I loved this article and its interactive elements. The potential field is an elegant abstraction which really elevates this from a math puzzle into something new.
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.
Cool problem. I always doing it a bit unsatisfying that magic hexagons so trivially disallowed solutions that aren't order 3. Starting at a different index is a nice modification, especially since for magic squares it's an equally hard problem.
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).
You're absolutely right, technically the title should have said "every order other than 2" or "every order larger than 2". The case for 2 is impossible, because it immediately forces equal numbers on the outer layer.
Interesting observation. That's simply part of the usual definition of a magic square, and it does indeed feel arbitrary once you compare it with the hexagon case, which is very symmetrical. I first learned about magic squares from math books as a kid, about 25 years ago, and just accepted those rules as given.
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
Why not every line of knights moves too? Because the cells aren't adjacent I'd say. (Both immediately make the problem unsolvable since all corners must match).
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.
If you were given something like $10mil, invested in index funds over the last 10 years, in nontax advantaged accounts. And your wife has health issues.
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'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.
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).
(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.
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.