GoKawiilWhat is so special about the number 19?
The question came up last month in a conversation among YSDA alumni, when the school turned 19. Someone pointed out that 19 is a twin prime. Someone else replied that 19 is the number of cells in the only non-trivial normal magic hexagon.
Wait, what is a “magic hexagon”? Let us start there.
A note on AI-assisted mathematics Recently, we have heard a lot about AI miraculously proving and disproving long-standing conjectures, often without much explanation of how it was done. This story offers a look inside the process of making such a mathematical discovery.
Magic Squares and Magic Hexagons
You probably know about magic squares. A magic square is a square grid of numbers in which every row, every column, and both main diagonals add up to the same total, known as the magic constant. We also usually require the numbers to be consecutive - typically from $1$ to $n^2$ - and call it a normal magic square. Otherwise, we could simply put the same number into every cell, which would be a very boring way to fill a square.
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Magic squares have been known for millennia and are now very well understood. We have algorithms for constructing normal magic squares of every order $n>2$.
A magic hexagon applies the same idea to a hexagonal grid. Its cells form straight lines in three directions, and every such line must have the same sum. As with squares, a magic hexagon is called normal if it contains the consecutive numbers from $1$ to $3n^2-3n+1$, the total number of cells in a hexagon of order $n$.
Above you can see the only non-trivial normal magic hexagon in existence - apart from its own rotations and reflections. The proof is straightforward. In each of the three directions, the cells are partitioned into $2n-1$ lines. Therefore, the sum of all the numbers must be divisible by $2n-1$. For every order $n>3$, the sum of the numbers from $1$ to $3n^2-3n+1$ fails this divisibility test.
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