There Are Magic Hexagons of Every Order

Magic hexagons, much like magic squares, are grids of numbers arranged in a hexagonal shape where each line of cells adds up to the same total. Unlike magic squares, magic hexagons exist in various orders, with the only known non-trivial normal magic hexagon containing 19 cells when the order is 19. A normal magic hexagon of order n contains consecutive numbers from 1 to 3n^2 - 3n + 1, but it appears that for every order greater than 3, this arrangement is impossible.

However, abnormal magic hexagons allow for more flexibility by relaxing the starting number constraint. These hexagons have been challenging to find, with the largest known solution being a hexagon of order 9 discovered in 2024. Researchers hypothesize that the difficulty lies in the vast search space required to find solutions.

To tackle this issue, researchers explored the concept of antisymmetric magic hexagons, where numbers are mirrored across a central point, effectively reducing the search space by imposing constraints on the possible arrangements. By using a potential field representation, researchers aim to search for solutions within a smaller space of hexagons where each line automatically sums to zero, potentially making the search more efficient.

To further simplify the search, another structure, antisymmetry, is employed, where cells opposite each other under a 180-degree rotation contain opposite values, simplifying the constraints even further.