What's The Largest Sofa That Can Fit Around a Corner?

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Chapters
0:00 The Moving Sofa Problem
2:06 Hammersley's sofa
3:15 Gerver's Sofa
3:55 Why is it so hard?
5:34 How Gerver came up with his sofa
9:50 Thank you Brilliant!
11:23 Will you find a bigger sofa?
Creator - Jade Tan-Holmes
Script - Alexander Berkes
Animations - Daniel Kouts and Simon Mackenzie
Music - epidemicsound.com

[@upandatom]

Sign up to Brilliant to receive a 20% discount with this link! brilliant.org/upandatom/ Recommended course: Advanced Geometry Puzzles brilliant.org/courses/advanced-geometry-puzzles/

[@joelklein3501]

Is his solution related to the stationary action principle from analytical mechanics?

[@axiomfiremind8431]

But your hallway has no height. All you have solved is the limit of the base that will navigate the base of the hallway. A bigger sofa can simply be stood on its end. And because your hall has no celling then we know the largest sofa that can fit around the hall is one with infinite height. Like wise why limit yourself to 3 dimensions? An nth dimensional Sofa will peek at around 5 dimensions before becoming smaller again. You know because you live in a mathematical world with mathematical objects and not the the real world with real objects.

[@lyrimetacurl0]

Could you have a slight extra spike on the corner of the phone shape? (along the length of it)? Since the bottom edge is slightly less length than the full width?

[@olli3686]

12:14 & 12:21 these are just two 90 degree angles, so we can use shape, but we rotate it about it’s longest axis in the hall way between corners! As for the 45 degree, we could use the same shape, just cut the couch in half and move it in two pieces 😂

[@fredashay]

I bet'cha that Presh Talwalker from _Mind Your Decisions_ can solve it!

[@guidoferri8683]

I tried to explain this problem to my friend, but he continued to scream things like "I'm not interested!", "I don't care about math!" or "Nobody asked you to cut off the edges of my sofa!"

[@SaHaRaSquad]

When you just want to move sofas around corners to further humanity's understanding and then people ask "who are you" or "how did you get in here, I'm calling the police"

[@beyondobscure]

@@SaHaRaSquad hate when that happens

[@jackhandma1011]

Most people don't care about math. We just have to deal with it.

[@kidredglow2060]

Ur friend is mean

[@MandleRoss]

HAHAHAHA!

[@blodpudding]

I'm Swedish, so my solution is of course to disassemble the sofa and bring it through in pieces. IKEA beats math every time.

[@R24_--___---]

Right😂

[@stevenarvizu3602]

I know it’s a hypothetical but the whole time I was thinking just disassemble the sofa lol

[@samueldeandrade8535]

By writing "IKEA beats math every time" you probably lost a thousand likes.

[@noxlusus]

IKEA 13B yearly income, Google 280B, math makes the difference 267B (subtraction) Sorry I was using math to explain that.

[@michaelmichalski4588]

​@@samueldeandrade8535I can just imagine a serious of solutions. The simple cube. The half circle. Gerbers etc. and the swedish solution where the whole sofa is broken down into a flat box.

[@b0nes95]

4:02 sofa we've just been guessing at shapes

[@remi1771]

Yes

[@duvasrealm]

On a fun note: Yes if you give the property of disassembly, there do exist a plethora of larger objects/sofa that can move over the hallway. Ikea still lurks around because of this.

[@DevtheViolinist]

The size possible is only limited by the parts allowed. IKEA teaches this the hard way.

[@SgtSupaman]

Seems like the problem now is less "can you find a bigger sofa?" and more "can you finalize the proof for Gerver's sofa?"

[@amegatron07]

Not necessarily. The fact that it hasn't been proved yet can mean two things: either it's just hard to prove, or it can't be proved, if it's not the biggest possible sofa.

[@GIRGHGH]

@@amegatron07 That's the same as finding the null proof. I feel it'd be more productive to try and prove or disprove it as a launching off point as opposed to starting from scratch.

[@Christian-mf4jt]

It seems more likely that Gerver's sofa is really the biggest possible one, because it is locally optimal. Any better solution would need to have a significantly different shape, and that probably would have been found already.

[@error.418]

@@GIRGHGH Usually you'd start with trying ti disprove.

[@error.418]

@@Christian-mf4jt "probably would have been found already" is the same trap other problems have fallen into. this happens in a lot of fields, and a fun one is speedrunning where "this run is optimal, no run will be faster" and then it's trounced by a logical leap a decade later. it's just not a good way of thinking about a problem.

[@YoungGandalf2325]

I didn't think a degree in mathematics was needed to to become a furniture mover. This is why sectional sofas exist.

[@sicfxmusic]

Exactly, I'm not good at math so guess that's the easy way out! 😂😂

[@wearwolf2500]

I bought a large chair and was worried about then carrying it downstairs. It came in 5 parts...

[@axiomfiremind8431]

What is the largest a section of a sectional sofa can be before it gets stuck?

[@AdelaeR]

@@axiomfiremind8431 The answer is: 2.

[@BobBeatski71]

I'm not a furniture mover, I'm a corridor arranger.

[@huzzzzzzahh]

This is such an incredibly elegant example of how math is actually done in real life. I wish all the students who “hate math” could see and really internalize this. Math isn’t about solving equations (although you gotta get your hands dirty sometimes) it’s about finding new perspectives and massaging hard problems into successively more tractable ones

[@chekhov-and-his-gun]

No school will tell you that though. That's why people tend to dislike math because it's shown as only equations.

[@emissarygw2264]

My teachers did try to frame math concepts in terms of applications. I didn't give a shit at the time though, because none of those applications mattered to me. Luckily I paid enough attention anyway, but honestly the biggest motivator to learn anything is when you need it to solve an actual problem. Unfortunately in today's environment, often (but jot always!) that's too late to start learning something.

[@GamezGuru1]

except the reason this problem is considered 'unsolved', is because it lacks a proof by the very sorts of equations you disdain... proving - in fact - that maths is about equations...

[@emissarygw2264]

​@@GamezGuru1 Of course equations are important. They're how you calculate numbers like the actual size of the sofa. Being able to calculate and/or prove things exactly is really important and useful. But there's a big component of just problem-solving and coming up with ideas.

[@Kaanin]

I thoroughly enjoyed the Numberphile video on this problem. When I saw you had posted one on the same topic I was skeptical you could add anything worthwhile to the discussion. I was wrong to doubt you! Your explanation of how a balanced shape has no space to gain through small movements was really intuitive! Your ability to turn a complex and difficult to explain concept into something easy is on another level. You are a terrific science educator!

[@Bunny1sAw3somesauce]

I gotta say as someone who worked for a moving company as a grunt for years, I find this fascinating. But it's not really the problem in the real world as we have 3 dimensions and most places have upwards of 12 foot ceilings. So you flip whatever your moving up on a side and then rotate it around, of course this breaks down and becomes complicated when the 90° turn is in the middle of a stairwell but there's ways to work it out. Even without touching the walls ( we put that to the test moving one family out and another into a place that was freshly painted so we couldn't touch a single wall) it's difficult but do able. The real problem is doorways.. like moving a L shaped couch that isn't sectional through a doorway into a hallway. Fun stuff Lol

[@mrdraw2087]

True, this problem only considers 2 dimensions. It would be interesting to see what happens once you consider a third dimension, although that will be even harder to solve.

[@benjaminpedersen9548]

@@mrdraw2087 What is required to be a sofa? The largest volume may simply be the largest in 2d filling the entire height.

[@jamielondon6436]

Meh, all you have to do is pivot. Piiivoooot! ;-)

[@doncarlodivargas5497]

Mathematicians: struggle moving a sofa for decades A random moving grunt:instantly move the sofa on the side

[@Malekariel]

Mover bro is right. Stand er' up take the legs off and spin the much easier to manuever L shape around the corner. Sofa's as big your ceiling minus like 5 inches or so depending on the shape. Don't scrape the fabric on the ceiling or you own it. Jordans Furniture customer service will replace anything damaged during delivery at the mover's expense so you learn to not touch the sides faster than a game of operation.

[@MedlifeCrisis]

Shout out to the Douglas Adams fans who remember Richard Macduff’s staircase sofa. Stuck ever since delivery men couldn’t get it round a corner but then couldn’t get it back out the way it came either

[@RussellChapman99]

Dirk Gently detective. Read it years ago and this video immediately reminded me of the story.

[@MonkeyJedi99]

Yep, and it was solved in another book when a character opened a door that was accidentally on the landing of the stair, and let the movers turn the couch around before closing the door (making it disappear forever).

[@bpj1805]

But can you get a swallowed toy sofa around the first corner of a person's small intestine?

[@quantisedspace7047]

@@MonkeyJedi99 Was the resolution of the sofa thing done across two books ?

[@MonkeyJedi99]

@@quantisedspace7047 IIRC, it was across two different book series.

[@kbrown4ou]

We moved a large oak desk (with no real effort) into our home office. When it came time to move the desk out we found it impossible to get back out as the doorway was narrow and led to a 90 degree hallway. Ended up sawing the legs off the desk to get it out. Still confounds me how we managed to get it in with no real problem.

[@MateusSFigueiredo]

This is some Douglas Adams plot

[@MohsinExperiments]

You should rotate its legs towards the inner corner. That's how you can easily get it out from your home.

[@julien1426]

6:24 "sofa we've been..."

[@JosephBlanch]

What makes the sofa problem even more complex is that you can rotate the sofa in a 3rd dimension (pitch, yaw, and roll as they name them in aviation). Additionally, real life sofas can also squish at the edges and corners. Sofas are so complicated 😂

[@testales]

I've a better idea: Move it in the 4th dimension! Go back in time to it's disassembled state, move it to the target destination and then bring it back to the assembled state. ;-)

[@curtisowens750]

They never mentioned that the sofa could be stood on it's end .... I've had to do that before

[@-YELDAH]

​@@testales as the proposed problem is in 2d, extending it to solve in 3d makes as much sense as 4d so you're not wrong lol

[@error.418]

The original sofa problem is strictly a 2-dimensional question. Extending to the 3rd-dimension is an extension of the problem, but not the original.

[@testales]

@@error.418 Chopping out a part of the sofa or turning it into a weird object that you can't buy anyway and many people won't even identify as a sofa, is more of a cheat then just using the 3rd dimension with a sofa you can actually buy. I guess the problem is solved for regular rectangular sofas, so in fact having some online tool that calculates this for 2D or even 3D would probably be actually useful.

[@DeclanMBrennan]

That was very enjoyable with great graphics. A stuck sofa also plays an important part in Douglas Adams "Holistic Detection agency". There, a computer simulation proved it could not be freed by going backwards or forwards. For the resolution of this paradox, read the novel. 🙂

[@bjornmu]

-""Eddies in the space-time continuum!" -"And this is his sofa, is it?" 😄

[@iriswaters]

This was the FIRST thought I had when I saw the episode.

[@renedekker9806]

Yeah, I immediately thought of "Dirk Gently's Holistic Detective Agency" as well. Excellent novel!

[@vylbird8014]

The specific situation is this: During the attempt to move the sofa around a corner, it got stuck. It could not be advanced, or withdrawn. This eventually lead to such frustration that one character turned to computer modeling to try to calculate the solution - only for the program to calculate that not only is it geometrically impossible to get the sofa past the bend, but equally impossible for it ever to have been placed in the current position. The answer to this problem is part of the resolution of the novel's several interconnected mysteries.

[@I.____.....__...__]

@@vylbird8014 Interestingly enough, this plays out in the real world with things other than sofas like kids sticking one's head between the balustrades of a staircase or sticking one's finger in a hole (I mean like rings and stuff that are just barely big enough).

[@saiganeshmanda4904]

Always a pleasure to be an audience in your family, Jade! I have been studying optimization in CS for my degree for 3 years now (about to graduate in the next summer!), and your content like this manages to tickle that little tone of fancy in my heart for math and provokes me to admire the hidden beauty that most often goes rather disappointingly unnoticed. But, I believe content creators like you, Derek, Diana, Henry, Brady, Grant (just to name a few) are always there to keep igniting those burning little sparks of curiosity ... Keep going and never forget that enthusiasts like me are always watching (in awe) the essense and value you bring upon in our community. Cheers :)

[@stacksmasherninja7266]

This is the best mathematical puzzle I've ever seen ! Such a simple puzzle yet difficult and quite an elegant solution !

[@randysmith9715]

At a music store I worked at we had pieces of plywood that had the same dimensions as our most common pianos and organs. We carried the plywood into the delivery location. Assuming plywood fit around corners, up stairs and through landings. Then the instrument would be delivered. Remember a baby grand that needed to be swung onto a 3rd floor balcony and rolled into the apartment.

[@leftmono1016]

Great comment. There’s a Laurel and Hardy scene similar to this. Hilarious 😂

[@Shamazya]

Fun topic! I've seen Numberphile tackle this topic but the part where you showed how lateral movement of the hall allows for an increase in area was really cool!

[@davidshelton1898]

This problem also looks like minkowski sums and differences! Very useful for checking for intersections and getting resulting shapes efficiently. And the intuition of "draw a vector along another vectors path." Looks similar if not the same to drawing the shape of no collisions when drawn along a path! 😀

[@SirRandallDoesStuff]

I love your channel. I have degree is Astrophysics and I love to watch others and how they teach and explain topics. You are a great teacher and one of the few channels that does it right. Keep up the great work.

[@quantisedspace7047]

Degree in Math and another in Astrophysics. Must be back to the dole queue on Monday.

[@alanhilder1883]

Douglas Adams brought up a similar problem in his story " Dirk Gently's Holistic Detective Agency ". In this case it was a corner half way up some stairs, the movers got it to the corner, rotated it all over and then couldn't get it out in any direction. It was stuck. ( The use of a time and space machine finally solved it but that part is different to this puzzle...

[@bjorntantau194]

Yeah, I thought that was going to be the reason why this is unsolved. But it also explains why Gently had to let his computer run so long to calculate every possibility.

[@alanhilder1883]

@@bjorntantau194 It wasn't Gently's computer, it was " The Client " and old school friend whose name eludes me at the moment.

[@bjorntantau194]

@@alanhilder1883 Ah, been too long since I've last read the books.

[@alanhilder1883]

@@bjorntantau194 I will have to re find them, it was last century that I got to read them...

[@babelchips]

Dirk Gently’s “interconnectedness of all things” applies here! This video came out on Towel Day 😊

[@nightpups5835]

I do love how the couch created so far is actually a reasonable enough shape. giving this a mathematicians favorite accomplishment, practical application.

[@jamielondon6436]

You're thinking of engineers. Mathematicians scoff at applicability. ;-)

[@nightpups5835]

@@jamielondon6436 Indeed

[@daveslamjam]

i feel really proud that i had the idea of moving the hallway to make a shape before you mentioned how gerver approached the problem 😊

[@alextaunton3099]

i think I realized one reason why I like your videos. You are very expressive and you naturally emote well, which makes your videos more engaging and fun to watch. Love your content!

[@francois__]

I can already hear Ross yelling "Pivot! Pivot!"

[@JMannus65]

My first thought

[@gladitsnotme]

It's in the first 30 seconds of the video, so yeah, I'd hope you could hear it lol

[@francois__]

@@gladitsnotme I commented a minute after the vid was uploaded, hadn't seen it yet 🙂

[@johnchessant3012]

Modern math has become so specialized, so it's always neat when something discovered 'recently' (the Gerver sofa was found in 1992) is not just vaguely accessible, its main insight can be understood clearly

[@uselessgamedev]

Very nice! I like the visualizations of the paths of the hallway

[@BrainWeevil]

Thank you for this interesting insight, and for the history! I immediately thought of Douglas Adams' book _Dirk Gently's Holistic Detective Agency_ when I saw the title. I'd had no idea this was a problem entertained by real mathematicians before Adams (Requiescat in pace) used it. I know his sofa was on a staircase, adding the z axis, but the principle clearly relates to your problem here.

[@wetterschneider]

OMG. When I was introduced to the puzzle, that's the first thing I did (similar to Gerver's model) - I built a 3d version of the hall in software, duplicated it and rotated it, using the stacked series of clones to Boolean carve a chunk from a much bigger solid. I failed, it didn't work, but animating the hall around a static chunk to carve the chunk was my first idea. Again, it failed. But thank you for elaborating on the history.

[@matthewthompson6455]

I’m glad it failed

[@buzbuz33-99]

When confronted with a narrow hallway, we generally solved this problem by tipping the sofa on it's side.

[@boggers]

Yeah, I think this problem would be a lot more interesting in 3D with design constraints so that it remains a functional sofa (eg. max heights for seat, arms and back) while using things like standard roof height and door frame sizes etc.

[@secondarycontainment4727]

Wrong approach. Just cut the sofa to fit around the corner.

[@LineOfThy]

Issue is you can’t do that in a 2D world

[@LineOfThy]

@@boggersbeauty of most math problems lie in their simplicity. Being more complicated does not automatically make it a better problem

[@boggers]

@@LineOfThy Even so, it would be more interesting to me. I used to move furniture for a living, there is an art to getting long sofas up winding stairwells, around corners, and through doorways, to me the 2D math problem is an over simplification of a real world problem. :)

[@Sef_Era]

I feel the need to mention (even though this is more of a hypothetical), that in practice you’d just take your sectional sofa apart, and stand the pieces on one end to fit them through the corner space. That’s why they make sectionals to begin with, as well as the weight to volume restriction of two people being able to reasonably carry each piece. If it’s not a sectional, it’s probably shorter end-to-end than the standard hight from floor-to-ceiling in your country. So when standing it on one end, you just have to fit its hight and depth through the corner; and it’s a sofa. You need to be able to sit on it once you put it back down, so the hight can’t be *that* extreme.

[@esbenthomas9430]

I spotted the engineer.

[@WaffleRune]

I think the beauty of mathematics advancing through time was that I too, thought of a hallway moving around a sofa when I heard about this. It's intuitive to think this way because that's similar to the passage of a fluid around a given solid, and we already know that, we used it to make plane wings.

[@devluz]

This was a great explanation and the animation show the problem in a really intuitive way. Amazing that there isn't a proof for this yet!

[@saschaschneider9157]

Am I the only one who was immediately reminded to Dirk Gently's Holistic Detective Agency by Douglas Adams. Where in the staircase MacDuff's sofa is stuck since about 3 week. And when he wrote a program to get the answer it can get out, the result is that the sofa never could have get there in the first place. Definitely one of my favorite books.

[@grandetaco4416]

she had me at sofa and small hallway.

[@DeclanMBrennan]

@@grandetaco4416 Not entirely. There appears to be a countably infinite number of comments on "Dirk Gently", including my own. 🙂It's nice to know that Douglas Adams is still so fondly remembered.

[@williamtell1477]

Just found your channel, really like your style, subscribed!

[@anthonynorman7545]

Your prop making and work made this so much more digestible!

[@gamechep]

"Some serious math lover has probably already made/ has had this Sofa made", is what I was thinking until that furniture thing popped up 😂

[@danielyuan9862]

The problem is more applicable than other math problems I see. I would be surprised if absolutely no one had this sofa made.

[@keithmichael112]

In my experience it can be much larger than the corner, as long as you're willing to break it

[@AdelaeR]

This is math, though, not engineering.

[@vigilantcosmicpenguin8721]

Ross, is that you?

[@keithmichael112]

@@vigilantcosmicpenguin8721 pivot!

[@atomicdmt8763]

LOVE these videos~ and your sweet energy!

[@mathewmunro3770]

I remember trying to solve this problem in year 10 (final year of high-school), some 30-years ago. I thought the optimal solution was a little more like the bottom one at 4:24, but symmetrical and more optimised, although with two curves that meet at 90-degrees in the middle on the edge of the couch that pivots around the hallway corner. My highschool maths teacher thought it looked a bit like a butt crack, and joked 'ah, so that's how you get around a corner' and motioned like he was moving his butt around the corner of a desk LOL. The cutting bits off & adding bits on wasn't as genius as cutting out a semi-circle and elongating the sofa while simultaneously using rotational and translational motion in my opionion. I too was able to increase the size by cutting bits off & adding more in other areas. I started to try to define the shape algebraically, but the math became horrendus.

[@MathOrient]

Impressive! This mathematical problem is truly captivating. While it may appear simple at first glance, the solutions prove to be more challenging than anticipated. The explanation and visualizations provided are truly exceptional, making the learning experience even more enjoyable. Mathematics never ceases to amaze, showcasing its inherent beauty and complexity.

[@kanjakan]

That's an insanely creative solution. Never would have thought to rotate the hallway, but it makes sense since, if you think about it, the sofa's shape is naturally dictated by the hallway's shape.

[@techman2553]

Moving the hallway was the first thing that I thought of when she started describing the problem, because I suck at math and if I had to brute force a solution, I would do it graphically. Just create a big blob much larger than the hallway, then use the hallway as the clipping boundary and let it subtract from the blob as the hallway moves and rotates. Whatever doesn't get clipped away is the shape of the sofa. The hard part is figuring out the optimal combination of translation and rotation. When all you learned in math class was geometry and trig, then everything becomes a polygon nail.

[@Mueller3D]

The solution wasn't so much about the frame of reference as it was about considering the interaction between the sofa and the hallway. Another way to look at it is to consider at any point what is stopping a large shape from moving further. In order to allow the shape the move further, you must trim a bit off. Do you trim away where the hallway corner is bumping into the sofa, or do you trim away from where the outer walls are bumping into the sofa? It just so happens that these interactions are easier to consider if the sofa is stationary and the hallway isn't.

[@milesgould8288]

​@@techman2553based on that comment, you do not suck at math!

[@saadrashid8825]

Thank you very much for new perspectives

[@gustavoforestobritodealmei888]

Wow... Such a great content! Brilliant video! Thanks... 🙏🏾

[@me0101001000]

In ChemE, we learned two ways to think of movement. Eulerian, which has particles move through a slice of space, and Lagrangian, which models an object moving throughout space. I feel like a part of the problem would be trying to determine which process is less computationally taxing.

[@Szymmon614]

Love the altitude of Michał Batsch at the end, like everyone tried to make ideal shape, but he just made an actual sofa. And in looks really nice, especially wit that coffee table.

[@mrpennywize]

This. What a video. I love it. Thank you.

[@clockworkkirlia7475]

I'd call this a bizarrely fascinating video, but honestly it's not that bizarre for me to be fascinated by something like this. Thank you for the really interesting watch!

[@BrickTsar]

We put a sofa upstairs and it had to include 3 dimensions - hitting the ceiling happened.

[@Tiqerboy]

Really enjoyed this one! You should make a follow up video, solving the problem with doorways at either end which are NOT as wide as the hallway because that is a more realistic problem to solve in the real world. At least the front entrance should be like that.

[@bunchofenergy4188]

Ikea has entered the chat.

[@LeoAngora]

Great video, thanks! The visualizations helped a lot. Adding to the joke answers: - Sofa? In this economy? I can fit a larger bean bag for less money - "As big as the apartment" (Ant-Man)

[@koopalad4]

step one: create a machine that can determine if a given program will stop depending on it's instructions, step two: create a program that searches for a better sofa, step three: analyze the given program using the machine defined earlier, If the machine says the program will stop, it means it's not the optimal sofa, if it says the program will loop, congrats, you found the optimal sofa

[@usvalve]

My first thought was a sectional sofa too. Then I asked myself whether I really want the biggest possible sofa cluttering up my room, especially when houses in the UK tend to be fairly small. I'm now working on finding out what shape and size of sofa combines minimum size with maximum comfort and intimacy.. 🙂

[@svz5990]

Minimum size can just be a size of a person for you to fit

[@SAOS451316]

I think of gear geometry with this problem. Optimal force transfer requires sliding planes and said planes can't be interrupted without losing efficiency. The slight corner rounding is also involved in optimized gears.

[@vylbird8014]

Then you learn that the involute gear pretty much perfected the practical engineering of gears.

[@bluesque9687]

Well done!! me is new subscriber, and the more I watch on your channel, the more I feel like I have found something really wonderful!!

[@karthik999x-narrowone8]

You come up with the weirdest complicated problems that are completely random and I absolutely love it.

[@diskritis2076]

We were giving a very hard applied calculus test and had similar questions like this and my friend just blurted out "sofa problem is still unsolved", I was confused but that day I got the taste of optimization problems for the first time

[@bwillan]

This is similar to the firefighter's problem of what is the longest ladder that you can get through a 90 degree corner of a hallway.

[@pranavid]

A collapsible one. 🤯

[@DaP84]

@@pranavid a rope ladder rofl

[@pingnick]

Yeah also is both the area and volume in the 3 dimensional thought about this identical almost challenge essential to consider-probably but maybe not!? I’m confused just thinking about volume but yeah probably volume doesn’t matter yeah just huge couch reaching ceiling of hallway!?!?🤯

[@dimaratosgeorgiadis2672]

One way to further enlarge the sofa maintaining him parallel would be to bend the inner curve while adding space to the inner corners, that so it will still be able to rotate through the wall. The question is if Gervers shape sits at the spot of this analogy where the area is maxed.

[@JulianOShea]

Excellent video, mate.

[@SmrutiDashiamironman]

Pivot!!!!!!!😂😂😂

[@stevendeans4211]

I was looking for this comment.

[@x7heDeviLx]

u missed a chance at a good pun. and i quote " Sofa weve just been guessing at shapes" lol love your content. keep up the good work

[@gordonwallin2368]

You're back! Yay!. In Canada we have Chesterfields, not Sofas-well used to. Cheers from the Pacific West Coast of Canada.

[@ertugrulsenturk8439]

Very interesting and unique video. I like your channel. I hope you'll grow more without losing your uniqueness.

[@leonardharris9930]

In the real world we operate in three dimensions and anyone who has had practical experience of moving large sofas ( or even beds ) around confined corners knows that the only way to do it is by tipping the sofa on its end and moving it vertically around the tight corner. That is by making use of the third dimednsion.

[@Frightning]

I have an idea for how to go about proving that Gerver's sofa is optimal. Observations: 1: Balanced solution is at least a local maximum (same perturbations of the hallway path will lower sofa area, as long as they are *small enough*) 2: Original search space at the beginning of the problem was *any* hallway path (so we didn't, at the outset potentially exclude a better solution). 3: Any hallway path must be continuous (note, sharp turns are fine here, but what's important to observe is the continuity of motion, this is a consequence of the motion being from the Special Euclidean group) 4: 3 implies that continuous perturbations of a hallway path can transform any hallway path into any other. I would argue that 4 can be used to show that the local maximum observation #1 actually implies global maximum (this is probably not easy, but I think, at least tractable).

[@jimhutton2390]

This is only considering two dimensions, when you add a third dimension, and flexible sections and latches to lock sections together you can get even more solutions.

[@dubleblitz]

I feel so smart after guessing the cutting of the corners to get more room at the top before it was mentioned in the video. I can't believe it took 24 years to be found. Maybe it comes from the fact that I once had an internship where I wrote a website about bodies of constant width and how you can transform any body with corners into a body of constannt width. Also I am surprised that there is no word for such bodies in english, whereas there is the word "Gleichdick" in german which translates to "evenly thick".

[@jeffwhite1334]

Jade, this probably isn’t what you intended for your videos… but they help me fall asleep. For me they’re like little scientific bed time stories to combat my insomnia. Your accent and voice are so calming. Thank you so much!! ❤

[@leftmono1016]

I think you’re probably right 😂💤

[@andrewharrison8436]

Watching real furniture movers putting a large sofa from a narrow hallway through a doorway using 3 dimensions is impressive. Glad I saw it because it turns out the manover is reversible to get it out again.

[@Blacksoul444]

well, every combination of translation and rotation can be reversed of course.

[@stromzardan5821]

Brilliant explanation

[@RyanStonedonCanadianGaming]

As a old furniture mover, Whatever ceiling height and width of the thinnest hallway is basically the limit of what you can fit around a corner as you have to tilt the couch up on an angle and hug it around the corner.

[@TheRAYviewYT]

I can’t believe math geniuses didn’t understand this😂

[@LineOfThy]

Try tilting your sofa upwards in a 2D world

[@husaynbootwala1729]

12:29 I think the bigger question is whether the largest sofa is waiting to be discovered or invented 🤔

[@b2gills]

Before I got to the part of moving the hallway instead of the sofa, I thought about using the hallway to sand down the sofa until it fit. Which may be sort of a precursor way of thinking towards moving the hallway instead of the sofa.

[@hippopotamus3025]

Best calc2 problem to learn optimization. thanks mr fomin

[@prabhakarrao4922]

Absolutely brilliant

[@markchapman6800]

"What's the biggest sofa we've come up with so far?" That pun was the inspiration for the video, wasn't it? Also, did you have a fit of giggles immediately after you cut there?😊

[@Flying0Dismount]

As a mathematician, I'm surprised that you didn't come up with the answer of the infinitely thin sectional, where you can literally make any size sofa you want by moving these thin slices and integrating them back into a whole sofa in the destination room...

[@erikburzinski8248]

I was thinking the ikea sofa

[@goldernie]

You've just invented bean bags

[@joachimfrank4134]

As soon as you split in into infinitely many parts and combine it back again, interesting things can happen. Theres the Banach-Tarski-Paradoxon, where a ball is doubled by splitting and re-assembling.

[@SmallSpoonBrigade]

The correct answer to this is the size of the room that it's going into and using an inflatable couch. Using an inflatable couch, you could even get a couch much larger than the final room to fit around the corner in most cases.

[@PJ-oe6eu]

@@SmallSpoonBrigade What if the hallway is much much smaller than the room. Maybe we should inflate the couch and then cut it up into infinitely thin pieces.

[@sylver369]

When considering the hallways, yes, you're working with only 2 dimensions. But when considering the couch, you are actually working with 3. A couch that can't fit around the corner, might fit if it's rotated forward or pitched up on one end. The doorways are actually the trickier part because you have less width to work with and more often than not a couch, love seat or lounge chair are bigger than a doorway, but people manage to get them through.

[@dylanparker130]

Had to do it for my Mum last year. It's so difficult to coordinate that I decided to do it by myself. Overestimated my abilities - do not recommend! First I used the front part of the base as a hinge to get it through a doorway but behind the door was a sharp turn. So then I had to use the side of the sofa as a base (by making the side the sofa's contact surface with the floor) and sort of tip it up so its other side was almost touching the ceiling. Reminds me of that viral meme of the popular children's toy. Putting differently shaped wooden blocks through differently shaped holes. But instead of putting the blocks through their intended holes, the kid puts them all through the square hole that is intended for the cube-shaped block. Navigating rectangular doorways with an L-shaped block is horrendus, but you get a lot more through than you'd expect!

[@RHLW]

As someone who once spent a few years moving furniture for a living, I can promise you that myself, and plenty of guys can do this in their heads. You can look at a given piece of furniture, in the back of the van, walk into the house (keeping a "picture" of the item in your head), look at a given hall/stairway, and be able to tell if something will go or not, and figure out the set and order of rotations needed to make it go. Ive seen guys who can do this, and be correct to the mm. So there must be a method or model, even if we don't have an exact analytical solution.

[@MikesTropicalTech]

The movers told me my desk is too big to make this exact turn in the hallway. I will have to prove them wrong with Math! P.S. Within 5 seconds of the start of the video I was shouting "Pivot!"

[@jamesphillips2285]

I have a large metal desk I have moved several times by taking it apart. May not work for cheap idea style furniture though (designed to be assembled once). Cleaning out of my grandma's house we found a bed frame that could not fit in the opening in the attic. Reasoned it was either assembled in place or the be frame was moved in during construction.

[@peetiegonzalez1845]

Came here just for this comment. PIVOT!

[@tomdekler9280]

I think what makes this problem fun is that unlike most unproven mathematics questions, this one had Gerver optimizing what was conjectured to be the best solution. With most unproven mathematics, it's a boring "yeah this is very likely the real answer but proving it is turning out to be annoying as hell". Now though? I don't see this sofa problem going anywhere anytime soon.

[@MongrelShark]

As an engineer I love the way of thinking presented here. As a removalist I'd like to point out 2 things I've noticed. 1.The biggest sofas I've moved into tight spaces, are always asymetrical. This makes sense somehow, but I can't explain the maths. its got to do with the big end usually going in first. 2. You can't ignore the 3rd dimension. In practice, Most sofas go in standing on their end. As this gives the smallest birds eye view for better cornering. Gervers solution can't be proved because it only looks at 66.6% of the solution. Unless you want to change the problem to: Whats the largest 2d shape that can fit around a 2d 90deg corner? 3. Gervers couch touched the walls so you lost money on that job due to patching the walls. In practive you need at least a few mm clearance from walls, or its a failed run.

[@LineOfThy]

Yes. That’s the literal problem. It’s meant to be 2D. Why are you changing the problem?

[@blue_champignon5738]

What if we could send blueprints to where we live and get AI generated furniture that can move through any hallway and designed optimally for the space we're in lol

[@AlexandHuman]

This reminds me a lot of the perfect wheel problem that "Morphocular" showed off. Is there a way to solve this problem for less complex hallways? Is there a way to transform the shape of the hallway into purely an equation and then use that to find the biggest optimal shape that matches up?

[@user-fx5hp4ru1l]

Thank you for 0:43, as soon as I saw your thumbnail I had to think of this.

[@jpr4747]

Interesting. I have searched this problem ( and some looking like) but I didn't know this has been a famous one. I drawed figures but thought thé solution should bé whether an algrebraic curve or inspired by pure geometric/mechanic. At first, I tried with a segment sliding in a right angle whose centre discribes a curve, then with a rectangle....but actually, I had first to solve the problem once I had to help a friend who move in a flat at the 12th floor...the lift cabine was a bit small for his sofa...that we turned in many angles. Another time we had to move out a 1/4 disc base aquarium ( 350 L ) in an helicoïdal stair...it left just 1 cm....

[@wilgarcia1]

Has anyone reimagined this in three dimensions? I bet it would be much more fun =D

[@karabenomar]

For a suitable definition of "fun".

[@tarmaque]

I tried to imagine this in four dimensions. There was a strange snapping sound, and now I can't remember my 7 multiplication tables and my Grandmother's maiden name.

[@BruceAFairchild]

End up the couch and it will fit around the corner. Usually the ceiling is high enough. Even if the ceiling is not high enough to allow for complete vertical position, it will most likely be close enough.

[@oliverwilson11]

When I saw the thumbnail I thought it must mean that mathematicians hadn't solved it for an arbitrary number of dimensions. I definitely didn't expect the 2D case to be unsolved.

[@bsjeffrey]

what if it's a gigantic plyable sofa, like a bean bag sofa?

[@glenneric1]

Nice video! I'd never heard of the problem.

[@King_Of_Midgard]

I propose that is is the sectional sofa with infinite middle sections that is the largest. Just cuz the hall puts constraints for moving any part of my furniture doesn't mean the size of my furniture is constrained by the hall; its confined by my willingness to move multiple pieces of sofa.

[@Its_just_me_again]

lover the intentional or unintentional pun of "what is the biggest sofa....so far"

[@DarkMatter1919]

The Gerver shape solution is Sofa King awesome.

[@vanpenguin22]

While I love your content, I find your exuberance and enthusiasm greatly uplifting 😊

[@ryanpmcguire]

Use finite element analysis to force a rectangle through the hallway. Wherever the sofa intersects with the hallway, shave it off. This ensures that only intersecting points are removed. Perhaps this will lead to the sofa with the maximum amount of remaining material.

[@ganrimmonim]

My maths abilities are very limited for those in the UK long, long ago I did A-level maths and the first year of an astrophysics degree before realising my maths wasn't up to it and switching to chemistry. That was way back in 2001, oh how time rashes on towards its inveritable conclusion. But for some reason I have become interested in unsolved maths problems. But from all of them this seems the most preplexing as to why it is still unsolved. As you say it seems as if it should be straught forward.