Solution: n = 14.  

The coordinates include n + 3, n, n – 3, n – 6, n – 9, and n – 12. Because (2, 14) is one of the points, then 2 must be the smallest value from that list, which is n – 12. Since n – 12 = 2,
then n = 14.

Note that all of the other coordinates have larger values. The next smallest value is n – 9, but
if n – 9 = 2, then n – 12 = 2 – 3 = -1. This is impossible, because all coordinates have positive integer values. All of the other coordinates have values greater than n – 9, so none of them can be 2, so n – 12 = 2, as stated above.

When the points are plotted on a coordinate graph, the following heart-shaped figure results: