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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 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!
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.
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).
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.
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 :)
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).
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!
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.
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.
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 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.
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
I do love how the couch created so far is actually a reasonable enough shape. giving this a mathematicians favorite accomplishment, practical application.
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.
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. :)
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.
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.
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.
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.
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.
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.
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.
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)
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.. 🙂
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!! ❤
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 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.
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!?!?🤯
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.
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'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 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.
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'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?😊
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
if we go to the 3rd dimension may be we can bring upto the height and width of doors. we can tilt to 90 degree angle in x,y,z axis. if i'am right give me a shot😎
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.
Working as a reupholsterer years ago, I was asked to split a folding bed base in half because they couldn’t get it upstairs. Happy days. Especially joining it together again.
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.
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.
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.
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
Slightly less than the height of your ceiling. When two people are navigating a sofa around a corner, the first person moving inward (lead) drops his end of the sofa to the floor, just as the second person (lag) lifts his end up so that the sofa is now vertical and resting on its arm. If the floor is carpeted, I like to place the sofa upon a small, but sturdy rubber backed rug so that the lead can grab the two ends of the rug and pull the rug while it moves the vertical sofa into the hallway after the turn. While the lead is pulling the rug and sofa down the hallway after the turn, the lag is holding the sofa steady at the top and if possible leaning slightly back toward himself. If the sofa does not have any legs, a rope or a strap lassoed around the sofa will give the lag something to hold onto so that the sofa does not fall over onto the lead. If the floors are bare, just use the same type of rug, however this time turn the rug over so that carpeted side is now down against the bare floor and rubberrized side is up. Then just stand the sofa up and onto the rubberized side of the rug and repeat all of the steps above. Now walk the sofa down the hallway after the turn until there is enough room for the lag to lower his end of the sofa back down to a horizontal position and complete the move. Repeat the process again at any other corners encountered.
When I was helping a friend move a fabric cutter's tabletop (from a remodel at the department store he managed) into his second-floor apartment we could not get it up the front stairs, and the back stairs were a sub-code afterthought being barely shoulder-wide and having a right-angle turn at the top. So we removed an entire window, framing and all, and hoisted it up and BARELY slid it in that way. When he moved out of that apartment, he took the smarter route and cut he table-top in half with a circular saw. - I currently have an upstairs neighbor who moved in everything except their queen-sized bedframe, because it would not fit all the way up the stairs, due to a 180 degree switchback with a support wall. For the several months since they moved in, the bed frame is just sitting against the wall in the hallway outside my door. - What does this have to do with the couch? Sometimes you need to give up on a piece of furniture and accept that you might need something that fits over something you have emotionally and financially invested in. Also, with the very specific hypothetical layout being used in this video, can you take out a window and move a big piece of furniture in that way?
Would be nice if at some point there will be a solver online where you can upload a step model of you house and ask what the largest sofa is to any room
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Is his solution related to the stationary action principle from analytical mechanics?
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 bet'cha that Presh Talwalker from _Mind Your Decisions_ can solve it!
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"
@@SaHaRaSquad hate when that happens
Most people don't care about math. We just have to deal with it.
Ur friend is mean
HAHAHAHA!
I'm Swedish, so my solution is of course to disassemble the sofa and bring it through in pieces. IKEA beats math every time.
Right😂
I know it’s a hypothetical but the whole time I was thinking just disassemble the sofa lol
By writing "IKEA beats math every time" you probably lost a thousand likes.
IKEA 13B yearly income, Google 280B, math makes the difference 267B (subtraction) Sorry I was using math to explain that.
@@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.
Seems like the problem now is less "can you find a bigger sofa?" and more "can you finalize the proof for Gerver's sofa?"
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.
@@GIRGHGH Usually you'd start with trying ti disprove.
@@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 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!
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.
@@mrdraw2087 What is required to be a sofa? The largest volume may simply be the largest in 2d filling the entire height.
Meh, all you have to do is pivot. Piiivoooot! ;-)
Mathematicians: struggle moving a sofa for decades
A random moving grunt:instantly move the sofa on the side
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.
4:02 sofa we've just been guessing at shapes
Yes
lol I was about to comment the same thing on another point in the video
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
Dirk Gently detective. Read it years ago and this video immediately reminded me of the story.
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).
But can you get a swallowed toy sofa around the first corner of a person's small intestine?
@@MonkeyJedi99 Was the resolution of the sofa thing done across two books ?
@@quantisedspace7047 IIRC, it was across two different book series.
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.
The size possible is only limited by the parts allowed.
IKEA teaches this the hard way.
I didn't think a degree in mathematics was needed to to become a furniture mover.
This is why sectional sofas exist.
Exactly, I'm not good at math so guess that's the easy way out! 😂😂
I bought a large chair and was worried about then carrying it downstairs. It came in 5 parts...
What is the largest a section of a sectional sofa can be before it gets stuck?
@@axiomfiremind8431 The answer is: 2.
I'm not a furniture mover, I'm a corridor arranger.
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
No school will tell you that though.
That's why people tend to dislike math because it's shown as only equations.
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. ;-)
They never mentioned that the sofa could be stood on it's end .... I've had to do that before
@@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
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.
@@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 :)
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. 🙂
-""Eddies in the space-time continuum!" -"And this is his sofa, is it?" 😄
This was the FIRST thought I had when I saw the episode.
Yeah, I immediately thought of "Dirk Gently's Holistic Detective Agency" as well. Excellent 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).
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 😊
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!
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.
@@bjorntantau194 It wasn't Gently's computer, it was " The Client " and old school friend whose name eludes me at the moment.
@@alanhilder1883 Ah, been too long since I've last read the books.
@@bjorntantau194 I will have to re find them, it was last century that I got to read them...
Dirk Gently’s “interconnectedness of all things” applies here! This video came out on Towel Day 😊
6:24 "sofa we've been..."
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.
I’m glad it failed
This is the best mathematical puzzle I've ever seen ! Such a simple puzzle yet difficult and quite an elegant solution !
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.
Great comment. There’s a Laurel and Hardy scene similar to this. Hilarious 😂
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.
I spotted the engineer.
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!
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
lover the intentional or unintentional pun of "what is the biggest sofa....so far"
I do love how the couch created so far is actually a reasonable enough shape. giving this a mathematicians favorite accomplishment, practical application.
You're thinking of engineers. Mathematicians scoff at applicability. ;-)
@@jamielondon6436 Indeed
My guess? 0.707 x 1.414 meters. Or any sofa with two diagonals of 1. This should work for all rectangular shapes with these criteria.
I can already hear Ross yelling "Pivot! Pivot!"
My first thought
It's in the first 30 seconds of the video, so yeah, I'd hope you could hear it lol
@@gladitsnotme I commented a minute after the vid was uploaded, hadn't seen it yet 🙂
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.
When confronted with a narrow hallway, we generally solved this problem by tipping the sofa on it's side.
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.
Wrong approach. Just cut the sofa to fit around the corner.
Issue is you can’t do that in a 2D world
@@boggersbeauty of most math problems lie in their simplicity. Being more complicated does not automatically make it a better problem
@@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. :)
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.
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.
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.
To prove the strength of flex-tape clear, I sawed this couch in half!
In my experience it can be much larger than the corner, as long as you're willing to break it
This is math, though, not engineering.
Ross, is that you?
@@vigilantcosmicpenguin8721 pivot!
You come up with the weirdest complicated problems that are completely random and I absolutely love it.
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.
Ikea has entered the chat.
If the ceiling is hig enough I would put the sofa vertically and move it that way
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.
@@techman2553based on that comment, you do not suck at math!
Your prop making and work made this so much more digestible!
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.
she had me at sofa and small hallway.
@@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.
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.
Yes. That’s the literal problem. It’s meant to be 2D. Why are you changing the problem?
We put a sofa upstairs and it had to include 3 dimensions - hitting the ceiling happened.
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)
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.. 🙂
Minimum size can just be a size of a person for you to fit
Excellent video, mate.
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 think you’re probably right 😂💤
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 can’t believe math geniuses didn’t understand this😂
Try tilting your sofa upwards in a 2D world
"Some serious math lover has probably already made/ has had this Sofa made", is what I was thinking until that furniture thing popped up 😂
The problem is more applicable than other math problems I see. I would be surprised if absolutely no one had this sofa made.
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.
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).
Very nice! I like the visualizations of the paths of the hallway
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.
A collapsible one. 🤯
@@pranavid a rope ladder rofl
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!?!?🤯
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.
Degree in Math and another in Astrophysics. Must be back to the dole queue on Monday.
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.
Thank you for 0:43, as soon as I saw your thumbnail I had to think of this.
12:29 I think the bigger question is whether the largest sofa is waiting to be discovered or invented 🤔
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 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.
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'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?😊
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.
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.
Came here just for this comment. PIVOT!
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
Pivot!!!!!!!😂😂😂
I was looking for this comment.
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.
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.
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.
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...
I was thinking the ikea sofa
You've just invented bean bags
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.
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.
what if it's a gigantic plyable sofa, like a bean bag sofa?
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.
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.
if we go to the 3rd dimension may be we can bring upto the height and width of doors. we can tilt to 90 degree angle in x,y,z axis. if i'am right give me a shot😎
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.
The problem by definition is 2 dimensional, so no the height literally does not matter
You're back! Yay!. In Canada we have Chesterfields, not Sofas-well used to. Cheers from the Pacific West Coast of Canada.
The real solution is to bring a lot of square sofas in and join them into one giant sofa
Working as a reupholsterer years ago, I was asked to split a folding bed base in half because they couldn’t get it upstairs.
Happy days. Especially joining it together again.
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.
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.
Michael Batch's comments on Gerver's findings boil down to "So-fa, so good!"
The largest sofa that can fit around a corner is always infinitesimally smaller than the new sofa you just bought :)
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.
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
My friend: trying to explain the masterpiece that is the sofa problem
Me: why don’t you assemble it inside the room?
Just found your channel, really like your style, subscribed!
“Let’s watch a movie”
“Sure thing, just let me move my hallway real quick”
That is a piece of cake, solving it is super easy.
Slightly less than the height of your ceiling. When two people are navigating a sofa around a corner, the first person moving inward (lead) drops his end of the sofa to the floor, just as the second person (lag) lifts his end up so that the sofa is now vertical and resting on its arm.
If the floor is carpeted, I like to place the sofa upon a small, but sturdy rubber backed rug so that the lead can grab the two ends of the rug and pull the rug while it moves the vertical sofa into the hallway after the turn.
While the lead is pulling the rug and sofa down the hallway after the turn, the lag is holding the sofa steady at the top and if possible leaning slightly back toward himself.
If the sofa does not have any legs, a rope or a strap lassoed around the sofa will give the lag something to hold onto so that the sofa does not fall over onto the lead.
If the floors are bare, just use the same type of rug, however this time turn the rug over so that carpeted side is now down against the bare floor and rubberrized side is up. Then just stand the sofa up and onto the rubberized side of the rug and repeat all of the steps above.
Now walk the sofa down the hallway after the turn until there is enough room for the lag to lower his end of the sofa back down to a horizontal position and complete the move. Repeat the process again at any other corners encountered.
If only this video came out before I tried to get my old desk into my attic.
What if the couch is inflatable? Then you could make a gigantic couch and move it as a deflated couch
Very interesting and unique video. I like your channel. I hope you'll grow more without losing your uniqueness.
Great explanation, thank you!
When I was helping a friend move a fabric cutter's tabletop (from a remodel at the department store he managed) into his second-floor apartment we could not get it up the front stairs, and the back stairs were a sub-code afterthought being barely shoulder-wide and having a right-angle turn at the top.
So we removed an entire window, framing and all, and hoisted it up and BARELY slid it in that way.
When he moved out of that apartment, he took the smarter route and cut he table-top in half with a circular saw.
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I currently have an upstairs neighbor who moved in everything except their queen-sized bedframe, because it would not fit all the way up the stairs, due to a 180 degree switchback with a support wall.
For the several months since they moved in, the bed frame is just sitting against the wall in the hallway outside my door.
-
What does this have to do with the couch?
Sometimes you need to give up on a piece of furniture and accept that you might need something that fits over something you have emotionally and financially invested in.
Also, with the very specific hypothetical layout being used in this video, can you take out a window and move a big piece of furniture in that way?
While I love your content,
I find your exuberance and enthusiasm greatly uplifting 😊
I would like to see The Irresistible Couch going around The Immovable Corner.
Would be nice if at some point there will be a solver online where you can upload a step model of you house and ask what the largest sofa is to any room