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Magic Hexagon

Most of us are familiar with Magic Squares and its wonders. But little is known about Magic Hexagons. This post is attributed for describing "Magic Hexagons".

A magic hexagon of order n is an arrangement of numbers in a centered hexagonal pattern with n cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant. A normal magic hexagon contains the consecutive integers from 1 to 3n^2 − 3n + 1. It turns out that normal magic hexagons exist only for n = 1 (which is trivial) and n = 3. Moreover, the solution of order 3 is essentially unique. These hexagons are shown for your reference. The first magic hexagon has Magic sum 1 and second has Magic Sum 38.

Note by Shivanand Pandey
3 years, 6 months ago

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This is interesting. Do you know why no other magic hexagon of higher orders exist?

With \(n=2 \), why can't we have

\[ \begin{array} { l l l l } & 1 & & 2 \\ 5 & & 4 & & 3 \\ & 6 & & 7 \\ \end{array} \]

The entries in each row sum to 12. Is there something that I'm missing? Calvin Lin Staff · 3 years, 6 months ago

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@Calvin Lin Higher order magic hexagons do exist, but cannot be 'normal'. Jonathan Wong · 3 years, 6 months ago

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@Calvin Lin I believe what you wrote down isn't exactly correct. Although a few of the rows sum to 12, not all of them do. I'll go through a few of the rows.

\(1 + 2 \neq 12 \)

\(5 + 6 \neq 12 \)

\(6 + 7 \neq 12 \)

and so on. Milly Choochoo · 3 years, 6 months ago

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@Milly Choochoo Ah, even 'partial' rows count. That makes sense now. Thanks! Calvin Lin Staff · 3 years, 6 months ago

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