Why this puzzle is impossible

[@3blue1brown]

For the question at the end, the intended answer is not "the handle lets you go in three dimensions", because for that matter a sphere is three-dimensional, but you could never solve it there. Think about what makes the surface of the mug (or a doughnut) distinct from that of a sphere, and how _that_ affects the argument. I think I went years knowing that Euler's formula looks different on different surfaces but had never really thought through why. In particular, the exercise will set good intuitions for learning about homology, if that's something in your future. Also, my apologies for two names typos here: Veritasium, and James Grime (evidently I accidentally pluralized him to "Grimes"). That's what I get for throwing on titles late at night, my bad! To everyone saying "I can't believe the math guys hadn't heard of this puzzle before". I agree that would be surprising! It's a very famous puzzle in math circles. Maybe I accidentally obfuscated this too much in the editing, but all the math guys most certainly were familiar with the puzzle. I mean, three of them make and sell the thing! This is why their contributions were either direct explanations or jokes. Derek and Henry had seen it before, but long enough ago that it still involved a little trial and error.

[@sen7859]

3Blue1Brown it is possible with a 2D plane

[@sen7859]

I have done it at my 2nd try! :)

[@sen7859]

3Blue1Brown and it is not like the mathloger's solution :D

[@imnotdaredevil3714]

One of your utilities reach 2 houses, your ninth line is a telephone line from the first to the last house hahaha

[@sen7859]

Jk srry 4 taking your time :D

[@abigailcooling9355]

This reminded me of something I heard a while ago: 'Mathematicians don't like to lose, so when they can't do something they just prove it's impossible to do it.'

[@dedwarmo]

Are you saying it’s possible?

[@gregvs.theworld451]

@@dedwarmo Not necessarily, more like if they attempt a challenge that looks like it can't be completed, they shift to trying to prove it can't be done, so they didn't "fail" at the task, more so they won by proving that it simply can't be done.

[@b4byj3susm4n]

Some may call that stubbornness or pride. Mathematicians may call it “certainty.”

[@thefoolthatdied]

But I solved it?

[@stewbaka4279]

@@LurkingAround nah maybe next time, but i also proved it

[@alwinpriven2400]

the parker square joke was hilarious. 10/10 brady.

[@caitlinryan]

i laughed so hard

[@MustardPipeLibrary]

And then, of course, Parker himself had a Parker Solution to the puzzle.

[@PeterAuto1]

that was the best solution

[@armu8282]

i dont get it??????

[@alwinpriven2400]

you have a parker understanding of jokes then.

[@kylerivera3470]

I love how almost everyone goes "draw over here and go around the handle" while one guy essentially went "just move the handle casuals".

[@bextomoose]

15:00

[@jordananderson2728]

I love Mathologer.

[@slevinchannel7589]

@Nexxol Ok

[@NullScar]

@@slevinchannel7589 Mathologer.

[@NullScar]

@@slevinchannel7589 Also, Tibees, very interesting angling of subjects. Especially her storytelling through painting.

[@TheAgentAPM]

I think this puzzle is so famous not just because it looks simple and is impossible. The secret sauce is that you're always precisely one edge short.

[@bentonrp]

Not me. I was THREE edges short! =)

[@illiji915]

it's not impossible. you can draw 7/9 lines without crossing then use the mug handle to basically bridge/tunnel the last 2. The lines don't "cross" because one goes through the loop of the handle while the other travels the handle itself

[@CrimmzZT]

@@illiji915 HOLY YOU ARE RIGHT! THIS IS THINKIN OUTSIDE THE BOX

[@illiji915]

@@CrimmzZT I figured it out

[@CrimmzZT]

@@illiji915 bro I was rackin my mind on how to get around it and didnt even think of the handle, thats very impressive and out of the box thinkin, and not mention it wasnt mentioned in the video at all, it is in the comments pinned tho, but I didnt read that and just went with what the vid said. very satisfying that you found this on your own!

[@WelchLabsVideo]

Huge thanks to grant for including me in this super fun video! It’s an honor to be edited back to back with some KZbin heroes!

[@AkhilNairjedi18]

Welch Labs You are one of the heroes! Your videos are amazing. Thanks a lot for creating such educational and interesting videos.

[@VincentZalzal]

I've just discovered your channel thanks to this video. I watched the "How to science" series and I have subscribed :)

[@budtastic1224]

Same

[@ThainaYu]

You sir are hero

[@yerrenv.st.annaland2725]

Dude, your series on Complex Numbers carried me through high school mathematics!

[@Stormingmonkey]

INFINITY WAR: The most ambitious cross over in history 3Blue1Brown: hold my mug

[@SanneBerkhuizen]

Most underrated comment!

[@jdao1sm]

That’s ironic because it has to do with lines not crossing over each other.

[@krazieecko]

TOP COMMENT OF THE YEAR

[@mojann1]

My thaughts exactly

[@MelonMediaMedia]

It doesn't cross over though...?

[@Ken.-]

15:07 No idea what Looking Glass was doing over here... Tries to solve a simple puzzle on a mug. Accidentally designs a working quantum computer instead.

[@nazishahmad1337]

Hahahahaaha

[@ziggyoickle3445]

So...I'm experiencing a bug where before I click on your comment, I'm seeing a comment on a previous video, but just yours "I wonder if dooku trained anakin..." Edit: wasn't even you who left the comment on the other video, left me thoroughly confused

[@99bits46]

she was doing meth

[@rafaelcorella1895]

@@ziggyoickle3445 interesting. I got that same thing when i first opened the comment

[@YardenAkin]

@@ziggyoickle3445 It's a bug with the KZbin app. Comments from previously watched videos show up randomly replacing comments on the video you're currently viewing. Hopefully it gets fixed soon

[@brooklyna007]

On a plane or sphere's surface any loop will split the space into two areas. But on a torus there are loops that do not split the plane into two areas. Specifically there are two sets of perpendicular loops, around the hole of the torus or perpendicular to it. Thus on a torus you can add an edge that neither lights up a point nor creates a new area. But you can only have two such loop of edges and they must be perpendicular. Any additional loop will split the torus into 2 regions.

[@aquinsvarghese9182]

In engineering class I would do the 8 connection and hope for partial credit.

[@thomy2562]

e = 3 = pi

[@aniruddhasanyal7625]

@@aidankwek8340 sin(π)=3

[@gsuaysuwgs]

@@aniruddhasanyal7625 The aproximation sinx=x is always taken when x is a very small angle, usually used in physics when doing calculation with an object that is slightly oscillating

[@ornessarhithfaeron3576]

sin(x) ≈ x for x

[@bradstevens4491]

As an engineer, you should have known to just drill a hole through the mug, "cross" any line you needed to, then drill back out next to the house. This puzzle can actually be done on a piece of paper using this method. Which just proves that pure mathematics stands no chance in the face of a determined engineer.

[@Wiebejamin]

I remember doing one of these in like, 3rd grade on a Flash game. The trick there was to right click it, and use the menu that the game doesn't register as a bridge to cross over.

[@dopperling2712]

Wiebejamin The impossible quiz

[@kABUSE1]

I might be 2 years late but I just wanted to point out that I love out-of-the-box puzzles, especially in videogames. Another great example for this is a game called Deponia. Your character had to remember a door code, then cross a market place with funky musicians playing music and enter it into a door lock. Problem is, he always forgot the code and began singing along the music beats instead. The solution was to mute the music in the game options... lol

[@pomelo9518]

Well there you have it, a bridge!

[@JediSteve-J3-]

@@kABUSE1 try a game called "there is no game" Well, you probably already have but if you haven't check it and it's sequel(?) "There is no game: Wrong dimension" out.

[@danilodjokic5303]

OMG I remember this

[@cherrywolf66]

I know this video is an old one, but I started watching your channel fairly recently, and as a gift for fathers day I got my dad (engineer) this mug. He texted me his progress with the puzzle, and its funny, he did the exact same thing, where he took the puzzle to paper and concluded it was impossible, then went back to think about why the puzzle was presented on a mug. I got a real kick out of watching this video, then having my dad text me exactly what these other mathematicians recorded themselves doing. Thank you so much for your channel making higher level math and puzzles like this more accessible to someone who's not as math minded or math educated as professionals.

[@redlok3455]

There is also an "engineer's solution". When you get to the point where you are left with the last edge yet to be drawn, just connect two houses instead, so they share their gas or water or whatever. No crossovers here =)

[@zargon7222]

Shared services for the win.

[@worldcolonyinitiativ]

exactly what i was thinking, you could also bundle water energy and gas into a single line and then use that line to connect to all three houses

[@xemnas577]

or just let one house don't have gas and let them heat up with electricity instead

[@redlok3455]

@@xemnas577 Right, but since electricity is pure exergy, it'd be a waste to use it solely for heating.

[@xemnas577]

@@redlok3455 I'd argue that gas energy isn't most cost efective and efficent let alone safe too but I wouldn't know that much tbh

[@MrHatoi]

Everyone else: oh i guess you just need to use the handle Looking Glass: _already 4 parallel universes ahead_

[@enzoqueijao]

She was using quaternions to explain how a mug works

[@MarkSmith-tu9qr]

she may not find the solution like everybody else but the she had an interesting approach 😅👌

[@mahindoescali]

She is too creative to solve this problem like everybody else

[@Hexagons7]

Actual mathematicians: This is hard 3blue1brown viewers: easy, what’s next

[@rogercruz1547]

had she just used a torus she would get it instantly, but she chose a sphere

[@crazyacorns1173]

As a kid in school we were presented with this problem, and incentivized with a pizza party if someone solved it. Our teacher made a fatal error though by drawing the problem on notebook paper, with no rules as to where the Gas, Power, Water, and houses had to be located. Note book paper has 3 holes on the left side by drawing 2 house on one side and the third one on the other side of the paper, I was able to use the holes to solve the problem.

[@benedixtify]

But did your teacher cough up the pizza party…?

[@benedixtify]

You’re thinking topographically 😁

[@kjl3080]

I mean that’s still a nontrivial solution so pretty cool

[@kjl3080]

Also damn that school is sadistic- like no homework if you prove FLT

[@crazyacorns1173]

@@benedixtify He did actually, one of my favorite school days lol.

[@Zarkonem]

I used to give this puzzle to my friends in highschool. I even made a poster and posted it around the school with a reward attached encouraging everyone to try it and come give me the answer. No one ever did. I had several people run up to me enthusiastically telling me that they solved it only for me to point out that they are missing a line. I had thought it was impossible to do it on a piece of paper for 18 years. Thanks for proving to me that i was right.

[@乇メ乇]

You really aren't right, neither him, it's pretty easy, the laws say "do not cross lines" so you can just cross the circles of utility with no problem!

[@Zarkonem]

@@乇メ乇 Except that's also an illegal move. I had multiple people try to do that too, you can't connect a house to a house or a utility to a utility.

[@乇メ乇]

@@Zarkonem well, the laws don't say "you can't cross utility" bruh, there is just one, just nobody think about it. And +, you are making this in a real situation, this is just hypothetical bruh.

[@Zarkonem]

@@乇メ乇 Well when i presented it back in the day, i stated the rules were that you had to connect the 3 utilities to the 3 houses without crossing any lines. That inherently insinuates that connecting houses or utilities to each other is not a legal move. Just because the rules in chess don't say that you can't pick the board up and dump all the pieces in the trash and you win, doesn't mean that is true.

[@TheJaguar1983]

What led me to figuring this one out was thinking: "If this puzzle was in three dimensions, it'd be easy". I thought of a line going out of the page, then realised the handle was doing just that.

[@MatsMatsuo]

the fact that it have 3 dimensions doesn't make it easier, because it stills a closed surface, you need a hole because a body with a hole (like the mug or the doughnut) cannot be seen as a closed surface. if you think about the doughnut is easier to visualize. Idk how to explain it better, i still nedd to think to make it more "formal".

[@TheJaguar1983]

@@MatsMatsuo When I say "in three dimensions", I'm referring to being able to "draw" in three dimensions, as if drawing in the air. I'm not referring to the mug being three-dimensional, but that the handle provides a way to draw "in the air" above the puzzle. A recent example I've had was soldering together an electronics project: The PCB is in two dimensions and has traces moving in 2D and I had to solder wires, resistors, etc in three dimensions. Much in the same way that the handle forms an arch, the wires and resistors form a bridge to connect two points that could not be otherwise connected if restricted to the 2D plane of the PCB.

[@parzingtheasian]

we need more people like you

[@ruffusgoodman4137]

The real question here is for what configuration of the problem in a 3D environment not possible to solve?

[@zakarylittle6767]

@@ruffusgoodman4137 Sphere. Cube. Anything without a hole maybe?

[@abipjo8173]

When all your favourite you tubers are all in one video . Best Christmas gift ever.

[@jamesfleming1155]

TT Cubed Agreed. This was awesome.

[@Daniel-rt4zz]

Only missing Vsauce

[@thefableparable215]

ViHart ;w;

[@chuzzywuzzy9545]

# when you're such a nerd you're already subscribed to all these people.

[@DVSnark]

I have a really simple solution. Just do the little ‘bridge over’ curve (as in an electronics circuit diagram) to indicate that the lines aren’t actually touching.

[@elgordobondiola]

Just use the power of topology to turn the mug into a donut and then just sit down and cry because of the broken mug pieces stuck in your hands

[@samlevi4744]

Quite literally the point of the handle.

[@bentonrp]

Or just have one line cut through another house on its way to its destination house. You'll find there's now enough room to draw everything to each one! 😊

[@Nnubbs]

@@samlevi4744 which makes the handle useless in accordance with the directions.

[@NocturnalTyphlosion]

this is why engineers are banned from philosophical debates

[@jankowalski-py1ey]

Where the proof breaks: on a plane, when you add a new cycle, you add a new region. On a mug, it is possible to add a cycle without adding a region. Have the cycle go around one of the legs of the handle.

[@next_thing_to_god]

yah i had the same ans as adding that vertex would lead neither edge increase or new region

[@whiz8569]

"I tend to make a parker square out of these...oops, see." I actually left the room after that.

[@beenaalavudheen4343]

Did u laugh or cringe? Lol

[@redstone8513]

I "cringed", per se. It surprised me out of nowhere but I still went along with it.

[@fiveoneecho]

I thought that part was great, because I thought he actually dropped it for a second... :P

[@cosminaalex]

beena alavudheen did you laugh or did you lose

[@TheMarkFeet]

My god that was a good one, Brady

[@applebombbob]

Nothing that they couldn't handle

[@memeislovememeislife3369]

Luke Alexander %😂😂😂

[@jaymalby]

Take your upvote.... 😆

[@the5thestate587]

Luke Alexander *ba dum tss*

[@awesomeguy9573]

Nice

[@Jartny]

I see what you did there 😂

[@mairisberzins8677]

When all hope seems lost. You remember of one dark and evil subject in maths... Topology.

[@mbrusyda9437]

That was the first thing I thought of when they use a cup with a handle, though, hahaha

[@thedoublehelix5661]

Topology is great

[@Alex-ud6zr]

Isn't this K3,3?

[@absolutezero6190]

Alex A. Yeah

[@CrittingOut]

@@Alex-ud6zr Kuratowski's theorem moment

[@DanielGonzalezL]

Gotta love Mathologer. It wasn't even a challenge for him. That man's a genius

[@SpiffyCheese2]

nor SingingBanna, Matt Parker is stupid

[@jmchez]

Kind of stacking the deck there. Also, I wish that Vihart had been invited. Pens, doodles and math are her thing.

[@Huntracony]

ThatMathNerd, Matt Parker is a comedian at heart. So considering it's partly his store that sold these mugs, he has done videos on klein bottles (and is clearly interested in topology), and he's a mathematician, I think it's fair to say he knew the solution and decided to be funny instead.

[@SpiffyCheese2]

I understand that, Its just a numberphile inside joke to make fun of him.

[@Huntracony]

ThatMathNerd, No, you make fun of his square. Not him. So you _could_ say that was a Parker square of a solution.

[@corlinfardal]

I think I've solved the homework. The main thing to note about the graph on a torus is that there are only three regions, two inside ones and an outside one. How the graph accomplishes that essentially relies on the fact that you can draw two lines starting from the same point on the torus and not actually divide the torus into different regions, by having them follow the "axes" of the torus. So, the last two lines of the graph pull the same trick, and don't divide the last region into the three that would be required on a plane. Ultimately, where I think the proof in the video fails on a torus is by assuming that any new edge added necessarily either hits a new vertex or divides a new face, which clearly isn't universally true.

[@fantasticphil3863]

Corlin Fardal Thank you for this comment

[@djbj1993]

Yes, you get something resembling a mobius strip :)

[@thomaspalazzolo5902]

Like most puzzles, this could be easily solved with judicious application of a power drill.

[@honourabledoctoredwinmoria3126]

On a sphere, sure. But the coffee cup already has a hole, so you don't need to drill another.

[@Omnomnomfish]

In the original pen and paper version the solution was to just punch the pencil through the paper and call it a day 😂

[@noobandfriends2420]

Klein bottle.

[@wren_.]

that’s what the handles for. I think a topologist would murder you if you made an unnecessary hole in the mug

[@DemonetisedZone]

​@@honourabledoctoredwinmoria3126witty 😂

[@zach11241]

It’s fun to think of how easily we can solve an “impossible” puzzle in a 2D plane by simply working the solution in the 3D plane. Then, taking this a step further, by thinking of the “impossible” in our own 3D world and how being able to manipulate solutions for then through the 4th dimension.

[@雷神索斯]

But I were able to complete it😂, just make a large line over a single house to make it😅(so the third line will not get block) (I wish I can post pictures😢

[@Darkerfoxtech]

Instructions unclear there are now 10 dimensions in the explanation.

[@alecvan7143]

mathologer definitely had the best answer

[@HHHHHH-kj1dg]

That's the only answer

[@shubhamtiwari5461]

@@HHHHHH-kj1dg I

[@jamesknapp64]

He has a lot of slick answers

[@JustWatchingVideo56]

*Everyone else solves the puzzle.* Matt: Ah... I love the taste of fresh dry erasable marker in the morning.

[@drcomrade]

On a torus, something unintuitive and interesting happens with one of the edges: it only touches a single region on both sides of the edge. All other edges touch two regions. Also, if you want to easily draw on a torus, you can just draw a rectangle and treat the opposing boundaries of that rectangle as periodic.

[@flametitan100]

Yep. I could visualise what was up with the Torus (you could draw a circle along the outside, and a circle going from the outside to the hole and back, and they'd only meet up at one spot, while trying to do something similar on a sphere would almost always have them connect at two points,) but was having a hard time coming up with a mathematical explanation for what that actually meant.

[@onecommunistboi]

Maybe Im simply drawing it wrong, but for me each edge touches exactly two regions:/ Also there are only three regions in total

[@seraphina985]

Yup thus why many 2D computer game worlds are actually toroidal, they often link the edges top to bottom and left to right. That is a 2D map projection of a torus right there for a sphere it actually moves half way across into the opposing hemisphere at the top and bottom and you stay on the same edge. I always found it funny seeing games that do the toroidal version on maps that were intended to be planets, it's like err that is not how spheres work.

[@awesomechaos4034]

I’m so glad I predicted the handle thing! My solutions are dumb most of the time so I’m glad I was able to actually figure it out!

[@JNUK9599]

The parker square reference by Brady at 1:40 is hilarious 😂

[@paperspock]

Final Fantasy helped me solve this one, or at least think through it. See, in the old final fantasy games, the world scrolls in such a way that it's like a rectangle where the top connects to the bottom, and the left connects to the right. And someone had joked that spheres don't work that way, so that the worlds of the old Final Fantasy games must be doughnuts. And I also remembered the joke about coffee cup = donut. So, not having a mug in front of me, I modeled the problem out on a sheet of paper with the added rule that the left border could teleport a line to the right border, an the top border to the bottom border. Once I worked it out there, I knew it would also be possible on a mug because a sheet of paper with warping borders like that is equivalent to a dount, and a donut is equivalent to a coffee cup.

[@purplenanite]

topology for the win!

[@GQSmoos]

I hate that I 100% remember that being a Final Fantasy rule (I’m thinking of IX) but can’t figure for the life of me why that isn’t how spheres work.

[@flyawave]

@@GQSmoos Consider an aeroplane, traveling around the world. If it goes all the way East on the map, it would be on the Left edge of the map. What happens if it goes even further beyond? It pops onto the Right edge of the map, or all the way West, so those two edges ARE connected. Where it breaks down is going all the way to the Top, or North. If it were to hit the North Pole, and go further beyond, it doesn't pop to the South Pole, but rather shifts to the opposite side of the North pole. If it was going North in along timezone 0 (the UTC/GMT line), upon going beyond maximum North, it would pop over all the way East/West and begin going South from the North Pole, along the International Date Line, right? In other words, going past the Top of the map, keeps you at the top of the map, but half way AROUND the world. I hope that helps you visualize how, in order for the Top and Bottom edges of the map to be connected like the Left and Right edges, the world needs a `doughnut hole,' where the outer diameter of the doughnut is the map's equator, and the inner diameter of the doughnut is maximum North/South.

[@slevinchannel7589]

@@purplenanite Hi. Want some scientific Watch-Suggests? Some Channel to check out?

[@seraphina985]

@@flyawave Actually you will reappear one half of the top of the page away, it's a 180 degree shift in longitude not 360 degree. For example if travelling due North along 90W (North of Canada) you would now be heading due South down 90E (Towards Siberia). If you went the full page around the top you would be heading back down the same longitude you went up which is not correct.

[@arforafro5523]

Everyone else: Making doodles on a mug Looking Glass: Studying alchemy or some other esoteric shit

[@shadesilverwing0]

Looking Glass: *summons Hermaeus Mora*

[@MrMessiah2013]

It looks like she topographically transformed the coffee mug into a donut through the law of equivalent exchange (them both being breakfast foods, after all), then solved the equivalent problem on a donut. I believe Matt Parker has solved this on a Bagel on his channel before.

[@ahitler5592]

She is in her period

[@m3lb0urn73]

I’m actually trying to understand what is looking glass doing ;-;

[@Xnoob545]

Mathologer: just move the handle Matt Parker: the coffee wets the marker and it doesn't draw, so no intersecting of the lines

[@zenedhyr7612]

17:02 for the homework: The handle of mug decrease the number of edges from 9 to 8 - the edge kinda like teleportery connected, an imaginary edge, thus making it required not 5 regions, but just 4 regions only. Therefore, Euler's Formula V-E+F=2 remains unbroken.

[@spiderduckpig]

That doesn't really answer the question for why it's possible on torus though, just explains away the extra edge. The reason why Euler's formula does not follow on a torus is because some lines can be drawn without creating new regions (For example, a line that goes all the way around a torus in a circle will not create 2 regions).

[@HeliosAlpha]

My teacher gave us this puzzle in grade 5. It was very frustrating. Years later I just thought that the solution had to be to draw through the houses like you'd do with actual utility lines

[@tristanheaton2127]

Yeah that's what I was thinking

[@j.c.k.8639]

i was pissing myself laughing when i realized that, whatching the vid, then wanted to like that exat comment.

[@nikkiofthevalley]

No, in real life you'd just put the pipes under/over the other pipes, and use straight lines.

[@MrZBoy-xr3gb]

When I first saw the mug, my mind started shouting “IT’S A TORUS!!!”

[@oweng8895]

Me too lmao It's like that classic joke: "a topologist doesn't know the difference between a coffee mug and a donut"

[@protoborg]

No. It isn't. A torus is a donut shape. The coffee mug is a cylinder with a ring attached. While it is true that the handle could function as a sort of bridge, it does NOT make the mug a torus. A true torus has ONE hole in it. As a system, this gives it a second pseudo-interior, but it is still a very different shape to a coffee mug.

[@protoborg]

@@oweng8895 That joke is wrong. An actual topologist would be easily able to distinguish the two as a mug is a cylinder with one capped end and a donut is a torus. The handle of the mug does NOT turn it into a torus in the slightest. If you were to connect the ends of the tube together then it would BECOME a torus, but it is not currently a torus, with or without the handle.

[@rjswonson]

@@protoborg A mug only has one true hole in it, that being the handle. A mug is perfectly homeomorphic (topologically equivalent) to a torus, as in you can deform one into the other without cutting, breaking, punching holes or gluing.

[@rjswonson]

@@protoborg In topology there is no such think as an cylinder with one capped end. In the example of a mug, the inside of the mug IS the top face. A bowl is topologically the same as a cylinder, and a mug is topologically the same as a donut, because they both only have one true hole( A hole that passes all the way through the shape). If you need a visual example, the Wikipedia page for "Homeomorphism" has a nice little gif of this specific example.

[@Sk1erDev]

I wonder how this problems comes up in writing for computers. The PCB can be many layers but there are only so many layers

[@Rex9594]

now this is fancy

[@whythosenames]

but there you have the full room to work with, if something has to cross just extend it to the next layer and cross it there, but nice thought to think of anyway

[@NFSHeld]

Yes, but the ability to actually cross solves everything. Two layers suffice to connect everything to everything else, you just need "unlimited" base space. Think about it, the task is basically "connect everything to everything else, but your lines MAY cross", so you just draw connections how you need them and whenever two lines cross, that's a bridge. The "difficult" part is usually just that you don't want to use up a lot of space. Furthermore, the more "bridges" you need, the more expensive production will get. Thirdly, there's certain areas where you want to avoid routing (e. g. below RF antenna or charging circuits). Then different routes need different wideness depending on the consumption of connected parts. For high frequency like RAM or CPUs on motherboards, certain routes need specific lengths accurate to nanometers of length (ensured by autorouters making squiggly patterns), plus for very sensitive bits, you need to take the capacity of the routes themselves into account. So the difficulty mostly arises from physical restrictions, not so much from knot theory.

[@angrydragonslayer]

@@NFSHeld i once had to buy a 32 layer motherboard due to special needs and i have to say The difficulty carries directly into price ($8k for that boards, $1500 for the processor)

[@adamrak7560]

it gets way more complicated! Multi layer does not solve everything: - some signals cannot cross layers, because of the signal integrity. - the wires are not infinitely thin, so they may not fit - sometimes the requirements are crazy, like certain wires cannot come close, or you need to treat _every_ wire as a coupled inductor and a lossy transmission line at the same time. - sometimes you make your capacitors and inductors and delay lines from the PCB wires directly. - optimizing the current flow through the ground and supply planes can a good idea too.

[@chielonewctle7601]

One of my guess to the given challenge is about whether a new edge will still create either a new lit vertex or a new region. The most unnatural thing for me in Euler's formula is actually the inifinty region. As for spheres, there can be one edge that goes to the infinity and back from the infinity. But that edge still has to create a new region, which is equlivant to have an actual vertex in a 2D plane representing the infinity for sphere. As for mugs, however, we can have a new edge through the infinity without creating any region, for which I can't construct an equlivant in a 2D plane. There have to be at least two edges to completely cut the infinity region into two parts. Or let's say, after adding an edge through the infinity, we can still add an edge through the infinity without "intersect" with the other one.

[@sebastianelytron8450]

*Everybody else:* "WTF???" *Mathologer:* "Amateurs"

[@88Nieznany88]

lol ikr

[@tomewyrmdraconus837]

James Grime has put out a video on the subject before: kzbin.info/www/bejne/hXXXqJiefN-igq8 Not to mention the mug is one of his items form Maths Gear: singingbanana.com/maths-gear/ Matt Parker and Steve Mould were almost certainly hamming it up for the camera, I'm reasonably certain they've been part of videos on the subject. The same goes for Brady Haran, he's filmed a LOT of videos on topology. Many of the rest of them looked like smart people that hadn't encountered the puzzle before, and they performed admirably.

[@risu2312]

"Ho, ho, ho." Sh-Shut up you monotone baldy! (JK, love the guy)

[@ThePotaToh]

Mathologer: *Ho ho ho*

[@agr.9410]

*Mathologer:* "Pathetic."

[@FeinryelRavenclaw]

Well, the next question has to be: “utilizing this puzzle on a torus, what is the shortest possible distance for each line connecting each house to each utility?”

[@vlad1209palovic]

If we use proper torus metric (not deformed by pushing it into 3D), it is same simple as on the Cartesian plane.

[@shadesilverwing0]

I imagine this could be solved by connecting strings to each house and pulling them as tight as they'll go.

[@adarshmohapatra5058]

Doesn't that depend on where the houses and utilities are located? So there isn't one simple answer to your question. Besides all this topology is done on surfaces where distance doesn't matter. Everything here is about position and orientation.

[@FeinryelRavenclaw]

@@adarshmohapatra5058 It shouldn’t. The houses and utilities can be anywhere on the torus, in any orientation, and the puzzle remains mathematically unchanged. Finding the shortest possible distance for every line here is a complicated question, but it should be possible to solve.

[@mtklass]

Actually, my next question would be, "How many handles would a mug need for us to hook up a fourth utility? A fifth? What if we add another house?" So, my next three questions I guess haha

[@domainofscience]

This is so cool! Happy Holidays everyone!

[@iopvixens]

noice hphld2u bai

[@ffggddss]

&u&u!

[@rijuchaudhuri]

Happy Holidays, Dominic! It would've been amazing if you were in this challenge

[@estebancorral5151]

This whole exercise is based on Leonhard Euler. He lived in St. Petersburg, Russia though originally Swiss. The city was never well planned. It is a city of islands, canals, and bridges, a logistical nightmare. The aim of his mathematics was to take the most efficient route any where in the city. Today, FedEx and Amazon trucks are routed through algorithms based on his mathematics. Billions of $ through the legacy of a man who died over two hundred years ago.

[@brooklyna007]

Euler lived in Russia for about 15 years but he lived in Berlin for the remaining 40 years of his life after that. Also, I worked on Amazon's supply chain systems for a while. Euler is undoubtedly one of the best mathematicians of all time and he indeed started some of the math but assigning everything that people are coming up with in supply chain to him (including AI integrated systems) is like assigning all of modern physics to Newton and Leibniz because they started Calculus proper. It is overkill.

[@estebancorral5151]

@@brooklyna007 Eratosthenes, Archimedes, Menaechmus, Aristarchus, Al-khawarizimi were no slouches either.

[@anonemoose7777]

Should have had on lockpicking lawyer (LPL) "I've got a line out of plumping, electricity is binding, false curve out of heating... and we're in! Now let's do it again to prove it's not a fluke. I'd like to thank 3blue1brown for sending me this today but there are a number of vulnerabilities with this mug detailed in the description thank you and have a nice day!" 🤣

[@Lance0]

I can hear his voice while reading this and I don't even have to try wtf

[@klausstock8020]

Using this mug handle which Bosnian Bill and I made...

[@rogogo1244]

Ok I love you.

[@qpSubZeroqp]

You have won the internet lol

[@mikeg5758]

"Lets see how this mug handles the Ramset gun."

[@sebastianelytron8450]

No fair! Wendover was confused because the puzzle doesn't involve planes :-(

[@mohammedjawahri5726]

Sebastian Elytron should've been "connect these 3 planes to 3 utilities" lmao

[@christianbro2]

He would just fly the lines so that they don't cross.

[@alphiek309]

underrated

[@skeeth2631]

Is that a pun

[@Huntracony]

These three highly remote houses need their utilities supplied by airplanes, and due to heavy FAA regulations their flight paths are not allowed to cross. Also, this scenario takes place on a torus world (which are mathematically possible!).

[@pvf6996]

15:00 THAT was outright badass!

[@Waermelon]

For the last section, I remember watching a video about the Klein bottle, where 2 lines can't cross on a 2D world but when entering another dimension [3D] it kind of overlaps the line without crossing it, the mug gives a 3D element to this puzzle, and allows the line to cross over each other, but not intersecting since one is 2 dimensional and one is 3 dimentional [on the handle]

[@nyroysa]

TOP 10 ANIME CROSSOVERS

[@BarackObamaJedi]

nyroysa 19 minutes too late

[@user-zu1ix3yq2w]

IT'S LIKE WE'RE IN ANOTHER DIMENSION

[@Danscottmusic]

TOP 10 MUG-HANDLE CROSSOVERS

[@U1TR4F0RCE]

You know, it was actually a light novel of an anime that first introduced me to Euclid's Formula, the Rampage of Haruhi Suzumiya has a problem that utilizes the Euclid's Formula.

[@NoNTr1v1aL]

U1TR4F0RCE the monogatari series introduced me to e to the iπ plus 1 equals 0.

[@slap_my_hand]

Even without knowing anything about topoloty i immediately knew that this would involve the handle. The exact same problem exists in PCB layout and you solve it by using multiple PCB layers. The handle of the mug is basically the same thing.

[@MrTridac]

That's exactly what I thought. I route PCBs all the time, it kinda felt obvious.

[@helveticalouie]

I'm dumbfounded and have nothing smart to say, but I'll leave a comment to make this more popular in KZbin algorithm. Thank you for a great eye opening video!

[@silvermediastudio]

howie Getants Needs more keywords like "gender fluid" and "progressive."

[@avinashreji60]

+800 Gorilla you just made a place about math have a slightly lower IQ

[@silvermediastudio]

Clearly then, you don't understand the YT algorithm.

[@lizzycoy1745]

800lb Gorilla can you just leave politics out of this math thing? Seriously you're just as bad as the sjw's.

[@silvermediastudio]

You don't understand machine learning through language-analysis algorithms?

[@ked49]

7:01 my solution, this but make the yellow baseline go under and between the handle, let the blue line go over it. This makes it cross over without intersection. Edit: wow, I got it right

[@gerostoumoria]

My great uncle showed me this puzzle ten years ago. He learned it while travelling throughout the world by his captain. Unfortunately he doesn't remember how the captain solved it, so thanks for making this video.

[@fccgrnp2968]

He was remember, the captain wasn't remember, but don't wanted to shoot the joke, because probably paid a price what we did... That's exactly the bulls it what the puzzle were covered with when I met with 20 years ago lol

[@RobotronSage]

@@fccgrnp2968 wow

[@Ket2cool4u]

15:24 is a physical representation of my coding projects

[@cristianmarint]

Hahahahahahahaha

[@sirsanti8408]

I feel like the looking Glass was more accurate

[@newkid9807]

Kid Punk i hate you

[@Slekejkwls-1819]

Just when read it it was showed up, so exact, it's crazy

[@somenamelastnaammee52]

This is sadly very accurate

[@Manabender]

Mathologer had the best solutions. Both of them.

[@arpitdas4263]

He's the Mathologer. Hes older than everyone else combined, and smarter as well

[@newkid9807]

Manabender they didn’t show any footage from him in the beginning because he got it in the start.

[@amitprakashjha1821]

Most of them are great math guys... I watch most of them... But Mathologer is my favourite

[@agentetaeko1422]

Share your opinion, ans his ideas

[@nameymcnameson1903]

Whole video invalid I solved the puzzle

[@larkefedifero]

The guy @ 14:50 has ONE of the two solutions I've considered: there's nothing in the initial instructions that says that the lines can't cross over the *houses* or the electric / gas / water *stations* themselves. But the more complex solution very much DOES involve so-called "going off the GRID" and doesn't require using (or having) a handle like the one on the mug at all. In fact, using a SPHERE very much IS conducive to solving said problem, at least for one of the solutions. Anyways, always have fun and PUZZLE ON, AMERICA! 😁😝😉 😊👍

[@IceMetalPunk]

Matt's solution is definitely the best solution. Math is wrong, coffee and wet pens win :P

[@pierrecurie]

Parker utilities

[@Jojoman103]

You can say that it was a "Parker Square of a solution"?

[@Henrix1998]

Me watching the video: USE THE HANDLE USE THE HANDLE USE THE HANDLE

[@MiaVilleneuve]

Henrix98 same

[@sadhlife]

ikr

[@drewkavi6327]

Yes there is the handle allows one line to go under and one over which if represented on paper would be line crossing but due to the topology of the mug allows two lines to cross without them actually crossing enabling the puzzle to be done

[@EricHallahan]

Me watching this video: It's a TORUS! Use the freaking handle!

[@WitherBossEntity]

They should have figured that there was a reason that they had to do this on a donut and not on a plane.

[@PaulPaulPaulson]

Screw the new avengers trailer, this is so much better! Also, thank you so much for intruducing two new channels to me! I was already subscribed to the other ones, and the two new ones will definetely get a try! Subscribed!

[@fuury09]

Paul Paulson So so..., also schaut der werte Herr doch nicht nur Pietsmiet :D

[@ophireden1751]

On a mug, you can draw a line to a house you have already reached without closing a shape

[@tasty8186]

I remember being told this puzzle back in 2006 or so when I was a kid, and it took literally 10 years and an electrical apprenticeship before I'd figured it out. Old circuit drawing notation to show a wire crossing over another perpendicular wire without connecting is to draw a "C" shape to signify that one wire bends over the other one. This is the solution to this puzzle.

[@youtubeiscorrupt3308]

No it’s not. The cups topology is the key to it. You draw on the handle and the other line goes under the handle. There’s no issues with any of this until the last two connections. So I mean yes this is the answer, but no it’s not. Unless you were using a metaphor.

[@TheGibby1973]

@@youtubeiscorrupt3308 yeah you just made his point if you think about it lol

[@CookerCeiling38]

@@youtubeiscorrupt3308 the handle is the C

[@youtubeiscorrupt3308]

@@TheGibby1973 that’s what I was saying. If he meant it as a metaphor then yes. I said that in the first reply lol. Re read it.

[@Charlotte-gm1hs]

@@youtubeiscorrupt3308 you literally said 'no it's not' but sure

[@NoriMori1992]

15:29 Typical Matt, Parker Squaring it as usual!

[@TheBrickagon]

I was dying of laughter when he said his genius solution 😂😂😂

[@iwansays]

Matt uses wireless power. What a chad.

[@AndreaCremoni]

Welch Labs guy has a face?!?!?

[@DDranks]

And a handsome one, even!

[@opiret44]

This is before the answer is revealed , but it's really clever that this was done on a mug since it isn't obvious but it's just a torus.

[@Tahgtahv]

Remind me not to have Matt install any utilities for me. He'd dig a tunnel for the utility, not install the line, and call the job done. (At least that's what I get from watching his Parker solution)

[@kyzer42]

1:39 Nice one, Brady! :)

[@thesoundofscience]

There's something deeply heartwarming about the my favorite youtube educators all being friends ...

[@DarkDragonLord]

You can solve it with 3 lines. Utilities can branch draw one line from each utility through each house.

[@lyrlwestrum3971]

This puzzle is actually solvable on a 2D piece of paper using only 3 lines, each connecting one house to all three utilities (or vice versa). The specific wording of the puzzle allows for traveling through houses and utilities. Just like in real life, one pipe can house several utility connections.

[@rackyphyr]

interesting solution!

[@miccool9ice363]

Actually that is not even needed They just forgot about how you can use the insides as well as tight corners to make it I was able to solve it on paper by doing this.(It is not at all impossible

[@milesobrien4231]

As usual mathematicians overthink a problem and the engineers have to clean up the mess lol

[@carlost856]

@@miccool9ice363 that's mathematically impossible unles you do what the lyrl did and you go through multiple vertices.

[@miccool9ice363]

@@carlost856 Note: I am not saying that I did not “think outside the box” I am only saying I used 9 lines not 3

[@Huntracony]

*Semi-spoiler* I got this pretty quickly, though I think it would've taken quite a bit longer if you hadn't told me it was a topology puzzle. Edit/note: The original title was "Science KZbinrs attempting a topology puzzle".

[@3blue1brown]

Ah yes, that is quite the context clue. The instructions I gave to people here (well, the people who hadn't seen it yet) made no mention of topology. Edit: Just changed the title, maybe that'll make it a bit less obvious.

[@Ouvii]

Huntracony yeah, having been told it was a topology puzzle I was just "YOU HAVE TO USE THE HOLE WHY AREN'T YOU USING THE HOLE"

[@oscarnemo8084]

Same.

[@petrusboniatus]

Same too

[@ashirizly]

Same, though I suspect that the importance of the puzzle being drawn on a mug would've made me suspect it. Despite my almost complete lack of topology knowledge, I did know that the cup was not homeomorphic to a sphere (it's one of the classic examples in videos much like this one).

[@Jesin00]

When I saw you put it on a mug with a handle, I immediately remembered that joke about the topologist who can't tell the difference between the mug he's dunking a donut into and the donut he's dunking, and how a torus is fundamentally different than the sphere everyone started off treating it as.

[@Ghost_Drive]

(4 minutes into the video) I saw this thumbnail so many times, thinking the point of the video would just be proving why this didn't work. I finally watch the video, and was confused as to why it was a mug for a minute until I realized topology. After getting fed up watching people use the handle incorrectly, I did it on pencil and paper and got it in a minute. Not saying I'm smart for doing it, I watched 4 minutes of attempts before then and I've seen several colloquia that mentioned topology, but it was still satisfying.

[@DawnBriarDev]

Just started watching. I want to point out an exception to this impossibility not resulting from math, but from the defining language: "At no point may two lines intersect." Well, just draw the lines through the houses and you get 3 curves that end up as parallel lines and never intersect. While this is, to a degree, mere sophistry.. It still has relevance in demonstrating the importance of clearly outlining the ruleset and conditions for solving a problem. In education, these kinds of exceptions only show up on IQ tests. In the workforce, they'll show up constantly whether or not you notice them. It's good to practice thinking outside of the box: Even for a mathematician. Because sometimes, the rules allow for unintended solutions.

[@malaven11]

mathologer solves it this way

[@akairis8920]

@@malaven11 Mathologer was thinking in that direction, but still needed the handle to solve it. So he wasn't quite there yet.

[@davidhines7081]

that would be my solution too. once you use an outside utility and connect the houses the center utility would connect to the houses on the lower side, and the last utility will connect on the top side.

[@kilbert666]

Bingo. This is a problem of functional fixedness, pure and simple. This video illustrates perfectly that sometimes trying to outsmart the puzzle just ends up making you look dumber.

[@pepijnstreng4643]

You gave me an idea: draw one point on the mug and call it the "no" point. At this point, the lines may intersect

[@jamesonuwu1346]

very smart math person: * *doesnt solve the puzzle immediately* * me the one dropped out of school watching: pathetic

[@portrand6654]

dont say subreddit names outside of reddit

[@ytsas45488]

+port Rand r/gatekeeping

[@1SSJA]

/r/Greekgodx

[@boredphysicist]

@@akarshrastogi3682 r/wooosh

[@boredphysicist]

@@akarshrastogi3682 he was joking how people watching these channels get an inflated opinion of themselves/assume it is easy from watching these channels.

[@troz3799]

0:32 how could you misspell daren from veristabilium's youtube channel?

[@usukandsarge]

SirMisteryYT Dirk from Veristablium

[@MrLompo123]

hi tim

[@MrTridac]

The Duke from Vatican?

[@bagelnine9]

(16:38) Because if you draw a path around a torus, you don't enclose any regions, so that means that on a torus, you only need to enclose 3 regions.

[@Eyalkamitchi1]

*BADABUM BADABING*

[@citiblocsMaster]

There you go

[@MrRisdar]

🅱️ADA🅱️UM 🅱️ADA🅱️ING

[@JoystuckTV]

George Carlin

[@ccgarciab]

*Squishifies*

[@ilikaplayhopscotch]

I was so happy to see many familiar faces in this video. Thanks for the inherent support/outreach this shows towards the YT platform a possible educational tool.

[@SirStik]

The first time I encountered this puzzle was in the mid 80s. I never knew how it was impossible until today. Thank you. I have over the years introduced it to many to see if they could solve it on a single plane. 30 years of my life finally resolved in a 20 minute KZbin video. 😆

[@nirhymeswithhi4849]

I first encountered it in the mid-80s, too. I believe I saw it in some sort of puzzle book which had most of the answers in the back, except there wasn't an answer for this one. -_- Thankfully it only took me 10 years of my life to find out it was impossible. Now I feel very fortunate! :D

[@Yuekitty]

It’s not impossible

[@thatrandomrecorder914]

Dude just set the ground on fire and everyone gets heat

[@JohnnyApplesauce1]

​@@Yuekittyreally!? How did you solve it???

[@Yuekitty]

@@JohnnyApplesauce1 I don’t remember

[@alexandros.samoutis]

16:52 So the reason that this porblem is possible on a mug is because there will be 6 vertices, 9 endges and 5 regions. 6-9+5 = 2. Problem solved.

[@abdelfiala]

I can't believe this is still on. I was first presented with this problem when I was a kid back in the 80s. Never managed to solve it and never thought about it for the past 30 years or more.

[@Zett76]

Same here. On the mug, though, you have the handle, as a bridge. 😁

[@EclecticSceptic]

Mathologer's smackdown near the end there was classic.

[@Devilogic]

A fun extra puzzle: the video tells you that you can't solve the puzzle on a plane or on a sphere (also not on a cylinder or a on regular strip of paper with ends joined to form a ring), but that you can solve it on a torus. Can you solve it on a (mathematical) Möbius strip? ;) P.S. By a mathematical Möbius strip I mean a mathematically thin one where points on both "sides" are actually considered to be the same points. For many properties of Möbius strips it doesn't matter if you make it out of paper and travel on the surface of the paper (a physical Möbius strip), or if you consider the mathematical version, which would correspond to traveling *inside* the paper; but for the posed puzzle it matters profoundly. Specifically, you have to circle the physical Möbius strip "twice" to return to the starting point, i.e. the curve on the surface has to sweep out 720 degrees in 3D space to return to the starting point (because after 360 degrees you end up on the other side of the paper from where you started), whereas the mathematical Möbius strip has to be circled only once to return to the starting point (i.e. after 360 degrees in 3D space you are back to where you started, and you didn't end up on the other side of the paper - you were inside the paper from the get go). So, to attempt the puzzle: glue yourself a real physical Möbius strip out of paper, but allow yourself to tunnel through the thin sheet of paper to its opposite side at any point (or points) you wish to (this is equivalent to being on a mathematical Möbius strip). Happy puzzle solving! ;) P.P.S. Extra extra: could you solve it on a physical Möbius strip without tunneling? Why/why not? :)

[@MewPurPur]

I got onto it (even though I have a ton of other things I have to do, but screw them for now). The P.P.S. is obvious. You can't do it on a physical Mobius strip, because in this scenario it's like a twice as long strip, though the length doesn't really matter. Maybe I'm not getting your extra puzzle. You move inside the paper, so you can't do tunneling through edges. You can represent it as a rectangle, where you can pass through only two of the edges and where the utility puzzle appears twice. But whatever you draw in one of the utility puzzles, happens in the other because you are moving inside the paper. So you have the same problem you have in the sphere with locking up one of the dots.

[@cardinalhamneggs5253]

You draw the final conduit up to the edge of the paper, back down the other side, and punch a hole through to the last house or utility.

[@sedfer411]

These are definitely the best channels on KZbin!

[@Alexc99xd]

Reminds me of an old friend that gave me this problem. He used a piece of paper, and I couldn't solve it. Then he showed me the solution which was drawing a hole and said the line went underneath lol.

[@harinandanrnair6768]

Alexc99xd Well yeah that is essentially like a torus

[@lnterest-ing]

For anyone who hasnt seen it yet, math youtuber Vihart did a response to this video where she added a 4th utility to the equation!

[@neonxvices]

that’s pretty much the end question but just changing a different variable edit: it was not

[@markhaddad9571]

Soloution: a region is a space where you cant connect a vertix from the inside to the outside without intercacting edges, euleras identity work because each edge either introduce a new vertix or a new region. In a mug you can put a starting vertix on the outside of the mug, then draw an edge from that point up to the handle crossing it like a bridge and going back to the same vertix. This process introduces a new edge without a new vertix (because you got back to the same vertix) and without a new region because you reach any vertix on the mug from any vertix where ever you choose because of the shape of the mug . Thus contradicting eulers formula.

[@adrian9270]

15:29 What a Parker Square of a solution

[@boristheantarcticunicorn8307]

someone else probably already solved this but the proof breaks down at the part where he declared that regions are created where a edge hits a vertex that already has an edge (7:26) This is not true for a torus. Because of the shape of a torus, a region can be 'divided' in half without actually splitting it into two separate regions.

[@xerzy]

@@theAstarrr exactly, it's still drawing over the line with a new one, but since the surface is three-dimensional, then it's a solution… except to me it just feels like cheating and an excuse to bring up torus

[@trumpetperson11]

@@xerzy No. This has nothing to do with 'going over' something by taking advantage of 3 dimensions. It is a puzzle that is possible on a torus (i.e. think of a donut). There is no obvious way to "hop over" another line on a donut. But it is still possible due to the topological nature of the torus/donut. Something that you cannot do on a sphere. Like the comment stated, this is due to the way that regions are divided differently on a torus.

[@xerzy]

@@trumpetperson11 Ultimately, and pardon my lack of mathematical proficiency, it's still two surfaces going parallel to each other in some axis, with some points on top of others, whereas something like a sphere can be treated sort of like an infinite plane. That's what I mean that feels like cheating.

[@MrVeps1]

@@xerzy thing is, a torus isn't two surfaces, it is a single surface. If you decrease the radius of the tube to zero, you are basically left with a circle. You can snip the circle once before you're left with a piece of string. If you deflate a sphere similarly, you can deform it into a piece of string without making a single snip. That's the simple reason why a torus can be divided once more than a sphere "for free", without blocking off a region.

[@amalantony8594]

Hey, can you explain how a region on torus can be halved without splitting it into two regions? I couldn't understand that part.

[@SeanSkyhawk]

15:48 "so when the line comes back out again the pen's not 😠 WORKING ANYMORE 😠 " also ew what a waste of good coffee

[@nullFoo]

I remember playing a pen-and-paper game kind of related to this concept. If I remember correctly you start with 2 points. On each player's turn they connect two points on the page and then add a new point somewhere along that line. You can't cross existing lines and no point can have more than I think 3 (maybe 2?) edges. If you can't make any legal move, you lose.

[@ericrobinson2611]

Thanks for this video. I remember working on this puzzle when I was a kid and never really *knowing* it didn't have a solution, though I suspected it. I'm guessing the proof breaks down because on a Donut, you can connect 4 points consecutively in a loop and still only have one region (for example, four points around looping around the outside of the donut), so therefore, the original proof doesn't hold? EDIT: Reading through more comments, and yeah, seems like this is correct :)

[@3blue1brown]

Yes, that's exactly the right instinct. A torus (wether in the form of doughnut, coffee mug, etc.) has two "intrinsic" loops, where you can close of a cycle in the graph without separating out a new region. Euler's formula gives an explicit way to compute how many such intrinsic loops there are.

[@alexrog6868]

Yeah, a shere is basically a 2d plane in 3d form, just wrapping in on itself, but the handle is a plane existing separate yet connected to the original sphere, so the natural conclusion from there was to systematically choose points of contact between lines and try separating them using the handle, also hearing it now, yeah thinking of it like a Taurus or a donut works too

[@stinkymart3173]

I remember one time in school, a substitute teacher got the class to behave by giving us this puzzle and telling us she'd give $100 to whoever could solve it, she even had a $100 bill for when we called her bluff

[@gamemeister27]

I remember first seeing this video and immediately pausing after understanding the challenge issued, specifically about the implementation on a coffee cup. Rather than thinking about it in math terms, I assume the puzzle was presented on a coffee cup specifically because the form factor of the mug was important. I guessed the handle lets you get around the obvious problems that emerge on a flat plane. I then forgot about it for 5 years, saw this thumb again today, and gave it a go on a coffee mug. It worked! The handle was the key.

[@JohnnyApplesauce1]

Yeah its like an overpass

[@whitherwhence]

Sitting here, watching them try and fail like, "use the donut! You've gotta take advantage of the donutness of the mug!"

[@Hedning1390]

I think the editing made them look more unsure than they really were and they were probably acting too, especially in the beginning. When explaining problems it is usually good to act as if you don't know the solution beforehand so that you can reason to the solution rather than dictate it. I think that's what they were doing (except maybe the guy who said it took him 25 min unless he lied for comedic effect).

[@jonathandpg6115]

no I think they didn't know. Not everything is immidiately easy to solve.

[@ephemeralvapor8064]

I'm pretty sure the mathematicians mostly knew right off, but the physicists took a bit longer on average. That's my estimate not knowing the unedited videos.

[@thomasharris1090]

Ephemeral Vapor Although topology has important applications in physics! Calabi-Yau manifold!

[@RodniOcomments]

In MIPT this problem is a part of one of the home tasks for second year students.

[@IceMetalPunk]

The proof breaks down on a torus at the part where you show that an edge can only add at most exactly 1 new region. On a torus (or other equivalent surface), an edge could potentially add more than one region. Consider a graph, on the surface of a torus (donut), with a single vertex. Now draw an edge that starts at the vertex and loops around the donut the long way back to the same vertex. You've split the space into two regions from one; that added 1 region, like we've been doing. But now add another edge, again from the only vertex back into itself... only this time, go around the smaller circumference of the torus. You can do that without crossing the other edge. By doing this, you've split the surface into 4 regions... but before we had 2! So you've actually added 2 regions with one edge, whereas the proof on a plane depended on being able to only add a single region per edge. QED?

[@dalehall7138]

If I read your construction correctly, it appears you're cutting the torus around an "equator", or the arc of contact the torus would make in rolling along a plane, and claiming to have cut the torus into two regions. That is incorrect. Instead, you obtain a single patch that could be stretched into a rectangular shape. Cutting the torus again along one of the meridians (the small-diameter circle that would be a radial ply in an automobile tire) does finally separate the surface, but now only into two patches. If I misread your construction: oops, sorry.

[@TavartDukod]

Actually, you're both wrong. After cutting torus for the first time you will have a side surface of a cylinder (a surface homeomorphic to that, to be honest). And after the second cut you will have a rectangle, so there will be still only one region. But that's true that the problem with the proof is in adding exactly one region.

[@Guckmalparty]

The task was to combine all icons with those house-images, no other restrictions were mentioned. So basically we can use a hub and it should work.

[@zakarylittle6767]

Kind of. They did also specify no overlaps. I would think one central hub would count as an overlap of lines. Now arguing doing it in series that I can get behind.

[@abucket14]

I drew an upside down L shape connecting through the center of the three houses to the Right most utility using a single line (the prompt says not to cross lines, nothing about crossing houses); Then from there its easy to connect the middle utility to the bottom of each house, and the far left utility to the tops of each of the houses.

[@Mxxx-ii9bu]

@abucket14 Yeah, no.

[@abucket14]

@@Mxxx-ii9bu no why? I'm literally following the prompt; thr fact that thr houses and utilities are represented by things that use lines is either not properly addressed in the prompt or is in line with my own answer (from my perspective understanding).

[@Arikayx13]

Everyone else: *Solves Puzzles* Looking glass: "I've designed a new particle accelerator!" 15:06

[@tungom]

Am I the only one to think her graph is clear? It's a torus; the dotted lines are one the back of the torus.

[@carlettos2623]

XDDDDDDDDDDDD