Six-decade math puzzle solved by Korean mathematician

https://www.koreaherald.com/article/10648326

by mikhael • 7 months ago

260 77 7 months ago

gnabgib• 7 months ago

(2024) Source is Scientific-American https://www.scientificamerican.com/article/mathematicians-so... (https://news.ycombinator.com/item?id=42946052)

Discussion on the paper (131 points, 2024, 36 comments) https://news.ycombinator.com/item?id=42300382

svat• 7 months ago

For what it's worth, the article posted here is not from 2024, but from this week: "Published : Jan. 4, 2026", and contains original reporting (quotes from the researcher).

The mention of "Scientific American" in this article refers to something more recent:

US magazine Scientific American named the research by Baek Jin-eon among its top 10 mathematical breakthroughs of 2025

This is a reference presumably to https://www.scientificamerican.com/article/the-top-10-math-d... "The 10 Biggest Math Breakthroughs of 2025" (dated December 19, 2025). It's more recent than your link https://www.scientificamerican.com/article/mathematicians-so... "Mathematicians Solve Infamous ‘Moving Sofa Problem’" (dated February 4, 2025).

November 2024 is when he posted the preprint on the arXiv: https://arxiv.org/abs/2411.19826 "Optimality of Gerver's Sofa" (submitted on 29 Nov 2024).

There has been other reporting since then, such as in Quanta Magazine: https://www.quantamagazine.org/the-largest-sofa-you-can-move... "The Largest Sofa You Can Move Around a Corner" (dated February 14, 2025). (Contains quotes from his adviser Michael Zieve, and from Gerver himself.)

The paper is still under review at the Annals of Mathematics, so there will be likely be another round of reporting when it has finished peer review and is published.

gnatman• 7 months ago

> “You keep holding on to hope, then breaking it, and moving forward by picking up ideas from the ashes,” [Baek] said in an interview with a web magazine published by Korean Institute for Advanced Study.

“I’m closer to a daydreamer by nature, and for me mathematical research is a repetition of dreaming and waking up.”

beautiful!

yearoftherabbit• 7 months ago

Professional mathematician here. Jin's description is spot on. Each repetition of the cycle he describes above feels like you're able to see things in progressively higher resolution. Then one day you wake up and realize you're now an expert.

jey• 7 months ago

Dan Romik has a nice intro on the moving sofa problem: https://www.math.ucdavis.edu/~romik/movingsofa/

lostlogin2• 7 months ago

Yikes. When you’re taking the window frame out to get furniture in, I’m arguing for smaller furniture.

modeless• 7 months ago

Is the equivalent problem in 3D harder or easier? Seems like it would be harder but you never know with these things.

shermantanktop• 7 months ago

My intuition says that with vertical walls, the problem collapses to 2d, but that 2d is a cross section of a 3d shape, and an infinite number of 3d shapes can hold that as the max cross section envelope.

avadodin• 7 months ago

As in a hyper-sofa moving around arbitrary unit hyper-square corners?

The volume has to be greater than 1 unit in order to be a solution so it could be harder in higher dimensions.

gcanyon2• 7 months ago

I think it depends on what you mean by "equivalent"

1. If the walls are vertical and you are maximizing the horizontal area of the sofa and you are not allowed to reorient the sofa, then the problem collapses to the 2D case and the solution is the same.

2. If you are allowed to reorient the sofa while moving it, and the sofa doesn't have a minimum height, and the ceiling is arbitrarily high, then the sofa can be arbitrarily large by angling it to an arbitrary degree and then setting it flat for measuring.

3. So the in between case is where the ceiling is of some limited height and/or the sofa is of some arbitrary height, and then you have to decide whether the sofa has to be an extrusion of a 2D shape, or if it can be an arbitrary shape, at which point you're maximizing volume rather than area. And for that the obvious lower bound is the 2D maximum shape * the ceiling height. But maybe there's better?

hugh-avherald• 7 months ago

What if you to traverse a tunnel that turns 90° in both the x and y directions?

VorpalWay2• 7 months ago

My natural interpretation of "equivalent" would be a 1 wide, 1 tall corridor, and the goal is to maximise the volume instead of the area.

murkle• 7 months ago

Google have just released a numerical solution to that one!

https://colab.research.google.com/github/google-deepmind/alp...

OsrsNeedsf2P• 7 months ago

Actual paper: https://arxiv.org/pdf/2411.19826

cm2012• 7 months ago

So Gerver’s 1992 curved “sofa” (area ≈ 2.2195) is not just a good guess but actually optimal.

The problem asks for the largest 2D shape that can be slid around a right-angle corner in a unit-width hallway.

Here is the perfect fitting sofa: https://en.wikipedia.org/wiki/File:Gerver.svg

plagiarist• 7 months ago

I think it would be cool to have a sofa in that shape as a joke. It could go in a room tiled with a single shape non-repeatable pattern.

hyperion2010• 7 months ago

Discussed at the time: https://news.ycombinator.com/item?id=42300382

throwawayk7h• 7 months ago

This is the famous sofa problem! It's hard to believe it's finally solved; I've spent many evenings staring at the wikipedia article wondering at how even what seem to be the simplest of problems defy the reach of mathematics.

sdenton4• 7 months ago

There are times when mathematics is a bit like a full set of wrenches trying to take apart a problem held together by screws.

ukuina• 7 months ago

*PIVOT!*

moi23882• 7 months ago

That’s tables /s

layer83• 7 months ago

The sofa solution happens to also apply to tables.

cousin_it• 7 months ago

I love the kind of science reporting on display in this article! It stays at a consistent, objective level of detail throughout (no "imagine a vector space as a block of jello" or whatever it is that Quanta and other publications are always doing). It allows specialists to understand exactly what's being claimed, and at the same time stays accessible to laypeople. It feels like it's written for the kind of reader that I aspire to be: not necessarily a specialist on every topic under the sun, but someone who has finished high school and is paying attention.

Though I guess writing like this doesn't pay off in the modern world. Most readers don't consistently pay attention when reading, and to be honest, I don't either.

trehalose• 7 months ago

I agree that Quanta can be irritatingly stretchy with the metaphors sometimes, but to be fair, "What's the biggest couch you can fit through this hallway corner" is inherently easier to explain to laypeople than like, the Riemann Hypothesis.

Y_Y3• 7 months ago

ζ(z)=0⇒-z/2∈ℕ ∨ Re(z)=1/2

i.e. if you apply the zeta function to a complex number, and you get zero, then that number must have been either a negative even integer or had a half as its real part.

What could be simpler than that? Those are all fairly simple concepts, and the definition of the function itself is nothing too exotic. I think any highschooler should be able to understand the statement and compute some values of zeta numerically. I'd like to see a statement about couches written so succinctly with only well-defined terms!

(I'm being intentionally a bit silly, but part of the magic of the Riemann Hypothesis is that it's relatively easy to understand its statement, it's the search for a proof that's astonishingly deep.)

penteract• 7 months ago

For many publications you could be critisizing, I'd agree with you, but Quanta usually reaches a higher standard that I feel they deserve credit for. Here's the Quanta article on the same thing [1]. It goes into much more detail, it shows a picture of the perfect sofa, and links to the actual research paper. They're aimed at a level above "finished high school", and I appreciate that; it gives me a chance to learn from the solution to a problem, and encourages me to think about it independently.

I agree with you that Quanta doesn't always "allow specialists to understand exactly what's being claimed", which is a problem; but linking to the research papers greatly mitigates that sin.

[1] https://www.quantamagazine.org/the-largest-sofa-you-can-move...

And here's how they clearly explain the proof strategy.

First, he showed that for any sofa in his space, the output of Q would be at least as big as the sofa’s area. It essentially measured the area of a shape that contained the sofa. That meant that if Baek could find the maximum value of Q, it would give him a good upper bound on the area of the optimal sofa.
This alone wasn’t enough to resolve the moving sofa problem. But Baek also defined Q so that for Gerver’s sofa, the function didn’t just give an upper bound. Its output was exactly equal to the sofa’s area. Baek therefore just had to prove that Q hit its maximum value when its input was Gerver’s sofa. That would mean that Gerver’s sofa had the biggest area of all the potential sofas, making it the solution to the moving sofa problem.

gniv• 7 months ago

Agree, but I wanted more. What is the intuition behind the optimality proof? I realize you cannot summarize a 119-page paper in two paragraphs, but still.

Edit: This article from September has a bit more: https://www.popsci.com/science/gervers-sofa-problem-solved/

kelipso• 7 months ago

It’s a simple problem that you can explain to kids, hence the no jello. And they don’t even begin to describe how the solution even looks like!

So I don’t think this article can even qualify as a good example for explaining math problems to laymen.

noduerme• 7 months ago

Looking at that graphic... it almost seems obvious. The outer corner radius would be relative to the inner curve radius in some fixed relationship, wouldn't it? The shallower the inner curve, the larger the outer curve has to be. A completely convex outside could have a flat inside. An inside that was concave from end to end could have a flat outside. Kudos to the guy for writing a proof! I wish this article explained better why it took 60 years to solve this...

kadoban• 7 months ago

It's usually hard to explain why proving something is hard, because it's often just: existing/known/obvious approaches didn't succeed. Not terribly satisfying. Often you just have to try doing it and see, and even that won't be satisfying.

xarope2• 7 months ago

Proofs need to be comprehensive.

Here's a silly one: since 1, 3, 5 and 7 are primes, it almost seems obvious that all odd numbers are prime. Naturally, they are not, and there are countless proofs about various prime number generators to show that they can generate prime numbers, which are really prime.

saagarjha• 7 months ago

1 is not prime.

jonas21• 7 months ago

The "obvious" solution that I think you're describing is the Hammersley sofa (1968), which has A=2.2074.

For a long time, it was thought this might be the optimal shape, but it was never proven. And it couldn't have been because it turns out that you can do better: the Gerver sofa (1992) is a more complicated shape, composed of 18 curve segments and has A=2.2195.

Nobody knew whether there might be an even better shape until now (assuming the proof holds up).

See: https://en.wikipedia.org/wiki/Moving_sofa_problem

hinoki• 7 months ago

That Wikipedia article has a nice picture comparing the two shapes which shows how they’re more complex than you’d think.

https://en.wikipedia.org/wiki/File:Gerver%E2%80%99s_and_Hamm...

dirkgwntly• 7 months ago

The so-called moving sofa problem asks how large a rigid shape can be while still being able to pass around a right-angled corner in an L-shaped corridor of a constant width of 1 meter.

This was also the problem in Dirk Gently’s Holistic Detective Agency, so fictionally this problem had already been solved.

SecretDreams• 7 months ago

Now I might need to rewatch that very odd, charming, detective show :).

lproven• 7 months ago

There were at least 2 separate and independent TV series, but I highly recommend going to the original: the 2 later novels from Douglas Adams, after he wrote the Hitchhiker's Guide to the Galaxy.

https://en.wikipedia.org/wiki/Dirk_Gently%27s_Holistic_Detec...

https://en.wikipedia.org/wiki/The_Long_Dark_Tea-Time_of_the_...

In the novels, a stuck sofa is revealed to have got there because a retired Time Lord, Professor Urban Chronotis, briefly materialised his TARDIS in such a place it provided a doorway. So the books are extremely loosely tied into the Dr Who universe.

Don't let this put you off.

ahazred8ta• 7 months ago

The DG novel was partly based on the Doctor Who story where there were six copies of the Mona Lisa, and the bad guy wanted to go back to primitive Earth before multicellular life.

"The atmosphere is poisonous. I'm not sure what's in it but it would certainly get your carpets nice and clean."

calvinmorrison• 7 months ago

This is why i keep whittling at the squaring a circle ;)

teekert• 7 months ago

Now I want a sofa of that shape to go with my ein-stein tiled floor and decorative Knuth’s dragon. I’ll add some nice art in a shape that can’t pass through itself as well.

turbonaut• 7 months ago

Ironic. In Korea sofas would often bypass the corridor by way of a ladder lift (which can be scarily high).

https://centers.ibs.re.kr/html/living_en/housing/moving2.htm...

seoulbigchris• 7 months ago

yep, I see those often here -- scary. Another sofa-related irony is that a lot of Koreans don't actually sit on their sofas. Rather, they sit on the (heated) floor and use the sofa as a back rest.

https://koreajoongangdaily.joins.com/news/2024-05-18/culture...

cubefox• 7 months ago

This won't be popular [1], but research breakthroughs in theoretical mathematics seem to be often useless in a way that useless science is not. Scientific breakthroughs are also often useless (nothing practical is gained from the first detection of a gravitational wave, or from finding out how flight first evolved in insects) but scientific insights still have more information content: they tell us facts specifically about our world, while mathematical proofs merely tell us about all possible worlds. About some consequence of made-up assumptions we happen to find interesting.

It's a bit like finding the fastest way possible to beat Super Mario Bros 3 while collecting the minimum number of coins. A solution to a neat puzzle, but it doesn't carry the epistemic weight of finding out how the universe works, even if both pieces of knowledge are equally useless.

1: And of course this point doesn't apply to applied math.

srean• 7 months ago

Number theory was deemed useless and celebrated for it being so.

leonvoss• 7 months ago

I agree with you. I would rather this brain power go to modeling genetics, politics, and evolution. Fake problems like the sofa problem are overcelebrated while important issues like theory of eugenics are villified and undercelebrated.

gn4d• 7 months ago

dw, you're not alone in this. Researchers like this are extremely impressive, but it seems like an absolutely massive misallocation of his brainpower. Sure, people can say the same about art/literature/chess/etc., but I would argue that more people benefit from viewing or experiencing the latter than will benefit from working through all 119 pages of this paper. This guy could be doing medical or other scientific research, but instead is working on some contrived problem. Even here, let's assume there is some remote application for a nanobot for targeted drug delivery transiting a capillary... "rough" computational solutions will be more than adequate, especially when taking into account wall elasticity and other variables. I do wonder why some of the top institutions in the world like KIAS are even funding this.

webdoodle• 7 months ago

Looks like a couch, should have just given the prize too a furniture mover. /s

1over137• 6 months ago

I love how the article ends with an email address as signature. No bloody social media, just normal universal email.

Six-decade math puzzle solved by Korean mathematician

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