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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!"

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

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

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

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

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

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.

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!

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

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 😂

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

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

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!

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...

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.

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 :)

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. 🙂

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

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 😊

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.

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.

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

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.

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

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.

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

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)

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

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!

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

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.

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! 😀

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

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

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!

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

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.

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.

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

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.

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.

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

“Let’s watch a movie” “Sure thing, just let me move my hallway real quick”

My friend: trying to explain the masterpiece that is the sofa problem Me: why don’t you assemble it inside the room?

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.

Pivot!!!!!!!😂😂😂

Mathematicians: *working on it for decades* Engineers: make it modular and join it in the room Ahahahah just joking. Great video, really enjoyed the subject,your presentation and passion about it. Keep it up

To prove the strength of flex-tape clear, I sawed this couch in half!

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.

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.. 🙂

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.

I think Gerver already proved his solution, but then he realized that the real solution was the friends we made along the way, and then decided to leave the problem unsolved.

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

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.

I recently ran into this conundrum when I had to move a rolltop desk from one room, into a hallway and then into another room. The solution was easy - set it on its end to move it. Sometimes a 2-D problem is not longer a problem in 3-D.

This is why no one wants to help mathematicians move. "How big is your sofa?" "1.67225472m^2." "Alright, do it yourself then."

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.

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!! ❤

I’m sure there must be an IKEA engineer out there that after watching this was like: “Hold my SÄLLSKAPLIG of beer!” And we’ll see the most optimal solution on their next year’s catalog

There's nothing like a perfect 90° trench you need to furnish with a couch. It is very satisfying to think so mathematically that you're trying find the ideal couch for a trench. The Polish dude added a coffee table. Big trench design wins again.

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).

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.

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!

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.

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

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.

i can imagine some quirky math teacher taking a class on a field trip to ikea in order to tackle this problem and introduce it to his students

Mathematician approach: What's the largest sofa I can fit through this 90 degree hallway? Engineer approach: Mate just buy Ikea, you can take it apart into a hundred pieces.

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.

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

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.

Michael Batch's comments on Gerver's findings boil down to "So-fa, so good!"

"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?😊

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...

Jack where's the cure for cancer I asked you? Jack: idk but look at this big sofa I found that can fit perfec...

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.

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.

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!"

Mathematicians solution: What this video explained. Engineers solutions: "Uhhh, idk cut it into pieces and call it a sectional."

The largest sofa that can fit around a corner is always infinitesimally smaller than the new sofa you just bought :)

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

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.

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.

What if we pushed a large, completely rectangular, plastic substance that's rigid enough to keep its shape, but plastic enough to deform when pushed through? Then see what shape comes out on the other side.

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

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?

Tord: "hold my vodka" Tord again: "No really, my only arm is getting tired"

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.

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

The answer is an infinitely long rectangle. There is no mention about the speed of the shape moving to the right and then down. So take an infinitely , or at least extremely long rectangle of height 1 and move it close to the speed of light. The length contraction will turn it into a flat vertical line. When it reaches the corner , it can easily rotate because it is only "1" in height while the corner is 1.414. As long as you maintain the speed as C the speed of light , you can fit an extremely long rectangle. After you make the turn, then you can slow down again and let the rectangle expand its length. This is the problem with mathematicians , they are divorced from reality. You can't state the question without the fact that the object has to move. Moving involves speed. If the object moves too slow then the contraction is not enough for it to rotate at the corner. You can calculate the slowest speed that still allows the rectangle to rotate.

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

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

The bigger sofa is a giant beanbag sofa. It’s amorphous, so you just push it through the hallway.

I was gonna comment "PIVOT" , you did it before i reached for the keyboard 😂.

If only this video came out before I tried to get my old desk into my attic.

Unpopular solution….just bring in more and more square cushions and turn the entire room into one massive sofa.

The video is incomplete -- there's been a new development *since 2018:* The *maximum THEORETICAL area* of the sofa has now been determined to be able to reach *2.37 m^2...* (up from Gerver's 2.22) ...now what's left is *to find the actual SHAPE that fills this area... :P* .

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.

As a nerd, geek, and former furniture mover, I can tell you with certainty that if the sofa fits in the hall, it will go around the corner, but you may need to stand it up, or be open to electric monks.