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In this video it is said that $nM$ for a given magic square is equal to $\sum_{i=1}^{n^2}i$, and then the result is also used for magic hexagons.

Why does this have to be the case, both for squares and hexagons? I haven't found any answers or clues on the Internet.

  • In a magic square which uses the numbers $1,2,3,\dots$, there are $n^2$ total spaces in the square and so the total of all entries is $\sum\limits_{i=1}^{n^2}i=\frac{n^2(n^2+1)}{2}$. Meanwhile the sum for an individual row (which is what I would have called the magic number, not the sum of all numbers) is the overall total divided by the number of rows, i.e. $\frac{n(n^2+1)}{2}$ – JMoravitz Jan 19 '18 at 05:32
  • Could you let us know what "magic number" means please. – Angina Seng Jan 19 '18 at 05:35

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In a magic square, you have an $n\times n=n^2$ square to fill in, using the numbers $1,2,\dots,n^2$ (one for each box). So if you added them all up, it's $1+2+\dots+n^2$, which you can write as $$\sum_{i=1}^{n^2}i$$.

Cbjork
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  • So, the entries have to be consequitive integers starting with $1$ for something to be a magic polygon? –  Jan 19 '18 at 05:29
  • @AlexAdamov not necessarily, but those are the types of magic polygons that they are talking about in that video. If you are looking at a different class of magic polygons, then of course the total of all numbers, or the total of a specific row, need not follow the same pattern as these. For history, details, and variations of the magic square see the wikipedia article – JMoravitz Jan 19 '18 at 05:33