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Yesterday, a friend of mine asked me for a number between 11 and 99 (not 100% sure about the boundaries). I had no idea what he was up to and called 38, about half a minute later he had written down the following:

$$\begin{pmatrix} 18 & 1 & 12 & 7\\ 11&8&17&2\\5&10&3&20\\4&19&6&9 \end{pmatrix}$$

At first, I thought it was already pretty impresive that all rows, columns and the two diagonals sum up to 38. Then he showed me that also the four 2x2 squares in the corners sum up to 38, and so do

$$\begin{pmatrix} 1 & 12\\ 8 & 17 \end{pmatrix}, \begin{pmatrix} 8 & 17 \\ 10 & 3 \end{pmatrix} \text{ and } \begin{pmatrix} 10 & 3\\ 19 & 6 \end{pmatrix}.$$

Does this "thing" have a name? How does one produce such a matrix in such a short amount of time?

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    These are called "magic squares" and there are some constructions given at the wikipedia article.2012-07-20
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    A wild idea came to my mind after reading your question: did you ask *your friend*??2012-07-20
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    Hint, replace 17 through 20 by 13 through 16 in your matrix.2012-07-20
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    @DonAntonio: He always impresses me with some "magic" in any possible way when we meet and for the last six years he would never explain anything. :)2012-07-20
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    You may also like to look at [Create a magic square for any number](http://www.mrexcel.com/tip069.shtml).2012-07-20
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    I see, @Huy, but unless your friend does that for a living or unless he's working in a very secret proyect, I can't understand the reason for his teasing. Mathematicians don't usually conceal the reasoning behind their stuff (au contraire!)...well, perhaps he isn't a mathematician.2012-07-20
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    The four corner entries (18,7,4,9) also add up to 38, as do the two 2×2 squares that wrap around from the middle bottom to top (1, 12, 19, 6) and the middle right to left (11, 2, 5, 20).2012-07-20
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    @DonAntonio: Clearly his interest in magic squares suggests he's a magician... and magicians never reveal their secrets...2012-07-20
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    @DonAntonio And magicians never do. The OP never claimed the friend was a mathematician.2016-04-08
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    Well if you remember a base one that adds to 55 and then every other one you add or subtract the difference between the 55 and the number called you can always get a result. If the original had odds and evens in a pattern the adding or subtracting x for every other position wouldn't have any duplicate numbers. It might take a while but I bet if you add or subtract 17 to every other number you'll find an original square.... or he simply remember 88 entirely different tricks for each number and would have done something entirely different had you chosen another number.2016-04-08
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    Dammit. I hate it when a 3 year old question gets bumped into active mode because someone edited it recently! ... Um, out of curiosity, what the heck is the incentive to edit a post that has been dead and buried for three years anyway? Why was anyone even reading it in the first place?2016-04-08
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    @fleablood: I don't think anything was edited at all?2016-04-08

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(A compilation of some comments...)

These are called "Magic Squares".

Wikipedia

Here's how to create a magic square for any number