Why this puzzle is impossible

Featuring quite a few science/math YouTubers!

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

Wendover Productions

Welch Labs:

MinutePhysics:

Ben Eater:

Mathologer:

Singing Banana:

Numberphile:

Looking Glass Universe:

Veritasium:

Steve Mould:

Special thanks to MathsGear for providing the mugs.

Music:

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld

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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that).

Various social media stuffs:

[bluebrown]

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.

[abigailcooling]

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.'

[kylerivera]

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

[TheAgentAPM]

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

[brooklyna]

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.

[alwinpriven]

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

[Stormingmonkey]

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

[Zarkonem]

I used to give this puzzle to my friends in highschool. I even made a poster and posted it around the school with a reward attached encouraging everyone to try it and come give me the answer. No one ever did. I had several people run up to me enthusiastically telling me that they solved it only for me to point out that they are missing a line.

I had thought it was impossible to do it on a piece of paper for 18 years. Thanks for proving to me that i was right.

[WelchLabsVideo]

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

[redlok]

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 =)

[cherrywolf]

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.

[jankowalski-pyey]

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.

[Ken.-]

15:07 No idea what Looking Glass was doing over here...

Tries to solve a simple puzzle on a mug.
Accidentally designs a working quantum computer instead.

[MrHatoi]

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

[zach]

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.

[awesomechaos]

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!

[aquinsvarghese]

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

[crazyacorns]

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.

[zenedhyr]

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.

[estebancorral]

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.