Why this puzzle is impossible 

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.

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.'

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

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.

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.

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

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

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.

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

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.

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.

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.

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

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 =)

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.

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!

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

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.

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.

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!

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.

Nothing that they couldn't handle

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

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.

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

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.

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.

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

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.

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

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

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.

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

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

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

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

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.

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.

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

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

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.

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]

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

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!" 🤣

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.

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 👍 👌 😀

mathologer definitely had the best answer

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.

This is pretty cool ! thank you for the video

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.

15:00 THAT was outright badass!

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

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

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.

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.

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

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.

I just saw 3 of my favorite You Tubers and Matthew. 😆 love it!

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.

TOP 10 ANIME CROSSOVERS

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

(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.

15:24 is a physical representation of my coding projects

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

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

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.

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!

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

I used to play a lot of flash games when I was a child, and one of them was this exact puzzle (on a flat surface). I tried for hours at the time to find a way to solve it, before realising it was impossible. The game was coded so that every time 2 lines cross each other, every single line you've drawn would disappear, but if you somehow managed to beat the thing, a victory screen would pop up. As a matter of fact, it was possible to beat it by exploiting tools at hand to trick the memory of the game into thinking you didn't release the click, and linked the last source to the last house, even tho it went straight through one of the other lines. The idea was to draw every line but the last one, making sure those never crossed, and then for the last one, hold the click, draw till you're realy close to the line you need to go through, and while holding the left click, press the right click so the window with options would pop up. Once it popped up, release the left click, and simply move your cursor to the last house, and click it. This method was the easiest, but didn't work consistently, so another method would have you clicking on the "zoom in" option in the right click pannel instead of going straight for the house. It tricked the computer more consistently, but this time the issue was that the zoom would sometimes move your cursor right into the line you were trying to go through (or sometimes another one), overlapping both lines, and resetting the game. It was pretty fun for me as a child to realise it was impossible, even tho I didnt understand the reason at the time, and it's realy cool to revisit this problem through this video. Thanks to you, I can now understand why it was impossible.

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

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.

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

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.

When I was an apprentice electrician, a journeyman friend of mine showed me this puzzle. Of course, I was doing it on paper, not a mug, but I came up with a similar solution as the handle. In electrical prints, lines do cross that dissociate. I used the same method to "solve" the puzzle on paper.

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

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

I have never been more disappointed in a youtuber and channel in my life. I'm not a mathemetician, I just love the graphics! Where's the damn answer? :)

i knew this puzzle for more than 10 years and all i needed was a mug. What fun!

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?”

Mathologer had the best solutions. Both of them.

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.

the reason the handle changes things is that it sort of connects two regions. so one region you create will not actually be separated, due to the handle, and will just cancel it out. you can almost think of the hole in the handle as "negative 1 region", because each vertex will now either: connect to a new point create a new region, or cancel out a hole

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.

This is so cool! Happy Holidays everyone!

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

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

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

1:39 Nice one, Brady! :)

Mathologer's solution was brilliant and hilarious.

couple of ways to solve this, one is connect the the utlities to each other since they will all need power to run, then to a hub (at the bottom inside the mug (the hub counts as four new edges)) and then to the houses, breaking the (edges) into more, or simply with the last house needing gas to use the handle to go over everything then inside the moug, and inside the mug to the house.

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

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

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.

6:37 This is where it breaks down when a mug is used. Imagine area A around one side of a handle, while area B around the other side. If you take the handle away, there is no way you can draw a line from one area into another. However, using the handle, you can bridge between the two. If you stretch the mug out into a doughnut shape, then areas show up as rings around the doughnut. There is no boxing in.

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.

Mathologer's smackdown near the end there was classic.

Wow, this is a serious collaboration

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

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