# Magic 7

Can you complete this magic square with the numbers from 7 to 15?

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In this magic square, which uses all the integers from $7$ to $15$, each of the rows, columns and the two main diagonals has the same total.

Which number replaces $n$ in the completed square?

If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.

**Answer**: 8

7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 = 99 = sum of all 3 rows

$\therefore$ sum of each row is 33

**Writing the numbers in terms of $n$**

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$\begin{align}26-2n\ \ +\ \ 19-n\ \ +\ \ 12\ \ &=33\\

26+19 + 12-33&=3n\\

24&=3n\\

8&=n\end{align}$

**Writing the numbers in terms of $x$**

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Say $n$ + 12 + $x$ + 12 = 33,

then $n$ + 14 + ($x-$2) = 33

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Same idea gives $x-$4 at the bottom

$\therefore$ 3$x$-6=33, so $x$ = 13

And $n$ + $x$ + 12 = 33 $\Rightarrow n$ + 13 + 12 = 33 $\Rightarrow n$ = 8

**Using reasoning and not algebra**

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33 $-$ 14 = 19

19 = 7 + 12

19 = 8 + 11

19 = 9 + 10

no more options so these 4 numbers are 8, 9, 10, 11

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$n$ + middle + 14 = bottom + middle + 12

difference between $n$ and bottom = difference between 12 and 14

This gives 2 options:

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