Pairwise Distance Patterns

You have an infinite grid, and $n$ pieces. We can measure the distance between two placed pieces using Taxicab Distance.

With $n$ pieces, there are $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ possible pairs. Your goal is to place the pieces in such a way that there is exactly one pair of distance 1, exactly one pair of distance 2, exactly one pair of distance 3 and so on until exactly one pair of distance $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$.

An example for $n=3$:

Place the pieces $A=(0,0)$, $B=(0,1)$, and $C=(0,3)$.

Then, $A\to B=1$, $B\to C=2$ and $B\to C=3=\left(\genfrac{}{}{0ex}{}{3}{2}\right)$

Is this possible for all $n$? Why or why not?

Hint

It is not possible for all $n$. There is exactly one $n\le 6$ with no solutions. Can you find which one, and prove that it has no solutions?

A solution can be found here.