There Are Magic Hexagons of Every Order
gukov.dev
gukov.dev
I'd love to see just how Lipschitz continuous, how smooth, the potential field can be; how adding features puts it closer to or further from solutions that fit the consecutive-no-duplicate constraint, say. Adding a smooth 'hill' is probably viable; is a 'river'?
One little critique I have is in the latter third. Using LLMs for proof is pretty standard now, but the way the text focuses on their tribulations was distracting. It might have been cleaner to use the mathematician's "we" after introducing the 'co-authors', so that the casual reader might sink their teeth into the math rather than be reminded LLMs can sometimes cost money and go around in loops.
But otherwise this is a really gorgeous article! The visualisation is really powerful, and something about how the symmetries impose a kind of conservation (which looks like hot soup but actually has a smooth potential) are very exciting, and curiously very physicsy.
The potential fields are a curious phenomenon. I feel like - and I haven't verified this in any way yet - the smoothness is dictated by the fact that all values are within a not-so-broad range. Then as you "peel" the hexagon from the outer layer using those 6-rings, building the potential field, each next layer inwards shouldn't changed too much.
I had also explored a case of n->inf where the sums turn into integrals, and consecutiveness turns into the uniform distribution. Then it's not so hard to find a solution. It's not in the article, because it turned out to be a dead end for solving the discrete case, and I didn't want to make the article too heavy - but if I remember correctly, there I also observed the smoothness.
I would guess that the smooth mountain-looking structure come from a very simple observation: a gadget consisting, in potential space, of a 2 surrounded by a ring of 6 1’s fully cancels at the center and in the ring immediately around the center and leaves a nice pattern of +1 and -1 residuals in the ring around it. I suspect that, fairly generally, as you try to build out small numbers around the outside of the magic hexagon, you end up with a large pile of things like this in the center, and a sum, even a very noisy one, of things that even vaguely Gaussians, tends to produce Gaussians. (That’s the central limit theorem.)
The "wide ring" gadget is an interesting idea. I think it will be linear pyramids and not gaussian. A set of these gadgets can also form a basis, and in such a different basis the potential fields may look very different - almost flat, perhaps?..
(In the hexagons, all lines are considered even if they don't have the maximum length)
(PS: make sure you hover your mouse over the diagrams)
Upon a quick research, there are many variations, and the closest to what you're describing is the "pandiagonal" magic square, where additional diagonals are considered (except they wrap around the edges of the square in a slightly funny way, so that every diagonal still contains exactly N numbers): https://en.wikipedia.org/wiki/Pandiagonal_magic_square
Seems "unfair" that hexagons have multiple line lengths to consider. I think this article's modification is a good one in that framing: Shifting every number up or down doesn't make the magic squares any easier, but it certainly helps with hexagons.
Then again, I do mostly remember this in terms of the yet unsolved magic-square-of-squares problem, not the standard magic square.
No kids.
Not to mention, a world where nature is taking a beating year after year.
Do you think you'd be interested in working in front of a computer day after day researching, script writing, etc. for people you don't know and probably will never meet?
Now take that answer, and ask yourself - if you were a content generator for your career, and your content got consumed into every AI model early on. Now making any submission have the added hurdle of being content flagged for possible AI usage. Ultimately taking a huge hit to your revenue model...
Or perhaps, spending your time with your wife. Enjoying nature. And just generally disconnecting from the rat race. Especially the rat race 3.0 (Now with AI!)
Like yeah, do some projects here and there for yourself. But do you really want the added burden of pleasing others?
I was just genuinely asking whether there was some specific reason or he just withdrew himself from the public because life happens.
No need to get argumentative.
"AI" cannot imitate the personality or perspective, and that's all anyone ever really cared about. He can vent in any format and people still listen, hence the success with podcasts. That would be effortless and enjoyable unless there was a serious life event to cause a change in perspective (the money isn't it). That's what the leading speculation is.
You're projecting a type of struggle that may exist for you, but was never there for him. He pretty loudly stated multiple times in multiple ways that he was losing his edge, and that's just life.
It makes more sense that he thinks his new authentic self might be more damaging to the brand than abandoning it. He shouldn't be afraid to share the next chapter, but that's his struggle and choice.
He says "every order" is solvable this way but I don't think any solution could work for an order 2 hexagon, even without his simplifying constraints (since fixing any side cell to x requires 2 cells to be set to sum-x).
By the way, I'd like to note that it will likely be a formalization of not the last result in that conversation, but of one of the initial proofs that relied on the Langford sequences and theorems of their existence.