There Are Magic Hexagons of Every Order

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)