It is asked to find the very special matrix where you put the numbers from $1$ to $16$ in it such that;
- The sum of these $16$ numbers in any horizontal, vertical and diagonal (not only main) line (explained below) will be the same number at least $14$ times (there are $4$ vertical, $4$ horizontal, $8$ diagonal lines in $4$x$4$ matrix)
- In that matrix no number will have any consecutive neighbor, for example if $a_{22}=5$ then $a_{12}$ or $a_{21}$ or $a_{23}$ or $a_{32}$ cannot be $4$ or $6$.
This is actually very like magic square matrix but a little bit more complex.
$\begin{bmatrix} a_{11} &a_{12} &a_{13} &a_{14} \\ a_{21}& a_{22} &a_{23} &a_{24} \\ a_{31} &a_{32} &a_{33} &a_{34} \\ a_{41} &a_{42} &a_{43} &a_{44} \end{bmatrix}$
The diagonal case could be in two directions, with 4 lines for each (colors represents which numbers you are supposed to sum)
(This figure represents only for one direction)
for example;
$a_{13}+ a_{24}+ a_{31} + a_{42}$
As a result, there are 16 possible vertical, horizontal and diagonal lines and you need to have 14 or more same result from these lines!
So you can find such a matrix?