Each of the integers 1 to 7 is to be written, one in each circle in the diagram. The sum of the three integers in any straight line is to be the same. In how many different ways can the centre circle be filled?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-077.jpg?height=371&width=403&top_left_y=232&top_left_x=1292)
We label the circles from $a$ to $g$, as shown:

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-131.jpg?height=371&width=412&top_left_y=2243&top_left_x=911)

Let $S$ be sum of the integers in any straight line.

Therefore, $S=a+g+d=b+g+e=c+g+f$.

Thus, $3 S=(a+g+d)+(b+g+e)+(c+g+f)=a+b+c+d+e+f+3 g$.

Since the variables $a$ to $g$ are to be replaced by the integers from 1 to 7 , in some order, then $a+b+c+d+e+f+g=1+2+3+4+5+6+7=28$.

Thus, $3 S=(a+b+c+d+e+f+g)+2 g=28+2 g$, and so $3 S=28+2 g$.

Since $3 S$ is an integer divisible by 3 , then $28+2 g$ should also be divisible by 3 .

Since $g$ must be an integer between 1 and 7 , we can try the seven possibilities and see that the only values of $g$ for which $28+2 g$ is divisible by 3 are 1,4 and 7 .

We must verify that we can actually complete the diagram for each of these values:
![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-132.jpg?height=374&width=1282&top_left_y=710&top_left_x=476)

Therefore, there are 3 possibilities for the number in the centre circle.