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 ${n \choose 2}$ 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 ${n \choose 2}$.
An example for $n=3$:
Place the pieces $A = (0,0)$, $B = (0,1)$, and $C=(0,3)$.
Then, $A \rarr B = 1$, $B \rarr C = 2$ and $B \rarr C = 3 = {3 \choose 2}$
Is this possible for all $n$? Why or why not?
Hint
highlight spoilers to read them
It is not possible for all $n$. There is exactly one $n \leq 6$ with no solutions. Can you find which one, and prove that it has no solutions?
A solution can be found here