Six-decade math puzzle solved by Korean mathematician | Hacker News Reader

Six-decade math puzzle solved by Korean mathematician

koreaherald.com

260 pointsmikhael85 comments

>> “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!

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.

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.

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.

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.)

Have you talked to a high schooler recently...?

Fair point, I was basing my comment on what the curriculum expects of students, rather than the bleak reality.

You need analytic continuation to define the zeta function at the places you are asking for zeros.

That's a good point. I do remember doing problems related to extending formulae outside the radius of convergence in my final year before university, but I don't think it's fair to ask for proper complex analysis from 17-year-olds.

As penance I did go an have a look for suitable numerical techniques for calculating zeta with Re(s)<1 and there are some, e.g. https://people.maths.bris.ac.uk/~fo19175/talks/slides/PGS_ta...

>What could be simpler than that?

At risk of being tongue-in-cheek, a monad is just a monoid in the category of endofunctors, what's the problem?

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.

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/

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.

(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

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.

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

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

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

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

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.

A very functional sofa joke as it's the optimal shape for moving.

Not really, because it's only optimal in 2d.

Most of us don't live in Escheresque labyrinths

If you've ever moved any furniture at all, you'll notice that it's often much easier to get around corners (or through doorways), if you can turn them sideways.

That's especially easy to imagine with tables, but sofas also count.

There are also sofas that can be easily taken apart. Eg one of our sofas at home, an L-shaped sofa, comes apart into two pieces.

You don’t have to move house many times to realise that yes, we do.

Not a problem, we can just shape it like the space covered by rotating the whole thing and make the sofa kind of "bone shaped", then we should be set for 3 dimensions. The only remaining issue would be, where to put the actual sitting area on the bone, but that problem I leave for homework.

3dimensions still allows for more freedom than that though since the couch can stand on end.

I would contend that it's still useful since you'd be able to turn the corner without over-complicating it by getting it into some weird tilt position.

> Not a problem, we can just shape it like the space covered by rotating the whole thing and make the sofa kind of "bone shaped", then we should be set for 3 dimensions.

That might give you a feasible solution, but I doubt it's optimal.

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

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.

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

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...

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...

> 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.

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

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.

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."

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...

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.

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.

1 is not prime.

I agree, modern definitions exclude 1 since "we lose" unique factorization. It's interesting to note [1] that this viewpoint solidified only in the last century.

[1] https://mathenchant.wordpress.com/2025/04/21/is-1-prime-and-...

No, 1 is excluded for reasons closely related to, but not conceptually identical with, the one you mention.

The "intuitive" argument that 1 is prime is that, as with prime numbers, you can't produce it by multiplying some other numbers. That's true!

But where the primes are numbers that are the product of just one factor, 1 is the product of zero factors, a very different status. The argument over whether 1 should be called a "prime number" is almost exactly analogous to the argument over whether 0 should be called a positive integer.†

It's more broadly analogous to the argument over whether 0 should be called a "number", but that argument was resolved differently. "Number" was redefined to include negatives, making 0 a more natural inclusion. If you similarly redefine "prime number" to include non-integral fractions (how?), it might make more sense to consider 1 to be one.

† Note that there is no Fundamental Theorem of Addition stating that the division of a sum into addends is unique. It isn't, but 0 is the empty sum anyway.

“ But where the primes are numbers that are the product of just one factor, 1 is the product of zero factors, a very different status.”

What do you mean?

The factors of 3 are 3 and 1. The factors of 1 are 1?

6 is the product of the members of the set {2, 3}.

3 is the product of the members of {3}.

1 is the product of the members of the empty set.

3 is also the product of the sets {3, 1}, {3, 1, 1}, etc.

We’re excluding the unit when defining these factor sets (ie, multiplicative identity) because it removes unique factorization.

That 1 is the unit is also why it’s the value for the product of the empty set because we want the product of a union of sets to match the product of a product of sets. But we don’t exclude it from the primes for that reason.

What.

Oh! So it’s like Python’s `reduce(multiply,s,initial=1)`, such that s={} still gets you 1. Alright, that makes sense.

No, you're wrong. The factors of 3 are 3. 1 has no factors.

To be clear, you are talking about "prime factors". 3 and 1 are both "factors" of 3, but 1 is not a prime factor.

"1 is the product of zero [prime] factors"

This seems to be circular since it assumes that 1 is not prime

darn, I just dated myself back to 1914...

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

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...

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

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.

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

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.

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.

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

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.

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.

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?

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

I'm not sure what you mean by this?

A screw thread type thing?

I have not read the sofa problem I am not qualified to even talk about it.

Given a rectangle tunnel, twist it so the walls become the ceiling, and also turn it in the same time. Seems like the same problem..

What is solved here is maximum shape, it feels like moving a object like this in 3D is basically a robot planning thing which in it self obviously magic for me. Finding a continuous path in a configuration space is the easy part, but then you have to understand how that changes when your shape changes.

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.

It seems pretty obvious that the 2D solution is still optimal in that case, by making the sofa maximally tall (1) at all points. Though I don’t know if it has been proven yet.

That does seem like a reasonable guess, yet it was only in 1992 that the current (now proven optimal) solution to the 2D case was proven so.

You could make the 3D case more interesting by having two corners: one in the plane and one out of the plane. It seems to me that that would need a new solution.

Certainly, and so would a 2D solution that has to support both left and right turns.

To make 3D relevant for a single turn, one of the hallways could be rotated 45° lengthwise.

Google have just released a numerical solution to that one!

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

*PIVOT!*

That’s tables /s

The sofa solution happens to also apply to tables.

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.

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

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

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