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

6 жыл бұрын

3Blue1Brown it is possible with a 2D plane

@sen7859

6 жыл бұрын

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

@sen7859

6 жыл бұрын

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

@imnotdaredevil3714

6 жыл бұрын

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

@sen7859

6 жыл бұрын

Jk srry 4 taking your time :D

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

2 жыл бұрын

Are you saying it’s possible?

@gregvs.theworld451

2 жыл бұрын

@@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

2 жыл бұрын

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

@thefoolthatdied

Жыл бұрын

But I solved it?

@stewbaka4279

Жыл бұрын

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

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

@caitlinryan

6 жыл бұрын

i laughed so hard

@MustardPipeLibrary

6 жыл бұрын

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

@PeterAuto1

6 жыл бұрын

that was the best solution

@armu8282

6 жыл бұрын

i dont get it??????

@alwinpriven2400

6 жыл бұрын

you have a parker understanding of jokes then.

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

Жыл бұрын

Not me. I was THREE edges short! =)

@illiji915

9 ай бұрын

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

9 ай бұрын

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

@illiji915

9 ай бұрын

@@CrimmzZT I figured it out

@CrimmzZT

9 ай бұрын

@@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!

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.

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

2 жыл бұрын

15:00

@jordananderson2728

2 жыл бұрын

I love Mathologer.

@slevinchannel7589

2 жыл бұрын

@Nexxol Ok

@NullScar

2 жыл бұрын

@@slevinchannel7589 Mathologer.

@NullScar

2 жыл бұрын

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

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

@AkhilNairjedi18

6 жыл бұрын

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

@VincentZalzal

6 жыл бұрын

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

@budtastic1224

6 жыл бұрын

Same

@ThainaYu

6 жыл бұрын

You sir are hero

@yerrenv.st.annaland2725

6 жыл бұрын

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

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.

@exequielda6649

8 ай бұрын

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

8 ай бұрын

@@exequielda6649 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.

@exequielda6649

8 ай бұрын

@@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

8 ай бұрын

@@exequielda6649 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.

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

@holctomaz2562

4 жыл бұрын

e = 3 = pi

@aniruddhasanyal7625

3 жыл бұрын

@@aidankwek8340 sin(π)=3

@gsuaysuwgs

3 жыл бұрын

@@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

3 жыл бұрын

sin(x) ≈ x for x

@bradstevens4491

3 жыл бұрын

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.

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

@SanneBerkhuizen

5 жыл бұрын

Most underrated comment!

@jdao1sm

5 жыл бұрын

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

@krazieecko

5 жыл бұрын

TOP COMMENT OF THE YEAR

@mojann1

4 жыл бұрын

My thaughts exactly

@kishorekumarsathishkumar1562

4 жыл бұрын

It doesn't cross over though...?

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.

@user-fd3kn6wz3b

5 ай бұрын

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

5 ай бұрын

Instructions unclear there are now 10 dimensions in the explanation.

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.

Nothing that they couldn't handle

@memeislovememeislife3369

6 жыл бұрын

Luke Alexander %😂😂😂

@jaymalby

6 жыл бұрын

Take your upvote.... 😆

@the5thestate587

6 жыл бұрын

Luke Alexander *ba dum tss*

@awesomeguy9573

6 жыл бұрын

Nice

@Jartny

6 жыл бұрын

I see what you did there 😂

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

5 жыл бұрын

Hahahahaaha

@ziggyoickle3445

5 жыл бұрын

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

5 жыл бұрын

she was doing meth

@rafaelcorella1895

5 жыл бұрын

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

@YardenAkin

4 жыл бұрын

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

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.

@DEVIL_SAHARAN

13 күн бұрын

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

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!

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

5 жыл бұрын

Wiebejamin The impossible quiz

@kABUSE1

2 жыл бұрын

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

2 жыл бұрын

Well there you have it, a bridge!

@JediSteve-J3-

2 жыл бұрын

@@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

2 жыл бұрын

OMG I remember this

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

2 жыл бұрын

Shared services for the win.

@worldcolonyinitiativ

2 жыл бұрын

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

2 жыл бұрын

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

@redlok3455

2 жыл бұрын

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

@xemnas577

2 жыл бұрын

@@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

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

2 ай бұрын

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).

This is the ultimate cross-over that I never knew that I needed but once I saw it then my face lifted up with excitement

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

@enzoqueijao

3 жыл бұрын

She was using quaternions to explain how a mug works

@MarkSmith-tu9qr

3 жыл бұрын

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

@mahindoescali

2 жыл бұрын

She is too creative to solve this problem like everybody else

@Hexagons7

2 жыл бұрын

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

@rogercruz1547

2 жыл бұрын

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

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

2 жыл бұрын

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.

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.

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

2 жыл бұрын

But did your teacher cough up the pizza party…?

@benedixtify

2 жыл бұрын

You’re thinking topographically 😁

@kjl3080

2 жыл бұрын

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

@kjl3080

2 жыл бұрын

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

@crazyacorns1173

2 жыл бұрын

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

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

2 жыл бұрын

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

Жыл бұрын

Quite literally the point of the handle.

@bentonrp

Жыл бұрын

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

11 ай бұрын

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

What a nice way to understand bipartite and planar graphs. This came into my recommendations right after my discrete math lecture on planar graphs. Thank you!

Seriously cool video! I suspect that the mug’s topography allows the 2nd-to-last edge to connect two existing vertices without creating a region, thus allowing the puzzle to be completed without violating Euler

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

@mbrusyda9437

4 жыл бұрын

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

@thedoublehelix5661

4 жыл бұрын

Topology is great

@Alex-ud6zr

3 жыл бұрын

Isn't this K3,3?

@absolutezero6190

3 жыл бұрын

Alex A. Yeah

@CrittingOut

3 жыл бұрын

@@Alex-ud6zr Kuratowski's theorem moment

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.

@lucasmatsuoca

2 жыл бұрын

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

2 жыл бұрын

@@lucasmatsuoca 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.

@jettaeschroff6924

2 жыл бұрын

we need more people like you

@ruffusgoodman4137

2 жыл бұрын

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

@zakarylittle6767

2 жыл бұрын

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

Thanks for this video! Someone showed me this puzzle when I was a kid and it haunted me for years

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]

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

@jamesfleming1155

6 жыл бұрын

TT Cubed Agreed. This was awesome.

@Daniel-rt4zz

6 жыл бұрын

Only missing Vsauce

@thefableparable215

6 жыл бұрын

ViHart ;w;

@chuzzywuzzy9545

4 жыл бұрын

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

TOP 10 ANIME CROSSOVERS

@BarackObamaJedi

6 жыл бұрын

nyroysa 19 minutes too late

@user-zu1ix3yq2w

6 жыл бұрын

IT'S LIKE WE'RE IN ANOTHER DIMENSION

@Danscottmusic

6 жыл бұрын

TOP 10 MUG-HANDLE CROSSOVERS

@U1TR4F0RCE

6 жыл бұрын

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

6 жыл бұрын

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

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

Жыл бұрын

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

Жыл бұрын

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

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.

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

@honourabledoctoredwinmoria3126

Жыл бұрын

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

@Omnomnomfish

Жыл бұрын

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

@noobandfriends2420

11 ай бұрын

Klein bottle.

@wren_.

10 ай бұрын

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

@DemonetisedZone

9 ай бұрын

​@@honourabledoctoredwinmoria3126witty 😂

mathologer definitely had the best answer

@HHHHHH-kj1dg

3 жыл бұрын

That's the only answer

@shubhamtiwari5461

3 жыл бұрын

@@HHHHHH-kj1dg I

@jamesknapp64

2 жыл бұрын

He has a lot of slick answers

if you drew one line over the handle of the mug and one line directly underneath the handle, you can make 2 lines cross without touching

This puzzle reminds me of that sectioning technique in mathematics where you have to connect all vertices in one go without lifting the pen. Its use was for mapping out areas. And that sections created by the the technique you have to color in different colors without them adjacently repeating. Cool math

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

@beenaalavudheen4343

6 жыл бұрын

Did u laugh or cringe? Lol

@redstone8513

6 жыл бұрын

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

@fiveoneecho

6 жыл бұрын

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

@cosminaalex

6 жыл бұрын

beena alavudheen did you laugh or did you lose

@TheMarkFeet

6 жыл бұрын

My god that was a good one, Brady

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

Interesting. Also reminds me of a puzzle from about 50 years ago where it was drawn on a sheet of paper and to solve it you took a line across the reverse side of the sheet.

I actually learned about Euler's law and planar graphs in high school, this was a nice reminder, although I was taught a much simpler method of finding if a graph was planar or a K graph or not.

15:00 THAT was outright badass!

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

@SpiffyCheese2

6 жыл бұрын

nor SingingBanna, Matt Parker is stupid

@jmchez

6 жыл бұрын

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

@Huntracony

6 жыл бұрын

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

6 жыл бұрын

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

@Huntracony

6 жыл бұрын

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

(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.

Adding before it gets to the solution, I remember this on paper back in middle school. The handle on the mug definitely makes this possible.

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

5 жыл бұрын

Corlin Fardal Thank you for this comment

@djbj1993

5 жыл бұрын

Yes, you get something resembling a mobius strip :)

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

I figured it out in like 10 seconds, the idea of making the line on the handle just tapped into my head so fast.

@Skxull44

Жыл бұрын

Genius

came across this puzzle a long time ago and it's absolutely brilliant. when you realize that it's impossible, you're halfway there.

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

2 жыл бұрын

topology for the win!

@GQSmoos

2 жыл бұрын

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

2 жыл бұрын

@@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

2 жыл бұрын

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

@seraphina985

Жыл бұрын

@@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.

15:24 is a physical representation of my coding projects

@cristianmarint

4 жыл бұрын

Hahahahahahahaha

@sirsanti8408

4 жыл бұрын

I feel like the looking Glass was more accurate

@newkid9807

3 жыл бұрын

Kid Punk i hate you

@JoseRojas-hl7sn

3 жыл бұрын

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

@somenamelastnaammee52

3 жыл бұрын

This is sadly very accurate

It specifically breaks down where you say it has to light up a vertex or form a new region because there is also the option of enclosing the hole leaving the inside still connected to the outside through the other part of the hole and that works exactly once.

as soon as he said that those who were drawing it on paper would have a hard time, it clicked in my head

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

2 жыл бұрын

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

2 жыл бұрын

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

@seraphina985

Жыл бұрын

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.

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

@shadesilverwing0

2 жыл бұрын

Looking Glass: *summons Hermaeus Mora*

@MrMessiah2013

2 жыл бұрын

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

2 жыл бұрын

She is in her period

@m3lb0urn73

2 жыл бұрын

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

@Xnoob545

2 жыл бұрын

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

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

Жыл бұрын

@abucket14 Yeah, no.

@abucket14

Жыл бұрын

@@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).

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.

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

@mohammedjawahri5726

6 жыл бұрын

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

@christianbro2

6 жыл бұрын

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

@alphiek309

6 жыл бұрын

underrated

@skeeth2631

6 жыл бұрын

Is that a pun

@Huntracony

6 жыл бұрын

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!).

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

@88Nieznany88

6 жыл бұрын

lol ikr

@tomewyrmdraconus837

6 жыл бұрын

James Grime has put out a video on the subject before: kzread.info/dash/bejne/gXio2cehd93VfLA.html 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

6 жыл бұрын

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

@ThePotaToh

6 жыл бұрын

Mathologer: *Ho ho ho*

@agr.9410

5 жыл бұрын

*Mathologer:* "Pathetic."

I am impressed that you gathered so many KZread Math geniuses for this video. This is fun.

When they brought up the fact that they'd use a mug, I immediately recognized the fact that the handle would let you complete the puzzle

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

@oweng8895

3 жыл бұрын

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

@protoborg

2 жыл бұрын

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

2 жыл бұрын

@@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

2 жыл бұрын

@@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

2 жыл бұрын

@@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.

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

2 жыл бұрын

Yeah that's what I was thinking

@j.c.k.8639

Жыл бұрын

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

@nikkiofthevalley

Жыл бұрын

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

This reminds me of my graph theory class in my college and studying why k3,3 graph is non-planar. Just some theoretical explanation to the above puzzle great 👍 👌 😀

The sponsored spot at the end reminds me of that game lights out

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

2 жыл бұрын

now this is fancy

@whythosenames

2 жыл бұрын

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

2 жыл бұрын

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

2 жыл бұрын

@@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

2 жыл бұрын

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.

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

6 жыл бұрын

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

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.

This is pretty cool ! thank you for the video

*BADABUM BADABING*

@citiblocsMaster

6 жыл бұрын

There you go

@MrRisdar

6 жыл бұрын

🅱️ADA🅱️UM 🅱️ADA🅱️ING

@JoystuckTV

6 жыл бұрын

George Carlin

@ccgarciab

6 жыл бұрын

*Squishifies*

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

@MiaVilleneuve

6 жыл бұрын

Henrix98 same

@sadhlife

6 жыл бұрын

ikr

@drewkavi6327

6 жыл бұрын

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

6 жыл бұрын

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

@WitherBossEntity

6 жыл бұрын

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

If the only condition is to don't interesct the lines (pipes), that means that the 9th line can go from one utility trough another one and that solves the problem

I made it simpler. I drew a triangle around one of the points where my mug handle is connected to the mug. I have 3 edges and 3 vertices now, but ONLY ONE REGION! Thus, I end up with 1 instead of two. This holds true with another graph (draw a connected triangle around the other mug point), and you could draw a more complicated graph and it would still be the same. This is a cool puzzle.

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

2 жыл бұрын

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

@klausstock8020

2 жыл бұрын

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

@rogogo1244

2 жыл бұрын

Ok I love you.

@qpSubZeroqp

2 жыл бұрын

You have won the internet lol

@mikeg5758

2 жыл бұрын

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

Mathologer had the best solutions. Both of them.

@arpitdas4263

4 жыл бұрын

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

@newkid9807

3 жыл бұрын

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

@amitprakashjha1821

3 жыл бұрын

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

@agentetaeko1422

3 жыл бұрын

Share your opinion, ans his ideas

@nameymcnameson1903

2 жыл бұрын

Whole video invalid I solved the puzzle

I was given this puzzle when I was a little kid but solved it a bit differently. I had previously played a city builder game before, that had an isoliner view. With the though process I went into the puzzle with you don't even need nine lines. IRL, Gas and Water would be run through Pipes. Where to pipes go? Underground. Water and gas can be serviced to all three houses via two separate pipelines running UNDER each house. Power, then gets run to each house through powerlines, above ground from the station. 5 lines, none cross.

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.

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

@TheBrickagon

2 жыл бұрын

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

@iwansays

2 жыл бұрын

Matt uses wireless power. What a chad.

This is so cool! Happy Holidays everyone!

@iwaru_iopfox

6 жыл бұрын

noice hphld2u bai

@ffggddss

6 жыл бұрын

&u&u!

@rijuchaudhuri

6 жыл бұрын

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

I did this at home to see if I could get this before watching. It was confusing at first because I was drawing on a 2D object (My iPad) so it took me a few times to realize a cup has a 360 way to connect a line lol.

i know this video is 5 years old, but i just watched it for the 1st time! a very interesting puzzle, which i think can be re-stated as a map coloring question. it's well known that any map in a plane can be colored using up to 4 colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color. now, how many colors are required to properly color any map on a torus (which is topologically equivalent to a mug)? the answer is (?) 7. replace the countries of the map with houses and utilities and you know that you could have even 4 houses and 3 utilities or 3 houses and 4 utilities, and still solve this puzzle.

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

2 жыл бұрын

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

@shadesilverwing0

2 жыл бұрын

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

@adarshmohapatra5058

2 жыл бұрын

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

2 жыл бұрын

@@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

2 жыл бұрын

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

1:39 Nice one, Brady! :)

i just imagined the lines going over or under the houses and it makes sense enough to me.

I love seeing all the different minds converge on the same "eureka" moment in their different ways.

Mathologer's smackdown near the end there was classic.

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

@silvermediastudio

6 жыл бұрын

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

@avinashreji60

6 жыл бұрын

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

@silvermediastudio

6 жыл бұрын

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

@lizzycoy1745

6 жыл бұрын

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

@silvermediastudio

6 жыл бұрын

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

Holy shit unexpected Ben Eater! That guy taught me more about digital design than 6 years of schooling!

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

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

5 жыл бұрын

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

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

2 жыл бұрын

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

2 жыл бұрын

@@fccgrnp2968 wow

Great video as always, Grant!! When I realized it got something to do with the handle, I stopped and went to paper and draw the square version of a torus (identifying opposite sides) and solved it. But then I was rather impressed with the one who actually drew a 3D torus on paper to solve it and I felt kind of lazy for using the square

Wow, this is a serious collaboration

15:29 What a Parker Square of a solution

Welch Labs guy has a face?!?!?

@DDranks

6 жыл бұрын

And a handsome one, even!

(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.

I was working on this for three days because a friend shared this with me from his math teacher.

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

2 жыл бұрын

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

2 жыл бұрын

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

@truenorthtransparency5230

2 жыл бұрын

@@youtubeiscorrupt3308 the handle is the C

@youtubeiscorrupt3308

2 жыл бұрын

@@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

2 жыл бұрын

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