$3 \times 3$ Grid Square pt. 2

Question

I have this maths question that reads: In a $3 \times 3$ grid in which each number from $1 - 9$ are placed on with $9$ as the central number, show that when the $2 \times 2$ squares within the $3 \times 3$ square are added, the total sum of the $4$ squares added together is not a multiple of $8$. Could I please have some help?

Answer 1

HINT:

You would like to show that

$$ 8\nmid 9+a+b+c. $$

where $a,b,c\in\{ 1,2,3,4,5,6,7,8 \}$.

For the sum to be divisible by $8$ it has to be either $16$ or $24$ which means

$$ a+b+c = 15\qquad or\qquad a+b+c = 7. $$

Now if you can find any solution to the above equation then your claim is false.

Hope this helped

Answer 2

The claim you are trying to prove is false; consider $$\begin{matrix}1&2&3\\4&9&5\\7&6&8\end{matrix}.$$