Why this puzzle is impossible
Featuring quite a few science/math KZheadrs!
Vihart response: • Four Utilities Puzzle ...
Brought to you by you: 3b1b.co/mug-thanks
And by Brilliant: brilliant.org/3b1b
Timestamps:
0:00 - Featured guests
4:30 - Why it's "impossible"
12:20 - Surfaces with holes
16:27 - Your challenge
17:35 - Sponsorship and end
Thanks to all the following channels for participating.
Standup Maths
/ standupmaths
Wendover Productions
/ wendoverproductions
Welch Labs:
/ taylorns34
MinutePhysics:
/ minutephysics
Ben Eater:
/ eaterbc
Mathologer:
/ @mathologer
Singing Banana:
/ singingbanana
Numberphile:
/ numberphile
Looking Glass Universe:
/ lookingglassuniverse
Veritasium:
/ 1veritasium
Steve Mould:
/ steventhebrave
Special thanks to MathsGear for providing the mugs.
mathsgear.co.uk/
mathsgear.co.uk/products/gift...
Music:
Vincent Rubinetti: / vincerubinetti
Divertissement by Kevin MacLeod is licensed under a Creative Commons Attribution license (creativecommons.org/licenses/...)
Source: incompetech.com/music/royalty-...
Artist: incompetech.com/
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
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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.
3Blue1Brown it is possible with a 2D plane
I have done it at my 2nd try! :)
3Blue1Brown and it is not like the mathloger's solution :D
One of your utilities reach 2 houses, your ninth line is a telephone line from the first to the last house hahaha
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.'
Are you saying it’s possible?
@@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.
Some may call that stubbornness or pride. Mathematicians may call it “certainty.”
But I solved it?
@@LurkingAround nah maybe next time, but i also proved it
the parker square joke was hilarious. 10/10 brady.
i laughed so hard
And then, of course, Parker himself had a Parker Solution to the puzzle.
that was the best solution
i dont get it??????
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.
Not me. I was THREE edges short! =)
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
@@illiji915 HOLY YOU ARE RIGHT! THIS IS THINKIN OUTSIDE THE BOX
@@CrimmzZT I figured it out
@@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".
15:00
I love Mathologer.
@Nexxol Ok
@@slevinchannel7589 Mathologer.
@@slevinchannel7589 Also, Tibees, very interesting angling of subjects. Especially her storytelling through painting.
INFINITY WAR: The most ambitious cross over in history 3Blue1Brown: hold my mug
Most underrated comment!
That’s ironic because it has to do with lines not crossing over each other.
TOP COMMENT OF THE YEAR
My thaughts exactly
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.
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😢
Instructions unclear there are now 10 dimensions in the explanation.
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.
Hahahahaaha
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
she was doing meth
@@ziggyoickle3445 interesting. I got that same thing when i first opened the comment
@@ziggyoickle3445 It's a bug with the KZhead app. Comments from previously watched videos show up randomly replacing comments on the video you're currently viewing. Hopefully it gets fixed soon
In engineering class I would do the 8 connection and hope for partial credit.
e = 3 = pi
@@aidankwek8340 sin(π)=3
@@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
sin(x) ≈ x for x
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.
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.
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!
@@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.
@@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.
@@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.
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_
She was using quaternions to explain how a mug works
she may not find the solution like everybody else but the she had an interesting approach 😅👌
She is too creative to solve this problem like everybody else
Actual mathematicians: This is hard 3blue1brown viewers: easy, what’s next
had she just used a torus she would get it instantly, but she chose a sphere
Huge thanks to grant for including me in this super fun video! It’s an honor to be edited back to back with some KZhead heroes!
Welch Labs You are one of the heroes! Your videos are amazing. Thanks a lot for creating such educational and interesting videos.
I've just discovered your channel thanks to this video. I watched the "How to science" series and I have subscribed :)
Same
You sir are hero
Dude, your series on Complex Numbers carried me through high school mathematics!
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.
yah i had the same ans as adding that vertex would lead neither edge increase or new region
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
Nothing that they couldn't handle
Luke Alexander %😂😂😂
Take your upvote.... 😆
Luke Alexander *ba dum tss*
Nice
I see what you did there 😂
"I tend to make a parker square out of these...oops, see." I actually left the room after that.
Did u laugh or cringe? Lol
I "cringed", per se. It surprised me out of nowhere but I still went along with it.
I thought that part was great, because I thought he actually dropped it for a second... :P
beena alavudheen did you laugh or did you lose
My god that was a good one, Brady
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.
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).
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.
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 =)
Shared services for the win.
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
or just let one house don't have gas and let them heat up with electricity instead
@@xemnas577 Right, but since electricity is pure exergy, it'd be a waste to use it solely for heating.
@@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
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.
But did your teacher cough up the pizza party…?
You’re thinking topographically 😁
I mean that’s still a nontrivial solution so pretty cool
Also damn that school is sadistic- like no homework if you prove FLT
@@benedixtify He did actually, one of my favorite school days lol.
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!
Thanks for this video! Someone showed me this puzzle when I was a kid and it haunted me for years
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.
Wiebejamin The impossible quiz
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
Well there you have it, a bridge!
@@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.
OMG I remember this
When all hope seems lost. You remember of one dark and evil subject in maths... Topology.
That was the first thing I thought of when they use a cup with a handle, though, hahaha
Topology is great
Isn't this K3,3?
Alex A. Yeah
@@Alex-ud6zr Kuratowski's theorem moment
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
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.
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.
When all your favourite you tubers are all in one video . Best Christmas gift ever.
TT Cubed Agreed. This was awesome.
Only missing Vsauce
ViHart ;w;
# when you're such a nerd you're already subscribed to all these people.
Everyone else: Making doodles on a mug Looking Glass: Studying alchemy or some other esoteric shit
Looking Glass: *summons Hermaeus Mora*
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.
She is in her period
I’m actually trying to understand what is looking glass doing ;-;
Mathologer: just move the handle Matt Parker: the coffee wets the marker and it doesn't draw, so no intersecting of the lines
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.
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
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.
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".
@@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.
we need more people like you
The real question here is for what configuration of the problem in a 3D environment not possible to solve?
@@ruffusgoodman4137 Sphere. Cube. Anything without a hole maybe?
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.
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
Quite literally the point of the handle.
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! 😊
@@samlevi4744 which makes the handle useless in accordance with the directions.
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
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]
The parker square reference by Brady at 1:40 is hilarious 😂
Gotta love Mathologer. It wasn't even a challenge for him. That man's a genius
nor SingingBanna, Matt Parker is stupid
Kind of stacking the deck there. Also, I wish that Vihart had been invited. Pens, doodles and math are her thing.
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.
I understand that, Its just a numberphile inside joke to make fun of him.
ThatMathNerd, No, you make fun of his square. Not him. So you _could_ say that was a Parker square of a solution.
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.
This is pretty cool ! thank you for the video
mathologer definitely had the best answer
That's the only answer
@@HHHHHH-kj1dg I
He has a lot of slick answers
*Everyone else solves the puzzle.* Matt: Ah... I love the taste of fresh dry erasable marker in the morning.
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.
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.
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.
@@brooklyna007 Eratosthenes, Archimedes, Menaechmus, Aristarchus, Al-khawarizimi were no slouches either.
Like most puzzles, this could be easily solved with judicious application of a power drill.
On a sphere, sure. But the coffee cup already has a hole, so you don't need to drill another.
In the original pen and paper version the solution was to just punch the pencil through the paper and call it a day 😂
Klein bottle.
that’s what the handles for. I think a topologist would murder you if you made an unnecessary hole in the mug
@@honourabledoctoredwinmoria3126witty 😂
*Everybody else:* "WTF???" *Mathologer:* "Amateurs"
lol ikr
James Grime has put out a video on the subject before: kzhead.info/sun/gqitp8mgfqylf4U/bejne.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.
"Ho, ho, ho." Sh-Shut up you monotone baldy! (JK, love the guy)
Mathologer: *Ho ho ho*
*Mathologer:* "Pathetic."
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.
I figured it out in like 10 seconds, the idea of making the line on the handle just tapped into my head so fast.
Genius
Me watching the video: USE THE HANDLE USE THE HANDLE USE THE HANDLE
Henrix98 same
ikr
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
Me watching this video: It's a TORUS! Use the freaking handle!
They should have figured that there was a reason that they had to do this on a donut and not on a plane.
TOP 10 ANIME CROSSOVERS
nyroysa 19 minutes too late
IT'S LIKE WE'RE IN ANOTHER DIMENSION
TOP 10 MUG-HANDLE CROSSOVERS
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.
U1TR4F0RCE the monogatari series introduced me to e to the iπ plus 1 equals 0.
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.
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
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!" 🤣
I can hear his voice while reading this and I don't even have to try wtf
Using this mug handle which Bosnian Bill and I made...
Ok I love you.
You have won the internet lol
"Lets see how this mug handles the Ramset gun."
15:24 is a physical representation of my coding projects
Hahahahahahahaha
I feel like the looking Glass was more accurate
Kid Punk i hate you
Just when read it it was showed up, so exact, it's crazy
This is sadly very accurate
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 am impressed that you gathered so many KZhead Math geniuses for this video. This is fun.
No fair! Wendover was confused because the puzzle doesn't involve planes :-(
Sebastian Elytron should've been "connect these 3 planes to 3 utilities" lmao
He would just fly the lines so that they don't cross.
underrated
Is that a pun
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!).
When I first saw the mug, my mind started shouting “IT’S A TORUS!!!”
Me too lmao It's like that classic joke: "a topologist doesn't know the difference between a coffee mug and a donut"
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.
@@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.
@@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.
@@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.
came across this puzzle a long time ago and it's absolutely brilliant. when you realize that it's impossible, you're halfway there.
as soon as he said that those who were drawing it on paper would have a hard time, it clicked in my head
15:00 THAT was outright badass!
I love seeing all the different minds converge on the same "eureka" moment in their different ways.
(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.
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.
topology for the win!
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.
@@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.
@@purplenanite Hi. Want some scientific Watch-Suggests? Some Channel to check out?
@@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.
Mathologer had the best solutions. Both of them.
He's the Mathologer. Hes older than everyone else combined, and smarter as well
Manabender they didn’t show any footage from him in the beginning because he got it in the start.
Most of them are great math guys... I watch most of them... But Mathologer is my favourite
Share your opinion, ans his ideas
Whole video invalid I solved the puzzle
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.
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.
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.
Corlin Fardal Thank you for this comment
Yes, you get something resembling a mobius strip :)
*BADABUM BADABING*
There you go
🅱️ADA🅱️UM 🅱️ADA🅱️ING
George Carlin
*Squishifies*
Wow, this is a serious collaboration
On a mug, you can draw a line to a house you have already reached without closing a shape
I wonder how this problems comes up in writing for computers. The PCB can be many layers but there are only so many layers
now this is fancy
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
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.
@@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)
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.
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.
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.
Maybe Im simply drawing it wrong, but for me each edge touches exactly two regions:/ Also there are only three regions in total
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.
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.
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.
@abucket14 Yeah, no.
@@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).
very smart math person: * *doesnt solve the puzzle immediately* * me the one dropped out of school watching: pathetic
dont say subreddit names outside of reddit
+port Rand r/gatekeeping
/r/Greekgodx
@@akarshrastogi3682 r/wooosh
@@akarshrastogi3682 he was joking how people watching these channels get an inflated opinion of themselves/assume it is easy from watching these channels.
Matt's solution is definitely the best solution. Math is wrong, coffee and wet pens win :P
Parker utilities
You can say that it was a "Parker Square of a solution"?
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.
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 👍 👌 😀
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!
Paul Paulson So so..., also schaut der werte Herr doch nicht nur Pietsmiet :D
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.
That's exactly what I thought. I route PCBs all the time, it kinda felt obvious.
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.
i just imagined the lines going over or under the houses and it makes sense enough to me.
15:29 Typical Matt, Parker Squaring it as usual!
I was dying of laughter when he said his genius solution 😂😂😂
Matt uses wireless power. What a chad.
I'm dumbfounded and have nothing smart to say, but I'll leave a comment to make this more popular in KZhead algorithm. Thank you for a great eye opening video!
howie Getants Needs more keywords like "gender fluid" and "progressive."
+800 Gorilla you just made a place about math have a slightly lower IQ
Clearly then, you don't understand the YT algorithm.
800lb Gorilla can you just leave politics out of this math thing? Seriously you're just as bad as the sjw's.
You don't understand machine learning through language-analysis algorithms?
I was working on this for three days because a friend shared this with me from his math teacher.
Me: connects all houses together, and all utilities together, and creats a line. They didn't tell me I had to connect them individually
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
Yeah that's what I was thinking
i was pissing myself laughing when i realized that, whatching the vid, then wanted to like that exat comment.
No, in real life you'd just put the pipes under/over the other pipes, and use straight lines.
1:39 Nice one, Brady! :)
14:50 is essentially the solution provided in a puzzle book from 1946 I just happened to have in my bedside table.
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?”
If we use proper torus metric (not deformed by pushing it into 3D), it is same simple as on the Cartesian plane.
I imagine this could be solved by connecting strings to each house and pulling them as tight as they'll go.
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.
@@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.
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
Mathologer's smackdown near the end there was classic.
Holy shit unexpected Ben Eater! That guy taught me more about digital design than 6 years of schooling!
on the mug, the handle lets one line go over the other, essentially making 2 lines intersect without intersecting it, so it works
This is so cool! Happy Holidays everyone!
noice hphld2u bai
&u&u!
Happy Holidays, Dominic! It would've been amazing if you were in this challenge
0:32 how could you misspell daren from veristabilium's youtube channel?
SirMisteryYT Dirk from Veristablium
hi tim
The Duke from Vatican?
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 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.
the way i came up with the solution is how i remembered struggeling with designing the board of an arduino, becuase the tricky part is to align the conducting path so they dont intersect. That moment it came to my mind "somekind of bridge would be help ful ..... oooooh the HAAANDLE" The math behind all that is sill very fascinating. Love your work and passion for all of this, greetings from germany.
MatureFister you are awesome Not!!!
Oh, that’s cool! A neat real-life instance of this.
i think it's easier to think of a real life scenario for this instead of just viewing it as lines and vertices. but yeah, the math side was cool lol