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This problem was originally posed by Leo Moser in 1966.
Quickly after, Hammersley (1968) proposed a very simple construction.
Take a semi-circle. Cut it into two. Then fill the gap in between, but leave a smaller semi-circle of empty space to help maneuver the corner.
This construction works and gives an area of 2.2074.
It is a very simple-minded construction, however. And it turns out, it is not the greatest possible area you can achieve.
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But it wouldn't be until Gerver (1992) that a big step forward was made towards an optimal solution.
Not a big step forward in terms of area -- his proposed solution was 2.2195, just a tiny bit larger than Hammersley's -- but it was dramatically more clever.
Gerver's sofa was made from stitching together 18 different curve segments. But they were not just arbitrarily chosen segments.
Gerver set up an optimization problem whose solution would satisfy a necessary condition to solve the general problem.
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Gerver conjectured that his solution was optimal, but he could not prove that his sofa was the only one (and the one of maximal area) that satisfied this strong condition.
This new paper by Jineon Baek (2024) claims a proof that Gerver was indeed right -- his sofa was the optimal one.
The paper is fairly long, and it'll take some time before the relevant experts have verified the correctness of the proof.
The paper:
arxiv.org/pdf/2411.19826
Dan Romik solved this problem in the equivalent way as Gerver did. I.e., he constructed a sofa that satisfies this strong necessary condition (local optimization), but he couldn't prove with certainty that it was optimal.
arxiv.org/pdf/1606.08111
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Since this is all over my timeline: Is this actually impactful, ie are there real world applications that benefit from solving this?
The answer, known by anyone who has had to actually move a sofa in the real world, is to turn the sofa on end. A much larger sofa can then be scrooched around the corner.
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if is correct, is there really a sofa there, or is that just our brain's interpretation of sensation input?
I worked for moving and storage co. in HS and college. When going around 90° corner, we stood sofa verticle on a roller dolley.
Here all this time I thought nerds were trying to cure cancer or solve world hunger…nope. They’re trying to figure out how to get the biggest couch through a one inch hallway.
lmao
Posts like this inspire more people to learn math, myself included. Thank you, sir!
Dunno, have been solving those since my grad school days. Usually all it takes is a couple friends. Just recently though, I had the moving fridge on the second floor problem. That one was hard, not gonna lie.
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Thread about Education, Technology, mathematics, Science.
en.rattibha.com/thread/1865116 I am curious about the next unroll! 
Did the consider just standing the sofa on its end so it'll move through the turn without having to make a strange looking couch?
If the ceiling is taller than the length of the couch there is no issue.
You just tip on one end.
I was just listening to something that was talking about unsolved problems and mentioned this, then I chuckled to myself knowing it was, in fact, solved at that moment in time.
Now the harder problem is it’s a bed you’re moving into a tiny 2-storey Japanese house and the corner is a near the top of a stairwell. So you’re doing this while you can’t see your feet. (Yeah moving out I gave up, took the bed apart, and lowered it over the balcony)
If the sofa is turned on it's end, the area of the sofa is looking limited by the height of the space, which is not given in the question. Am I wrong? The corner-like shape of the sofa on its end would definitely help navigate the corner 
Did they "brute force try 5500 shapes then investigate shapes that did the most"?
If all sofa edges touch the wall during movement, it is optimal. If not, then there is some room for improvement.
Curious that the worm problem by Moser, which looks even simpler to pose, is still open afaik.
Douglas Adams solved the problem neatly in "Dirk Gently's Holistic Detective Agency."
You just need a time machine to show up on the corner while you're moving the couch. Then, you simply open the door...
This was a minor but surprisingly important plot point in "Dirk Gently's Holistic Detective Agency" by Douglas Adams.
It takes over 6,000 nails to properly secure a roof. Even one out of place can lead to leaks or void a wind warranty. That’s why we patented the industry’s first larger nailing area, called The Zone®.
#FeatureFriday #Weatherproof
This still doesn’t solve the problem of having one guy that is shorter on one end and the taller guy being hungover, so he keeps dropping his end, even though he is carrying less weight due to his height advantage. Just saying.
Yeah but this doesn't consider that 99.9% of sofas don't have that angle and are straight rectangles
I've seen this done in calculus as an optimization problem before, but with moving a metal pipe around a corner. It's probably a lot easier to compute the area of a rectangular object than whatever this is. 
how squishy is the sofa? theoretically you could maneuver a sofa with a much larger overall area so long as the padding can be successfully compressed into this shape by the pressure of the walls. are we assuming the sofa is being moved by humans of average size and strength?
