The Moving Sofa Problem - Numberphile
ฝัง
- เผยแพร่เมื่อ 22 มี.ค. 2017
- Featuring Dan Romik from UC Davis.
More footage from this interview soon on Numberphile2.
Dan's comprehensive page on this topic: bit.ly/movingsofa
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science.
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What do you do when you can't solve a problem? You create a new problem, that you can't solve either. Genius!
That may be true, but whoever's gonna solve this problem in the future will definitely have gained perspective because of this.
Check out Buffon's needle problem. One way to solve that problem is by basically creating a new harder problem similar to the original.
It's one of many examples that show the importance of creativity in mathematics.
+Schizophrenic Enthusiast You mean a *reduction*
Mandalore unneccessary
It IS genius. That's why we do math, because solving problems is fun. The more problems, the more fun. Quite unlike real life, I guess.
Claudio Melendrez in real life, it might not be fun because it might be personal, and to those people that hate to problem solve, don't take it personally. Always solve problems before people have mentioned them. Always seek to find and always innovatw
Solution: An endless beanbag sofa that's flexible like a snake and the exact size of the hallway.
SWIFTY_WINS I feel like this is the chaotic neutral response to the problem.
Lol
well if it's endless it may as well be half the thickness for easier handling, the area is the same
I see we have an engineer
@@LeafLeafy would cou consider the hecksaw as the chaotic evil solution?
I have discovered a truly remarkable optimal shape which this corridor is too tight to contain.
legygax Wait....
Angel Mendez-Rivera what
nice reference
@@joda7697 what
@@trickytreyperfected1482 That is a reference to Fermat's last theorem, where in his notes he wrote:
"I have discovered a truly marvelous proof of this which this margin is too narrow to contain."
where the margin is referring to the space in his notebook.
You just need to yell: PIVOT! PIVOT!
I saw the title of the video. I looked for "pivot" in the comments. I found you within seconds.
shut up! shut up! SHUT UP!
And then, you end up cutting the sofa in half and giving it back to the store asking for a refund … classic situation XD
@@bunbury4620 the same :D
This problem may be extended to n degrees of freedom, which is trivial and has been left as an exercise for the reader
It would be an open ball of radius one for the l1 norm.
one of my favorite comments ever.
Those statements were sometimes true, other times they were infuriating.
Classic Gauss banter
I flunked math.. but somehow I think you're pulling my leg with your comment.
*Solution:* Chairs
*Solution:* PIVOT!
Pivot tables are completely different.
Damn dude, you really excel at this.
PlayTheMind
Better solution: Floor
Word.
Never did I expect to see a Zoolander reference in a Numberphile video. I am not displeased.
Amazing edit
mindstormmaster Is that you Ernesto?
I am happily surprised
*moves sofa into 4th spatial dimension*
outstanding move
They didn't even use the 3rd yet!
*you weren't supposed to do that*
Dirk Gently?
I bet Clifford Stoll has a solution involving Klein bottles.
Brady, you do realise you're going to have to create the official limited edition sofa of Numberphile now, right? 😉
MojoBeast - your Modsauce and Redstone Yeti! +
+
It is clearly a love-seat, just add an S-shaped back to it. I'd go with a different color though.
A use for his brass leather stamp when the shoes are done.
I'm totally getting one if they do.
Now I want a Gerver Sofa. Maybe even a Gerver Desk. Where's the Numberphile store where we can buy these things?
Shad Sterling you will have to make it.
I'm getting into woodworking. Originally I wanted to try my hand at making a Chesterfield, but now I'm thinking a Gerver sofa next to my Morris chair, Chippendale highboy, and Sheraton desk.
Reminder for myself to make this in a few years
@@forloop7713 the time has come.
Contact IKEA! ;)
Professor Romik is one of my favorites! Anyone in math at Davis should definitely take the opportunity to have him!! He gave cute math puzzles to students with the highest score in linear algebra and would bring fruit from the farmer's market to differential equations
I love this channel. I can watch any video and feel like a mathematical prodigy even though I'm just seeing the results of someone else's hard work!
Interesting, but not what I was hoping for. The "real" sofa problem that I've experienced is not getting them through hallways but through doorways. Most of the time that requires standing the sofa on one end and teetering it around the doorjamb.
Exactly! My biggest problem understanding this stuffs point... it feels so theoretical to me, like my teacher trying to tell us in grade 5 about the carpenters 3,4,5 trick for finding square, she did not know how to use it in the real world so she made all these theoretical and un realistic settings, I find it so sad that i only started understanding numbers and amazing uses of math AFTER I quit school. 3 summers with a carpenter in his 60's taught me more math then 11 years of paid schooling.
3, 4, 5 rule is used in real life though. My old boss uses it to double check the flooring frames before pouring in the concrete.
+KyleRanger
The problem you guys are facing is a basic misunderstanding of the academic world. Theoretical research is just that, theoretical. It doesn't have to have any real world applications. If they had applications it'd be called engineering (just kidding)! Practical use comes later after people figure out how to use the research to build something worthwhile.
This problem, it seems to me, benefits from people with engineering mindsets. Engineers come up with real figures (for electrical components, for example) partly based on outside standards. You'd rather not have a 20k resistor when 22k is the normal, off-the-shelf value. You can either tweak the circuit to operate properly with the 22k, or you can increase the parts count (and expense) by using two 10k resistors in series. That's optimizing to a very specific set of conditions rather than the general terms mathematicians generally prefer.
That being the case, it also does not surprise me that the specifications are well-defined, but not very elegant. Git 'er done. Engineering mindset.
A fun little problem to solve. If you have a tall closet that's a rectangle on its side with width l, door with width w and height h. You can prove a theorem saying that in the case of closet height being maximum, all four corners of the closet must touch the sides of the doorway. Maximum closet height is either h itself or vertically √(w^2 + h^2 + l^2 - 2l(l * w^3 + h^2 * √(w^2 + h^2))/(w^2 + h^2)). Please somebody check this, don't know if it's right
I wonder what sort of solutions you get if the turn isn't a 90° angle, but something like 30 degrees
Freefle or one hall wider than the other
Huh. If one hall was wider than the other, does that necessarily mean that the couch takes the shape of a Gerver sofa but only limited to the width of the smaller hallway?
No, but it might mean that it's a Gerver sofa that's been elongated on the front end.
Freefle it would be the same shape because 30 degrees is less than 90 degrees
Ooh, the ratio of the two hallways might be able to apply to the functions of the gerver curves as a weighting factor
This is some groundbreaking stuff in sofa technology. When it first came out, it completely reshaped the sofa-making and sofa-moving industries. And this is what humans can achieve when they really put their mind to it. Quite inspiring.
I'm imagining mathematicians buying sofas with these precise shapes for their houses, and waiting for someone to get the joke.
i want to be that guy for some reason
The problem is that my mom doesn't know where to move it. :/
GOLD 1515
Hits way close to home.
That may be one of those equations that can't be described in closed format!
GOLD 1515
next: the decision conjecture
Pivot!
As some who is taking Linear Algebra, this triggers me.
typical oppressive 3 dimensional behavior
MAMAjAMAj8 i get the joke
Shut up! Shut up!shut up 😁😁
I came down here expecting to see this
Problem proposed by
Augustin-Louis Couchy
wow
Just cut it up, move each piece individually, and use flex tape to put everything together at the end
To show the power of Flex Tape, I moved this sofa through a narrow hallway
@@dboldersgaming7152 and so it fits, i cutted it in half!"
This seems like the kind of question a computer could optimise through evolution very well. Of course it wouldn't give the perfect solution, but if it can be seen to be converging to one of these shape that's very strong support for the conjecture.
I was thinking the same thing, this would work very well with an evolution simulation; the only problem I can think up is programming it to do the right motion around the corner, but I am sure someone more knowledgeable than me could figure it out hehe.
Wiki says: "A computation by Philip Gibbs produced a shape indistinguishable from
that of Gerver's sofa giving a value for the area equal to eight
significant figures.[6] This is evidence that Gerver's sofa is indeed the best possible but it remains unproven."
But where's the fun in that?
I'd say that the actually interesting part is how to solve such a problem rather than the solution itself. Simply because there can be many variations of the problem. Computer solution may confirm that your solution is indeed optimal, but if you want to know why in order to solve the other variants, you are exactly where you were before, holding no tools to approach the problem in general.
What about an evolutionary algorithm that is designed in a way that it wouldn't always return the same shape as being optimal?
I don't have any idea how it could be programmed, but suppose it'd work: either all optimal sofas eventually become Gerver's, OR it returns a variety of optimal sofas, all with the same optimal surface area that is greater than or equal to Gerver's, through which we can derive the criteria for an optimal sofa.
Dirk Gently wants to know if you can help him move his sofa.
I was thinking the same thing 😂😂
“More footage from this interview soon on Numberphile2“
No link? After more than two years?
This guy has a fascinating voice. No matter how loud I turned up my speakers, he still sounded quiet.
I remember looking through Wikipedia a while ago about unsolved mathematical problems and I saw this one.
Any fans of Douglas Adams in the house? Reminds me of Dirk Gently's Holistic Detective Agency
You just need a time machine for the door to appear. Or you need to pivot.
i LOVE hitchhikers guide to the galaxy I want them to review it on their podcast!
Also I feel the sofas in the swamp need to know about this
That was the reason why i clicked on this video
Andrew Knorpp I thought those were mattresses
Romik was really fun to listen to! I'd like to see more videos with him in them.
Joke reference to Zoolander but not the more pertinent Dirk Gently's Holistic Detective Agency "Sofa stuck on a stair" by Douglas Adams
Yes this is the main reason I watched this!
To me!
To you!
To me!
To you!
HitchhikersPie Oh dear oh dear oh dear.
I 'chuckled' at this one
What??
+HitchhikersPie I didn't get it :/
Chucklevision chuckle-chucklevision
Nah I just twist it every single angle and manage somehow
Sometimes, that's mathematically impossible
Just like trying to move a 2d sofa down a 2d hallway in a 3d reality?
you just didn't twist it enough
I recently moved a bed mathematically impossible to fit through the corridor. The mattress took some pushing. The bed took a window...
Wanting to shout "PIVOT" every time they start to rotate it around the corner.
See, this is why engineers invented modular sofa suites ;)
No matter the topic, these videos are always fantastic! Thank you for doing what you do Brady!
so nature designed ANT like this
This is a very inspirational video, i loved watching it! one of my favorite videos now!
Ooh this is really cool. I love the unsolved/unproven ones
Congratulations on 2 million subscribers!
thanks
International giveaway of 2 million pies?
636619.772368 Pi to make 2 million
i did 2000000/pi
Solution: you park your time machine in the stairwell, open the door, and then use the extra space you created to manoeuvre the sofa into place.
Woe woe woe, easy chair buddy.
robberynl Fantastic.
I shot a blue portal in the floor under one end of the couch. Then I shot an orange portal on the wall facing down the hallway in the direction I wanted to move it. Then I went back and pushed the couch into the blue portal and gravity did the rest.
John Reynolds 5schtzgjg zsfgb0ougmm kg xzzzvnm0yyb u
Don't forget your abacus!
This was one of the most interesting videos I've ever seen! The premise was entertaining, but in itself it was very exciting to listen how the problem was approached :D
Hi I’m a 76 year granny who can’t do algebra but find your explanations interesting , I am in awe of your mind ,
This is math solving real world problems.
That's not what math is about ;)
PlopKonijn But it should be. If math isn't applicable in the real world at all, then it is as useful (and at the same level) as the sci-fi writer dreaming up an ultraproto-gadget that launches laser guided shark-bots. Fun to think about, but ultimately useless.
Clearly you know nothing, Mark. Surprisingly often physicists and engineers come across obstacles in practice that were solved mathematically centuries prior. Take for example George Boole, in 1847 he invented what would later be called Boolean agebra. At the time it was perhaps a "useless" theorietical exercise, but about a century later it proved instrumental in the development of computers, and arguably kickstarted a whole new field of math that would become known as computer science, way before computers were even imagineable. Or maybe you just think computers are "ultimately useless." This whole idea that everything has to be applicable is just ridiculous.
Nope. In the real world you make your sofa in three parts you can take apart, not in an ugly shape.
Ze Rubenator In the end what he came up with was applicable though... He never said it had to be useful immediately.
2:34 one of the most satisfying animations I've ever seen
I love this channel so much!! Every video is excellent!
Very moving video. Thanks!
This reminds me a lot of Dirk Gently's Holistic Detective Agency. Love it! :)
It bothered me that they didn't mention it, nor Douglas Adams. (Especially when they've mentioned 42 in the past) Of course, Dirk's computer was working in 3 dimensions which they also didn't mention. So... maybe they're just not up for it yet. 😜
The reason I clicked into this was because of Douglas Adams! They have done a video on Douglas Adams's answer to Life, Universe and Everything (42)!
Yes! Forget Friends, I'm here for the endlessly spinning couch simulation
So how many people have these shapes of sofas? Mine is rectangle.
I've actually seen curved ones (stretched hemicircles) as well as ones where the inner angle of an 'L' shape is rounded off
Entropy Zero there is no entropy zero
Mine is
[_][_][_]
[_]
That's because you're a square.
Do you even know what entropy is? The entropy is defined as
S=- sum_i p_i log p_i , with p_i a probability mass function. If you now consider a (fair) coin flip the probability for each side would be 1/2 and therefore S=1, taking the logarithm with base 2 (measuring information in bits) and using some logarithm rules or a calculator if you aren't familiar with logarithms.
But now consider the completely boring case, that the coin is twoheaded, so the probability for head would be 1. Plugging this into the entropy formula S = - ln_2(1)*1 = 0 - zero entropy, meaning that we have complete information of the system.
To display this level of expertise, and still not be tempted to claim certainty without sufficient evidence. This type of math seems trivial to a lot of people, but the world could learn a thing or two from this humble mindset.
I watch numberphile videos. But it was a pleasant surprise to watch Prof. Romik here. It was a pleasure to have him as a professor in my grad school :)
Very well done. That said, everyone knows the largest surface area sofa which can go around corners is the liquid metal sofa developed by skynet.
Your channel got me through year 9, thank you.
This was amazing. Thanks for making this.
Ok, I was wondering which sofa should i choose. Now I'm gonna go to the shop and buy this sofa. Thank you for advice!
I thought this vid is about the problem with the sofa which is irreversibly stuck on the staircase to Richard MacDuff's apartment, in the book Dirk Gently's Holistic Detective Agency.
+
12:50 is that THE Tony Fadell scrolling by?
Who knew the roaster of CrossFit himself would end up on Numberphile.
ZERO
If I ever need to move a sofa, I'd call upon the helpful team at the Sofa shop.
this is really really interesting. thank you for posting!
congrats for 2 million subscribers
The smartest way to move a sofa is a Philips screwdriver
Go away you filthy engineer
Ugh, who uses Phillips srews anymore? Pozidriv is the way to go!
(Plus, to deconstruct most sofas I've ever seen you'd need a saw, not a screwdriver… Hey, maybe you could optimize the number of cuts you'd have to make to get an arbitrary shape through a corridor and around a corner? xD)
@@entropyzero5588 Pozidriv?
Entropy Zero
Nah the minus head screws
@@entropyzero5588 If you were Canadian, I'd forgive you for saying Robertson. But Pozidrive is right out. Torx or go home.
This guy looks like he's desperately holding back the giggles the entire time he talks.
This is very interesting! A math problem that fits a practical real life situation, I like it!
What I would like to know is this: if the path would have to be a unique path (from 'start' to 'finish'), would the Hammersley Sofa be the optimal shape?
I wonder if Douglas Adams wrote Dirk Gently with this problem in mind, since there is a sofa that got into a position from which couldn´t be removed or something like that
“Odd,” agreed Reg. “I’ve certainly never come across any irreversible mathematics involving sofas. Could be a new field. Have you spoken to any spatial geometricians?”
-Douglas Adams, “Dirk Gently's
Holistic Detective Agency”
This is from the introduction of Paper that present this solution of Ambidextrous Sofa by Dan Romik
DOI: 10.1080/10586458.2016.1270858
hehe I usually dont go through the description. Thats actually pretty cool, thanks for showing it to me! I didnt remember the exact text where the sofa problem is described, but seeing as how it is written is looks obvious that adams knew about this problem
I had Prof. Romik for Abstract Linear Algebra. One of the best teachers.
Mathematician is one awesome job, u can literally have problem with anything and the problem will confuse not only other mathematicians but also everybody in the world
where's the extra footage on numberphile2? I can't seem to find it, nor it is there in the video as a clickable link by the end. Please put in here as a reply to the comment, thank you! 🙏🙂
1:54 OMG THAT'S TOM HANKS
oops, my bad! I thought it was Dave Gahan from Depeche Mode 😂💀
Congrats on 2 million! Objectivity deserves the same love
If we want to upgrade the problem we could add third dimension for even more chaos.
Now to pose an even bigger problem. What if the length, width and *height* is also 1m now what is the largest object (doesn't have to be a sofa) in cubic metres that can go through the gap. Rotating, flipping etc is allowed. The object can be of any shape and have any length, width and height.
So you guys can thank me for adding yet another problem to the list of unsolved math problems.
A: 1 meter tall Gerver sofa.
B: the tardis
A 1m cube. You don't rotate it at all.
Surely you can do better with a semicircle with 1m height, but surely the semicircle can be beaten again...
Has someone tried this with 3 dimensions? Having a square tube with an X, Y, and Z turn. Im curious if it would just be his shape rotated along the longer axis 360 degrees.
Maybe this will be the next unsolved problem in mathematics.
When I was in college, our professor posed the question of what's the largest 3D prism you can you turn in a hallway that meets another hallway of different width but with the same height. Solution left as an exercise to the reader.
Thank you so much for saying "3D-printing" in the end! I was wondering the whole time whether those shapes are curved out of pineapple and mango or not. Now I can be at peace with that question.
Zoolander in a Numberphile video, i would never have imagine that!
What's the largest volume sofa if allowed to rotate in 3d. Assume 8' high ceilings 😉
James Cotter probably the exact same thing, but with a depth of 8’? Dunno, just my guess
Have a Question about measurements .. and will It Fit before I buy ..
Have 31 maybe 31-1/2 front foor
Looking at A big couch says 34" door opening 98ninch long
is that measure to just slide in ...
Is it possible to angle fit the arm first 45*it ... then ""roll """ it in
this is a very satisfying to watch, like an interrupted gear system.
I had Dr. Gerver as a professor twice at Rutgers University. Very interesting professor. He retired recently in December 2017.
What happened to the extra footage being added to Numberphile2?
This is what I want to know.
It can turn left! Lol
I really like the visual problems in the last two videos.
This is sofa is ALSO a solution to the famous problem, "What are the MINIMUM dimensions for a sofa-bed adequate for Siamese twins who are 1 unit tall?"
Did you guys talk to Dirk Gently about that problem yet? He might have some interesting insights.
Excellent video
Having tried in my youth to move three folding bed sofas down a circular staircase from the third floor of a Montreal walkup this hits home with me.
Was this inspired by Dirk Gently's Holistic Detective Agency?
Just need to have the sofa model in 3D and let the computer run simulations.
Other way 'round, I'm sure.
Although ...
there was a time machine involved, so...
I thought this was gonna be about dirk gentlys holistic detective agency
Excellent video. The exact amount of empiric content
Its cool that his ambidextrous sofa is still a little better than a semi-circle which can only turn one way.
It would make sense for an ambidextrous one to be way smaller but the fact that its actually bigger is cool.
Douglas Adams, Dirk Gently's Holistic Detective Agency, his solution was better!
Lawrence Cuthbert I've always wondered if Douglas Adams was thinking about this problem when writing Dirk Gently
numberphile=π
numberphile2= 2π
thanks for listening!
Hora da Álgebra you're welcome!
numberphile2=τ if I may...
2π or not 2π? That is the question.
I really like how that special any-corner sofa looks, but really at that point it's just two chairs with a frustrating interlinking bit.
Dear Numberphile, it is 1:52 AM for me, and i'm watching strange sofas... i will watch 'Incredible Formula - Numberphile'. i love your channel
How can you NOT mention Douglas Adam's Dirk Gently?
A number of elements in the novel were inspired by Adams' time at university. For example, one plot thread involves moving a sofa which is irreversibly stuck on the staircase to Richard MacDuff's apartment; according to his simulations, not only is it impossible to remove it, but there is no way for it to have got into that position in the first place. In a similar incident that occurred while Douglas Adams attended St John's College, Cambridge, furniture was placed in the rooms overlooking the river in Third Court while the staircases were being refurbished. When the staircases were completed, it was discovered that the sofas could no longer be removed from the rooms, and the sofas remained in those rooms for several decades.
Now I can have the perfect sofa!!!!
2 MILLION CONGRATS
This is why I love maths, it is the rules underlying whatever situation you can imagine.
As discussed in the wonderful Dirk Gently.
What is this? A sofa for ants!?
Adil Rahim the sofa must be......THREE times bigger!
I was going to comment that the ambiturner sofa is in the shape of an ant and wondered if ants were shaped that way for evolutionary reasons related to the sofa problem (navigating small passageways). Then the damn Zoolander quote about ants got stuck in my head.
Now for the real challenge: a sofa that also fits through the 3-D space that is the stairwell leading up to my apartment.
Fascinating.
can this work with people?
and hello internet