Each of the nine circles in the diagram has a distinct number 1 through 9. The sums of any three circles on a line are all the same and a multiple of 4. What number goes in the white center circle?

From: Derrick Niederman's
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Math Puzzles for the Clever Mind
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Let the sum is $4k$ as it is multiple of four.

There will four equations whose each sum is $4k$ .Let each number in the circle of outer side is $c_i$ where $1\leq i \leq 8$ and center circle is $c$

So, adding the four equations $\displaystyle \sum_{i=1}^{8} c_i +4c=4k \times 4=16k \\ (1+2+3+...+9) + 3c=16k \\ 45+3c=16k \\ 3(c+15)=16k \\$

So, $16| c+15$ .The only possible value is $c=1$

Therefore, the value in the center square is $\boxed{1}$

Now , using this value we get $k=3$ So, each sum is $4 \times3=12$ .The sum of two opposite square is $12-1=11$ .The pairs are

$\{(9,2);(8,3);(6,5);(4,7)\}$

So, we get the whole arrangement.