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Four Utilities Puzzle (and how to ruin a bagel)
Рет қаралды 408,032
6 жыл бұрынA new take on a classic.
In response to this collab: • Why this puzzle is imp...
"Topology Joke": • Topology joke
Susan Goldstine's bead crochet video: • Bead Crochet Demo--Sev...
Susan has also collected a page of other artistic renditions of the 7-color torus and links to instructions for beadwork, fiber arts, and other further reading: faculty.smcm.edu/sgoldstine/to...
This video not sponsored by or affiliated with mathsgear, but here's where they sell the official version of the mug: mathsgear.co.uk/products/util...
This video is supported by viewers like you on Patreon! Thank you especially to Caleb Wright, Albert Wenger, Pat Devlin, Jack Heidrick, David Perryman, Jade Bilkey, Chris Pierik, Donald "Chronos" King, Andew Romaner, Jodi Vezzetti, Carol Ghiorsi Hart, Andrea Di Biagio, Charley Sheets, Yana Chernobilsky, David A Smith, and Michael Tiemann!
Oh, and we put in the order for the Tau Day shirts, but we ordered some extra so here's a link to shirts and hopefully your size is still available! store.dftba.com/collections/v...
@HalfBlindPrince 6 жыл бұрын
"Oh, that's a nice little theorem you have. It would be a shame if someone were to generalize it."
@DillyRed 5 жыл бұрын
HalfBlindPrince ok
@shengshu3510 5 жыл бұрын
@@DillyRed k
@ew8381 4 жыл бұрын
sheng shu ok
@julie9337 4 жыл бұрын
MayMayYt ツ ok
@abl9643 4 жыл бұрын
Bunnies Rule ok
@Sam_on_YouTube 6 жыл бұрын
If you're going to bagel it, then really bagel it. There are places that make rainbow bagels by weaving colored dough. You could turn that into an edible version.
@DieAlteistwiederda 6 жыл бұрын
Sam I'm a simple person who also doesn't care or mind to ingest a few toxins so I would still eat that bagel. I mean it most likely won't kill me right after eating it and I have to die anyway but not eating would still probably kill me faster
@U014B 6 жыл бұрын
DieAlteistwiederda Sharpie ink is supposed to be non-toxic, so you'd probably be fine. It might taste like you're chewing on a big, squishy aspirin, but at least it won't kill you.
@DieAlteistwiederda 6 жыл бұрын
Noel Goetowski thank you for this little bit of extra knowledge. We don't have the sharpie brand over here and what we have is toxic so I just assumed that these markers are as well. Well, now I know that I probably wouldn't die much faster than we all already are if I ever fly over to the US and eat a bagel that someone colored in with a sharpie. 😂
@Sam_on_YouTube 6 жыл бұрын
DieAlteistwiederda Yes, that is useful knowledge.
@Arashmickey 6 жыл бұрын
Ezekiels Pretzels
@fealubryne 6 жыл бұрын
The fact that I can understand this, to a degree, even with ten years of medical issues and brain surgeries destroying my memory and fogging my thought process, is impressive. You make a really good teacher, even though my heart broke when you drew on the peach rings.
@Gotlyfe 6 жыл бұрын
I died a little inside when the sharpie hit peach ring
@thomaslecky5301 6 жыл бұрын
Hope you are feeling better :)
@masonloeffler8064 6 жыл бұрын
I hope you get better. And yes, if all teachers were as good as her learning would be much easier.
@octosaurinvasion 2 жыл бұрын
how are you doing now?
@fealubryne 2 жыл бұрын
@Costumed Creeper Better, thank you! Still have some physical difficulties, but the combination of video games and writing fiction have helped with memory and focus issues.
@saltedapplesticks2380 6 жыл бұрын
I used to just watch your videos because I liked the art but now I'm actually going to learn about this stuff so it's helpful. So thank you.
@toadsoup7507 5 жыл бұрын
All this info is too complicated for my stupid snake of a brain, but her voice makes me feel nice :D
@scarymonstaz1413 2 жыл бұрын
@@toadsoup7507 very true words
@rapophie9228 6 жыл бұрын
Its fascinating how many ways you find to somehow involve food into your video.
@kimberlys313 6 жыл бұрын
Rapophie Cuz they got the munchies. Why do you think they’re so crazy?
@Roxfox 6 жыл бұрын
Nice avatar, Commander.
@yaseen157 6 жыл бұрын
never forget the hexaflexamexagon
@jessdoritowhale 6 жыл бұрын
That’s it! Math= FOOD NO WONDER IM FAT AND GOOD AT MATH!
@TalysAlankil 6 жыл бұрын
Can I just say I love that you saw a problem and your first instinct was "how do I make it harder" (I mean yes I know you're using proven results to work your way down, but still)
@iblesstherains 6 жыл бұрын
*Vihart - the only youtuber that you can go to to draw on mugs and bagels*
@dextrician5932 6 жыл бұрын
Everyone's upset about the bagel, I'm here crying about the sugary goodness that was 4:20.
@7eartcat248 6 жыл бұрын
Same
@joeyzucchini 6 жыл бұрын
same
@hues- 6 жыл бұрын
mood
@adrianremo1988 6 жыл бұрын
Finally someone gets our pain.
@logicisnthere7523 6 жыл бұрын
420 :))) but why do i think about the sharingan whe puts the 3 dots on the sour gummy donuts and connects them
@fuzzypineapple5118 5 жыл бұрын
1900: “I bet there’ll be flying cars in the future” 2019: “Let’s do some puzzles with a bagel”
@noxfortes 3 жыл бұрын
Absolutely better than flying cars
@user-dr5nt2is7f 2 жыл бұрын
@@noxfortes Can't agree more
@megathecat375 Жыл бұрын
Flying cars do exist, they are called helicopters.
@robertsmith7627 6 жыл бұрын
Finally I've been looking everywhere for a non adult method to ruin a bagel
@JustinY. 6 жыл бұрын
Is this a subtle advertisement for your new mug merch?
@superwinner5010 6 жыл бұрын
Oh hey again
@finlayl2505 6 жыл бұрын
Either you watch every video on the internet or just the same ones as me and I'm not sure which is more unlikely
@MisterAppleEsq 6 жыл бұрын
+bred From what I've heard, he just *comments* on lots of videos, which isn't the same as watching them all.
@hoobslice 6 жыл бұрын
Hi neighbour
@chamoy5579 6 жыл бұрын
Yustin I didn't see you in a couple channels dunno why people hate on you :v
@squishlez 6 жыл бұрын
Even though I'm older than I used to be when I found vihart I still have no idea what she's talking about
@7eartcat248 6 жыл бұрын
I just like listening to her rambles
@Roxfox 6 жыл бұрын
Interesting sentence... Will you ever not be older than you used to be, I wonder?
@JuryDutySummons 6 жыл бұрын
All I know is that I'm getting old for my age.
@blazerlazer55 6 жыл бұрын
today i feel like being a bit older
@matemahe 6 жыл бұрын
Pielover1617 is a hairry leg world is fool of smart asses
@JoelDowdell 6 жыл бұрын
It took about 1.5 sheets of graph paper, and some distractions, but I graphed a 6 simplex on a 8*14 grid, looping around top and bottom. Seeing as I have no bagel or mug to ruin, I will leave the 4 houses and 3 utility buildings in their neat, evenly spaced row in the middle. On the 8*14 grid picking a row along the 14 axis and spreading utilities and houses spaced along intervals of 2. I drew lines with slopes of 2 and 4/3 and continued them through the connected borders of the rectangle. I can't believe such a simple solution took me so long, but I am glad I spent the time. This is why I watch vihart and 3blue1brown, it helps me keep these topics near the surface of my mind.
@ButCritically 6 жыл бұрын
i love the bridge you drew for the reductions into 4 color map theorem! it's so beautiful
@hexagondaddy1015 6 жыл бұрын
I live how you talk, and how you say your thoughts...for some reason it makes me calm...my friends tell me that "That's weird, but that's why we are friends" but why is that weird....I just like hearing you talk! Keep up the amazing work!!!!!!!!!!
@abigaylelynn1143 6 жыл бұрын
Oh yay I was waiting for a new Vi Hart video since I just finished reading a Martin Gardner book.
@zh84 6 жыл бұрын
Since Martin Gardner retired, and subsequently died, people like Vi Hart, Henry Segerman and Brady Haran have kept recreational mathematics going.
@7eartcat248 6 жыл бұрын
The good side of youtube
@jakeroberts8471 6 жыл бұрын
This is genius, I watched the original collab video and your method is so much more elegant. Love it!
@Tehom1 6 жыл бұрын
Might be worth mentioning that after you add your internet & secure network, you have the complete graph K_6. Not just theoretical, this actually has some bearing on the puzzle: You have all the simple edges that you can ever have. Even if you added 1000 more types of network, but no more houses and utilities, the puzzle would still be soluble, if not easy to physically draw.
@RickyCruzMedia 6 жыл бұрын
Whoa, never would’ve thought the color-map theory had that kind of a connection to graph theory.
@E_FoxSnowspirit 5 ай бұрын
Woah indeed
@pharynx007 6 жыл бұрын
"it gets squished when you do topology to it."
@sylvieshuu 6 жыл бұрын
R.I.P. bagel, but quite an interesting method to reach the solution
@TheMaskedDonut 6 жыл бұрын
Well, at least she didn't do that to a Montreal bagel. That would be truly tragic
@sylvieshuu 6 жыл бұрын
TheMaskedDonut True
@JaapVersteegh 6 жыл бұрын
Never mind the bagel.... The candy, she spoiled the candy!
@7eartcat248 6 жыл бұрын
Rest in pepperonis
@snowandcats12 6 жыл бұрын
Using bagels and mugs is certainly a creative way to solve it; including internet is also interesting. I love how all your videos holds up suspension of disbelief so well :D
@JoshuaHillerup 6 жыл бұрын
So, does the axioms available to you for doing this math create some sort of topology that could describe the number of valid ways to connect the 7 colour torus problem to the utilities and houses on a mug with internet problem?
@itstrudoe6450 6 жыл бұрын
Joshua Hillerup I have no idea what you just said. I suck at math
@JoshuaHillerup 6 жыл бұрын
Tropical 24 if it helps what I said basically has nothing to do with multiplying numbers and whatnot. Basically math is one big logic puzzle, or rather a collection of logic puzzles, and a given puzzle could be described as having a "shape".
@SakosTechSpot 6 жыл бұрын
Exactly what I was wondering 😶
@jfb- 6 жыл бұрын
I think what you're asking is "how many ways are there to do this?" First you need to define what it means for two ways to be different. The most natural way I think is a homotopy, where two ways are the same if you can continuously deform one into the other (i.e. you can "slide around the lines"). Then, the answer is infinite, because any time you use the handle, you could wrap around it as many times as you like which produce homotopicly inequivalent solutions.
@Blox117 5 жыл бұрын
if theres an internet problem then nothing is possible
@NO-ll4fp 6 жыл бұрын
I like how I heard about every topic in this video, yet you created your own problem, and used these facts to do something I never heard of before.
@FreyaTheGreen 2 жыл бұрын
I have routinely come back to your videos year after year since about 3rd grade, and now I’m in ninth. Every time I learn something new or relearned what I forgot. Thank you for making math so fun and inspiring me to keep trying with it.
@Walthanar 6 жыл бұрын
Words cannot express how much I love you Vi! But numbers can! I love you TAU
@yogee7068 6 жыл бұрын
3 VIDEOS IN A MONTH?! I AM LEGITIMATELY IMPRESSED!
@abigailcharneskie8918 6 жыл бұрын
This is absolutely incredible. I am always in awe of the way your brain works and it’s inspiring.
@AJFes12 6 жыл бұрын
Mic drop Bagel drop Mug drop Sharpie drop
@charliespinoza1966 6 жыл бұрын
AJFes12 +
@rabbitpiet7182 6 жыл бұрын
Bass drop?
@katherineshideouslaughter 6 жыл бұрын
Did you see my bag?
@rosiefay7283 6 жыл бұрын
Sharpie drop OK. Mug drop might be a bit messy though.
@rachelleb8238 6 жыл бұрын
The Masked Performer it's hella trophies and it's hella thick.
@lunaria45 5 жыл бұрын
**eating donut** dude, you’re _eating my mug_
@MonarchMercury 6 жыл бұрын
I don’t understand anything you say and yet I’m still subscribed to this channel.
@deepthisharma4653 5 жыл бұрын
You are my new favourite KZfaqr forever. I cannot stop watching your videos. They are amazing and you are amazing. And thank you
@aud654321 6 жыл бұрын
I've been watching your videos since I started middle school and now I've graduated ♡ you're so brilliant
@lizparkmakeup 6 жыл бұрын
You are so unbelievably brilliant. I haven't taken a math class in years and still enjoy your videos. I was wondering if you'd ever want to make another video about music theory? I have watched your Twelve Tones video countless times since I discovered it my sophomore year of school. It sparked my interest in theory, specifically 20th century theory, which has grown to be a deep love. I'm a music major now and your video really helped me in my theory 4 class. It helped a lot with 12-tone composition and helped give me a strong foundation in these concepts that made the material easier to learn. I'm now a tutor/grader for my theory professor and I think I owe you thanks for that, but mostly for helping me fall in love with theory ❤️
@sadiqmohamed681 6 жыл бұрын
I got lost in the middle, but when you drew the lines on the handle I had alightbulb moment! Thank you Vi for another great video.
@sugarcakezz 5 жыл бұрын
Math isn’t limited to a flat space so I love how you did this
@HollowPiero 6 жыл бұрын
Did she just destroyed everyone in the original video?
@hindigente 6 жыл бұрын
No, they are okay.
@Sci0927 5 жыл бұрын
Felipe Hindi r/whooosh
@grass4655 5 жыл бұрын
Yes
@daydodog 5 жыл бұрын
@@Sci0927 actually you got whooshed
@Olect 6 жыл бұрын
Two of my favorite branches of Maths in a single video! Always fun and informative videos.
@sierra6649 5 жыл бұрын
I looooove this puzzle. The whole world is transformed when you imagine space as a torus. Please, please make more torus puzzles or come visit my school and do some torus puzzles with me!
@dustyfella 6 жыл бұрын
I'm not sure why I subscribed, but I think it's because these videos are oddly mesmerizing.
@thepip3599 5 жыл бұрын
Hey vihart. Here’s something you might find useful. You can wash sharpie off of things using sunblock. Not the spray-on kind, but the stuff that squirts out of a bottle. No idea why. My family discovered this by accident a long time ago. I don’t know if it works with all sunblocks.
@NoahTopper 6 жыл бұрын
It's so sad that you missed that collab. You didn't just solve this puzzle. You obliterated it into tiny little pieces.
@nunooliveira2606 6 жыл бұрын
Genius, interesting solution! Thanks for your videos Vi!
@ninjakille316 6 жыл бұрын
YES! I understood for years that there was a topology solution behind this puzzle but the paper and mug applications always seemed too messy to cooperate with theory. Time to google 4 color map theorem and 7 color map theorem and have this solution cemented in my memory.
@flipppbook 6 жыл бұрын
I'm gonna cry, I love Vihart, I love math, the world is a beautiful place
@icky2748 6 жыл бұрын
Your videos make me feel really smart and yet so confused at the same time.
@phaze2010 6 жыл бұрын
When you spiraled the 3 utilities around the mug handle, my brain broke. Bravo, well done.
@minimooster7258 5 жыл бұрын
So before seeing your solution, I did manage to find one, it required about thirty minutes of scribbling with a marker on a handy not sentimentally important mug, and some stories about rebel houses who don't want their wires to play by your rules, but I found one!
@Bloodmuffin6 6 жыл бұрын
I have an intense physiological hunger response to peach rings, and seeing you put sharpie on them was so painful. But for math!!! And entertainment!!! Great video
@salzshakes4895 6 жыл бұрын
I recently acquired a copy of Robin Wilson's Four Colors Suffice and so I was excited to see the map problem introduced :D
@thesecretthirdthing 6 жыл бұрын
vi you're too smart i gave up understanding halfway and ate a bagel. it was a yummy bagel. thank you vi :)
@NotaWalrus1 6 жыл бұрын
Using the 7 color theorem was absolute genius.
@curry5136 6 жыл бұрын
Nice to hear from you again, Vi 😁
@Prof_Granpuff 6 жыл бұрын
This was so iconic like every single one of your videos is iconic and I love being a mathematician bc we think in this kinda way
@narwhalgirl12 6 жыл бұрын
I’m going into 7th grade so I haven’t really used the information in your videos yet, but they’re just so interesting to me for some reason that I can’t stop watching :)
@uelssom 6 жыл бұрын
Great video! When i saw the original mug video, my solution was to cram as many lines on the handle as possible and ended up with a solution like this!
@RtsShrtFrRtthw 6 жыл бұрын
I trust you know what celtic knots are?
@c1nn4m0r0 4 жыл бұрын
Thank you you helped me get better and start to like math so I went from failing to getting an A and liking math ❤️
@heyitsfen 6 жыл бұрын
Please continue the Doodling In Math Class series! I found the dragon one when I was 7 and was actually interested in the math because of the fun way you portrayed it! I will support you no matter what but I just really miss that series
@ZardoDhieldor 6 жыл бұрын
This is really awesome! I felt smart discovering the solution to the mug puzzle in 3Blue1Brown's video but this one really tops the other one.
@dvstcl0ud 5 жыл бұрын
OMG YOU’RE BACK YESSSSSS!
@MyAce8 6 жыл бұрын
this is the most "hard math" vi hart video imo and I'm really digging it
@elizabethneeley4692 6 жыл бұрын
hey vihart! if you want to keep the mug washable, you can bake on some oil based sharpie ink in the oven.
@livintolearn7053 6 жыл бұрын
I saw this problem on brilliant while working through one of the weekly challenges. It didn't mention the house and the utilities though, it just used vertices. But it's just the same thing.
@raftel3224 6 жыл бұрын
Wow! Your not dead! I love your videos!!!
@MelindaGreen 6 жыл бұрын
Ha! I took a topology course from Gerhard Ringel and at one point I painted a bagel with 7 colors to prove that the graph was embeddable in the torus. I called it "embagelable" which gave him a chuckle. No idea what became of the bagel.
@MaxencePlou 5 жыл бұрын
Man! I just discovered your channel and I am hooked! Your mind is so beautiful and goofy. I think I am getting inappropriately attracted to your brain.
@kappaross6124 6 жыл бұрын
That poor markered bagel.Its valiant sacrifice for Math shall not be forgotten.
@samueltortellini 6 жыл бұрын
Definitely the most aesthetically pleasing solution
@OceanBagel 6 жыл бұрын
I can never avoid commenting on a bagel video. That bagel's not ruined, it's a work of art!
@GarbageGamer74 6 жыл бұрын
Nice challenge at the end, I finally got it. To get the complete planar 7-graph on a torus, start with the complete 6-graph on a torus: / / / - A-B-C- /| /| /|/ -1-2-3- / / / Now take out the three edges in the middle of the box: / / / - A-B-C- /| |/ -1-2-3- / / / Replace them with an X connected to all six: / / / - A-B-C- /|\ | /| | X | / | /|\ |/ -1-2-3- / / / Now we just need to replace those three edges from before. Add 2-B by going straight top to bottom. Then add 2-C adjacent to 1-C, and 1-B also adjacent to 1-C. Looks something like: / |// // - A-B-C- /|\ | /| | X | / |/ | \|/ -1-2-3- // //|/ Translated to a coffee mug, that's six edges sharing the handle!
@davidsarea51 Жыл бұрын
Thank you for this explanation. I know this is a bit late, but I think your diagrams omitted the A-3 edge. Regardless, they were very helpful for visualizing the solution.
@GarbageGamer74 Жыл бұрын
@@davidsarea51 You're right! Fixed. :)
@davidsarea51 Жыл бұрын
Wasn't expecting such a prompt and polite response. Color me impressed. You rock!
@flatthreee 6 жыл бұрын
Ayyy Vi uploaded!
@RedStefan 6 жыл бұрын
Funny though I was subscribed to most of the math channel you mentioned, but only hsve stayed here I guess I like the drawing or your delivery more.
@etepeteseat7424 6 жыл бұрын
Finding out Vihart wasn't able to participate in that 3b1b video is like going to see Chicago in concert and then finding out one of the horns players was sick.
@plaustrarius 6 жыл бұрын
Serious maths here. You never disappoint me.
@dianamocanu574 5 жыл бұрын
This channel just blows my mind...
@nicoleblackwell2845 6 жыл бұрын
This is so cool! youre awesome vihart :’
@22kaybee22 5 жыл бұрын
Wait! I thought this was completely unrelated to me and I was just watching a fun video I didn’t understand but made me feel smart But then I realized this directly correlates to my job and is very familiar! I make 3D models, and 3D models are made up of faces. Kind of like papercraft. However there are rules. Like, usually you can only have triangles OR four sided polygons. And you have to make sure certain edges are clean, forming nice loops all around the object. You don’t want to have the edge of a cube cross through the middle of a flat face So you’re always ending up with these stray edges that don’t go anywhere. All you desperately want is to stick it on a vertices And sometimes the only vertices that makes sense is one you can’t reach, unless you snake all over the object and find it on the other side That part where the three lines are crossing over the handle side-by-side? Even though they started as being far apart? Very familiar!
@BrutusAlbion 6 жыл бұрын
I was simpler with the 3 utilities challenge, I simply bundled the 3 lines next to each other and had them pass through the houses in a straight line and return back to the utilities. Nobody told me that was forbidden. The only rule was that you're not allowed to cross them and bundling them was never forbidden yet people seem to easily forget that. The thing is, people think that the houses are single dots that you can only touch from one side but I treated them like boxes through which lines can be drawn. If you use this method you can solve the entire problem without any hassle at all. You just get a nice 3 color road going from the utlities to the houses and back. The internet would probably be harder BUT I'm sure I'd find a way. Nature always finds a way.
@howarthe1 6 жыл бұрын
(3:00) "In case your into mathematical beadwork..." Hilarious. I love it.
@belladeville2271 4 жыл бұрын
What reason does this have to for some reason become one of my favorite channels
@Daniel-br4nc 6 жыл бұрын
Dang you wasted a perfectly good bagel ;-; I mean, kids in -Africa- *Detroit* could've drank the ink you used on it.
@morgenstern4.669 6 жыл бұрын
2008 called, It wanted it's joke back
@Daniel-br4nc 6 жыл бұрын
Morgenstern k I'll change it up
@7eartcat248 6 жыл бұрын
Oof
@hasiumcreeper5384 6 жыл бұрын
In Detroit, vampires suck the ink out of sharpies and pens, so no one may write any topology.
@Tmastergamer1 6 жыл бұрын
Komaeda would be sad
@toobusytocreateaname 6 жыл бұрын
I love this, not because I love math, but because I love being reminded that problems that seem impossible might not be.
@CalvinsWorldNews 6 жыл бұрын
The mug actually makes it much easier to see how the handle is a bridge over the connections, something that's (obviously) not an option in the 2D variant
@cenedra20 6 жыл бұрын
I don't understand half of what you're saying, but I'm still facinated all the way to the end
@s6th795 6 жыл бұрын
I was the topology person when I first saw 3blue1brown's videos. I immediately thought of the joke and realised there was no reason why Grant would put it on a mug unless I were meant to model it as a torus.
@Strawberry-rz8gp 6 жыл бұрын
I'm 11, yet I still watch your videos.
@the_majesticc_potato_2521 5 жыл бұрын
I feel very stupid when I watch your videos cuz you know a lot more than I do
@adamklam1 6 жыл бұрын
I think Vihart wins this round of educational KZfaqr challenges. excelent video. please come up with a clever challenge/merch concept so i can buy them as mathsy christmas gifts =D especially as gifts to math teachers.
@scoutstrong372 6 жыл бұрын
I'm so confused, but I enjoyed watching the video
@mitzkonic5024 4 жыл бұрын
Who knew math and theorems were so artistic and fun
@pianoguy222 6 жыл бұрын
It's nice to see that my... erm, "half -baked" solution back in elementary school turns out to have some topological validation to it.
@gljames24 6 жыл бұрын
Yay, new Vihart video!
@SergTTL 6 жыл бұрын
You have a very cute mind. In a very good sense. Your videos are such a pleasure (especially the Pi/Tau series) :)
@morgenstern4.669 6 жыл бұрын
Came here for the maths, stayed for the bagel (and the candies of course)
@energiesoler 6 жыл бұрын
this bagel is like "meh, being eaten is so old school, I'm going to have math stuff drawn all over me, sounds cooler !"
@lilettex0421 6 жыл бұрын
The bagel and that candy looked really good 😭
@MCC900 6 жыл бұрын
There are a few companies that print mugs all over including the handle. Somebody should make a blueprint for the two versions, i.e. the houses and services with connecting lines and the rainbow color-split version with all colors touching each other.
@thecl0wnc0rperationo92 5 жыл бұрын
how are you so smart? My brain flew out the window as you explained this
@thepip3599 6 жыл бұрын
I started messing around, and I figured out that both versions of the puzzle are totally possible on a Möbius strip. I even made them. I don't know if that is interesting or not. I think tori and Möbius strips might be kinda the same. Both have a hole and only one side (unless your torus is hollow). I guess Möbius strips are bagels in disguise.
WIRED
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