Distances

In the above picture you can see a point and its distances to its sides. It is now clear that the sum of two of these distances is the side length of the square, and the other two distances also add up to the side length of the square. Thus, we have these three equations: $4 + 8 = 13 + n$ , $4 + 13 = 8 + n$ , and $8 + 13 = 4 + n$ . The solutions are $-1, 9,$ and $17$ . If $n = -1$ or $9$ , then the numbers will no longer be increasing. If $n = 17$ , then the numbers will be increasing. Thus, the only solution is $\boxed{17}$ .