This puzzle assesses the extent to which you approach problems by strategic attack, instead of simple trial-and-error. A careful reasoning process before you start writing down any numbers can reduce the matter to just a few possibilities. For example, we can focus on the number at the center of the matrix, which gets added to every other number in some combination or other.

We realize we can't use 9 for the center, because adding it to large numbers like 8 would overshoot the target sum of 15. By the same reasoning process, we can disqualify 8, 7 and 6. We can also eliminate 1, 2, 3 and 4 as too small to make sums of 15 in all directions. When we realize that only 5 will qualify for the center position, we can quickly figure out the rest of them.

This skill is also called fencing.