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Mathematics-Online problems:

Interactive Problem 63: Magic Square

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A magic square is a chessboard-like arrangement of the numbers $ 1, 2, \ldots , n^2$ in such a way that the sum $ s$ of the numbers of any column, any row, and any diagonal is the same.
\includegraphics{A419_bild1}         \includegraphics{A419_bild2}

Now, any (not necessarily different) natural numbers $ a, b, c, \ldots \in \mathbb{N}$ are accepted. How many possibilities are there for $ s=12$, if we have a square of 9 cells?


Answer:

    possibilities

Hint: Show that for a square of 9 cells the solution is unique, when $ b$, $ c$, and the sum $ s$ are given. To achieve this, replace $ a$ and $ d$ in particular by $ b$, $ c$, and $ s$. Which conditions have $ b$ and $ c$ to satisfy, so that all entries are integers? Which conditions have to exist, that all entries are positive?


   

(From: Day of Mathematics 1996)
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  automatically generated: 8/11/2017