watched the video, the first one, that stops before giving the solution... I WAS pissed again.. then clicked over to this to lay to rest a painful search for a solution and to question everything no matter how credible they may seem..

[@ScyrousFX]

The story of a lifetime, folks.

[@tksbilliards]

+Mike Pratali the solution is to put it on a sphere. draw it on a basketball then do it. with a 3d platform its easily possible

[@edcrSpeedruns]

no it isn't

[@barbelmeister]

@Mike Pratali - Reading your post, your experience is identical to mine - absolutely to the last detail!!! I was told there was a solution, the teacher wouldn't show us, he left us with the puzzle over a summer holiday. He did say that the solution looked like a 'teddy bear'. I never saw the teacher again so never did get to see a 'solution'. I am sure he couldn't have been that cruel...

[@trxllest98]

You were pressing down on that marker so hard when drawing the line it made me cringe lol

[@keiris2]

Found this puzzle a couple of years back and it is in fact impossible if you do it this way, but if you go through corners, where 3 lines connect (in an angle where all 3 are crossed) you aren't abusing the rules but you can still get a solution.

[@vTxpic]

I figured it out!!!! It's not impossible!!!

[@mixtraa]

Yes it is

[@vTxpic]

no

[@mixtraa]

+KittenGaminq then post a video and link it

[@vTxpic]

if I can get my phone to stand up, then okay

[@Gold31000]

+KittenGaminq Why is it that all the people who say they can do it are usually retarded 11 year old Minecrafters?

[@Pennysdad16]

next time tell me it is impossible first... before I waste an hour of my life

[@MrLoftyDave]

Not impossible

[@Asmobia]

Whenever you go through a wall of a room, you end up at the different side (inside or outside) of the room. If a room has odd number of walls and you go through all of them, you end up at the different side (inside/outside) of that room. There are 3 rooms (named A, B and C) in this puzzle have 5 walls, and you try to go through all of them: If you start inside one of them (say room A), which means you start outside room B and room C, so you must end inside both room B and room C if you plan to go through all walls, which is impossible. If you start outside all of them, then you have end insides all of them, which is also impossible.

[@DanLokar]

There's only one way to solve this: imgur.com/tkhXhwJ

[@edcrSpeedruns]

lmao xD

[@itsiwhatitsi]

OMG do not do that fkin noise with the markerrrrr!!! Aaaargh!

[@emeses7568]

At first glance I notice that the top two rectangles each have 5 required crossings, which in words would be: "In-out-in-out-in" ---> both of them require that you stop inside, since you need to stop inside of two different places it's impossible

[@bigmjeaaof]

The solution works took me 3 min to get, it never says u can cross corners and if u cross a corner u will automaticly cross 2 to 3 lines and then it's possible =) it never said cross 1 line at the time =) but if thats the case its impossible

[@HatsuneNeko01]

Mind blown

[@bigmjeaaof]

+Matthew L yes and its easy tbh try it out

[@Pete-Prolly]

Robin Lindsten that's total bullshit. It's a simple case of parity: If a light-switch, (or toggle,) is down and you turn it the other way, (or toggle it 180° every time,) then it will always be down after an even number of turns, and it will always be up after an odd number of turns. So, in any case where the toggle is down and you turn it an odd number of times it is impossible that it will end down▪ (That was the whole point of this exercise.)

[@josephnardone1250]

What is the answer? You didn't draw the curve. This is totally ridiculous. The title says solution. Am very disappointed in this video. It was very poorly done.

[@singingbanana]

The explanation was complete. The solution is the explanation of why it's impossible. I also gave you the name of how to solve problems like this, Euler Trails, which you can look up.

[@paperboatproductionsshow1720]

Check out my vid it has the actual answer

[@figureit-out6059]

you used the wrong brick, you silly sod

[@PLSKICKME_]

Its his fault that you cant understand a good and simple explanation, right?

[@evasetina1529]

I read your comment in a british accent. idk why :))

[@lexnellis4869]

When I was middle school I looked a brick wall in class and out of bordem decided to make a puzzle for myself. This was the puzzle. You have no idea how many years I spent trying to solve it and how many times I drew the damn the thing. I think was still attempting it as a Senior. Now I know I was never going to achieve my goal.

[@theuncalledfor]

Your approach was wrong. When you kept failing over and over, you should have stopped trying to brute-force it, and instead analyzed why you keep failing. That way, you could have figured that it's impossible, and why, instead of wasting your time drawing pointless lines over and over.

[@marlinmiller6762]

I "solved" this problem 45 years ago in 8th grade. My math teacher gave me an "A" for creativity. Every Friday, he gave me a puzzle to solve over the weekend. After 20 weeks or so, he presented me this puzzle. You should have seen his expression when I informed him Monday morning that I 'solved' it. PRICELESS! I adhered to all the rules, crossed through each line only once, and didn't lift my pencil off the paper. You see, my teacher didn't tell me it was a trap like the Kobayashi Maru. en.wikipedia.org/wiki/Kobayashi_Maru

[@ecmuppet9538]

If the rule is you can't take your pen or pencil off the paper you go through leaving 1 edge inside and your line on the outside, you bend the paper into a roll until the edge of the paper is next to the uncrossed edge and then draw over it, it's continuous from 1 observers perspective

[@livedandletdie]

James the only problem is that it's possible to cross 3 lines at the same time.. if you cross through an intersection. However you will start inside and end up inside or start outside and end up outside.

[@sunnylilacs]

If it must be done on a two dimensional plane with the requirement to cross through the edges, then it’s impossible. But if the wording says to intersect the lines, it’s really easy. (Think inside the box for that one.) It’s also feasible if you approach the problem using three dimensions. (Like taking a needle and thread to puncture/pass through each edge, then tie the thread outside the piece of paper.)

[@kaaajeee]

I came up with that solution put more simply in my opinion. For every area with 5 borders you may either start by entering and end by entering it, or start by leaving and end by leaving it. Because you must alternate leave/enter/l/e/... 5times. You can start by leaving only one, also end by entering only one. And you have 3 areas of 5 borders. What do you think?

[@zackarywertz4106]

the rules say that you have to get through so the wall ones by going through a corner you are go through two walls at once

[@oceanusprocellarum6853]

Well I noticed that this sort of thing can be simplified. If you take just those two top rectangles (and each of the bottom sides are divided into two) or actually just take 2 adjacent pentagons, it appears to be impossible. I believe it's because, going in and out, you take up an even amount of lines, but here there's an odd number of lines, so you are forced to go in a polygon and cannot come out. And yes I didn't finish watching the video

[@AKmacintosh]

Such an intuitive way to introduce the topic! Great video.

[@adammosher1874]

I've literally been trying this since I was 9 years old and this randomly popped up on KZbin

[@hobopotato]

It's only impossible if you draw it on a 2-dimensional surface. Try drawing it on a torus. If the torus is modified a bit, the drawing itself can even remain 2D, although the line you draw won't be. Still, I consider THAT to be the best solution, or at least, the most creative thinking-outside-the-box solution.

[@ThePotaToh]

Even knowing what Eulerian trails and Eulerian paths (and their distinction) are, and looking at the title of the video, I just couldn't think of a way to convert the picture into a representative graph until you showed it (except you didn't join the 'outside' as a nine-edged vertex). With the underlying graph drawn it can be easily shown how to change the original puzzle such that it _is_ possible to cross every line only once, and then there are quite a number of ways to do this-either by removing a certain line, or splitting a certain line, to respectively remove or add an edge between odd-degree vertices to make them of even-degree. The original problem is itself a graph, and it turns out that its 'dual'-where the faces are the vertices and the vertices are the faces-is the graph needed to solve the puzzle. (I use the properties of Eulerian trails a lot in my work, as I do embroidery and it saves thread to visit every edge only once, and if the Eularian trail does not exist, how to split the design into subgraphs of which it exists.)

[@xevedor8083]

Just wanted to offset all the negative comments and say that this is a great puzzle, feels like a more complex Königsberg bridge problem. Also, I find it awesome you're still replying to comments after seven years.

[@godzillazumagod9146]

Read the whole book "Journey threw GENIUS". Euler is mentioned in that book but changed because more books means you need to go down way down and lose to someone for the name.

[@kenhaley4]

I've seen this before. And I've wondered why the impossibility proof can't be explained more simply as follows: Consider the drawing a floor plan of rooms. Draw a door in each wall leading to an adjacent room or to the outside. The puzzle now becomes this: Walk through all the doors without going through any door twice. Notice that every time you enter a room you have to leave by a different door, unless you started or finished the trip in that room. So all the rooms except the first and last must have an even number of doors. But there are three rooms with an odd number of doors--five doors each. Hence it can't be done. The proof is equivalent to what was presented in the video, but without the extra steps of drawing the graph with a center point in each room, etc. The graph using the centerpoints of each room is useful in a sense, because it illustrates a correspondence with problems of drawing geometric figures without lifting your pencil. But that equivalence isn't necessary, and I think my version of the proof is easier to see..

[@magicart8840]

In fact, there's lots of solution. The problem says that the line may not cross the same line twice. But we're allowed to make the line crosses itself, think about it and u'll find a solution, there are many possible like that.

[@sqwama]

Draw a line following the edges,when you draw the last line, make sure to stop it before it touches the others. Then you solve it normally.

[@Madgearz]

5 square rooms + outside = 6 rooms 3 rooms with 5 doors 2 with 4 Outside has 9 Basic rules: Each room is in one of two states > 0=Empty, 1=occupied On only one room can equal 1 at a time Am odd number room (5 and 9) will always end opposite of what they started with Even number rooms (4) end the same as when they start. Senarioes: Door#: Start State > Fiinsh State Starting ng in odd number room: Odd(5): 1 > 0 Odd(5): 0 > 1 Odd(9): 0 > 1 Even(4): 0 > 0 Even(4): 0 > 0 Ends with 2 rooms occupied, impossible! Starting in even number room: (5): 0 > 1 (5): 0 > 1 (9): 0 > 1 (4): 1 > 1 (4): 0 > 0 Ends with 4 rooms occupied, still impossible.

[@theuncalledfor]

Beautiful.

[@kookeekwisp]

In order to cross every like, there must be less than 3 boxes with an odd amount of "doors". you MUST start or end in each one, but a continuous line needs 2 starts or ends to do this, and that's not possible.

[@ggstoney]

I showed this to all of my teachers...i started the year with an A+ for all of my classes. Don't ask how or why, I give the credit to this man here.

[@piesho]

The puzzle is impossible to solve with the given conditions because it's not a Hamilton circuit. Dirac’s Theorem: If each vertex of a connected graph with n vertices (where n> 3) is adjacent to at least n/2 vertices, then the graph has a Hamilton circuit.

[@p.singson3910]

You may be a singing banana or a weeping apple, but for me you'll always be the Numberphile guy.

[@tammambadran8314]

Adult Hintler It has a solution In 0:47 he could enter the left square and cross the down line then enter again from the outside throw the ledt line then the top line .. and finally the middle line in the picture ☺☺

[@safrussalmus9056]

I think (I stoped at 1:14) that it should be impossible because all 3 big rectangles have 5 sites, which means that either the begining or the end of the line should/must be here, but there are only 2 ends of the line for 3 rectangles.

[@Reptilligator]

I spent hours trying to solve it and now that know it's impossible I can actually continue on with my life

[@JosephGordon615]

As long as it's not required to bisect each line you can solve it by crossing at the intersection of 3 lines (the corner) which covers 3 lines with one crossing. Solved.

[@ade8419]

it can't just be me that infuriated by the noise of that pen. GET A BALL POINT PEN!

[@KiliTheKiller]

But you can bring it down to one missing line. Look at the beginning of the "snake", you can easily just add a line though the bottom left corner of the bricks.

[@jordanbudhu454]

The point where intersecting lines meet is called the point of intersection. There is no limit to the number of lines that can share a point of intersection.

[@yoska1420]

This is the only video I can watch on any thing for somereason