Magic Hexagons Exist for Every Size, Not Just the Classic One
There Are Magic Hexagons of Every Order
A new interactive visualization reveals that magic hexagons—hexagonal grids where every row, diagonal, and column sums to the same total—exist for all orders, not just the famous n=3 case. The author presents a potential-field construction that generates these arrangements, challenging the long-held belief that only one magic hexagon exists.
The potential field approach reveals a continuous family of magic hexagons, shattering the myth of their uniqueness.
- yunruse
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.
- sghiassy
Hexagon is the bestagon!
- ball_of_lint
Al Zimmerman ran some related contests last year:
- amelius
Why is not every 45 degree line considered for the rectangular grids?
(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)
- arjie
Huh, this potential technique seems fairly neat. Thank you for explaining the whole thing in a fairly accessible way. Enjoyable interactive bits as well. An aside is that the playground looked fine on my iPhone. The anticipatory objection of smallness did not materialize.