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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.
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?
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 😂
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!"
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"
@@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.
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.
@@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.
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.
@@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.
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
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.
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.
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
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).
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
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.
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...
@@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.
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 😂
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. ;-)
@@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.
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 :)
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!
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.
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. 🙂
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.
@@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).
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)
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 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!
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...
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.
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
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.
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
I used to work at a big hospital whose ER nurses station was a square shape with a hallway running along each side. Naturally the hallways had to be wide enough for gurneys to make it around the corners. One of my coworkers remarked that a hospital where he previously worked built a new ER with a similar layout, except that the blueprints called for narrower hallways, so when equipment was being moved into the new ER, they noticed that the gurneys weren't able to make it around the corners.
I do love how the couch created so far is actually a reasonable enough shape. giving this a mathematicians favorite accomplishment, practical application.
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.
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.
@@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. :)
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! 😀
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 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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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!! ❤
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.
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.
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.
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.
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.. 🙂
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.
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).
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!?!?🤯
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.
"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?😊
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!"
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.
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.
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.
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.
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.
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...
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.
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.
@@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.
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.
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!
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.
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.
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.
A better answer is.. depending on how tall the ceiling is. You're only looking a 2D.. you HAVE to look at it 3D.. length, width AND HEIGHT. You can always tilt the sofa upwards so you only have to worry about the thickness of the sofa which is about 3 feet wide.
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
First off you dont move a couch around a corner, you stand it up and walk it neatly around , done. Spiral staircases are a pita ! You have to stand up the couch, and lean , and rotate the couch as it goes up the spiral, its really fun! Especially brand new when you have to TLC the whole way.
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.
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.
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.
I think what you've left out is this problem can only be solved with an enormous amount of screaming. Anyone who's ever moved furniture inherently understands this basic truth.
As someone who has moved a lot of furniture the answer is a sofa as wide as the hallway and as long as the HEIGHT of the hallway. Get to the corner, tip it up on its end, then tip it down the other hallway.
