Parameterization of Natural Numbers

Question

Suppose we have 4 positive integers $a<b<c<d$ such that $a+d=b+c=n$, i.e. $a,d$ and $b,c$ have the same average.

Does there exist $p,q,r,s \in \mathbb Z$ such that

\begin{equation*} a=p+q, b=p+r, c=q+s, d=r+s? \end{equation*}

Answer 1

Sure. Rewrite $\begin{cases}a=p+q\\ b=p+r\\ c=q+s\\ d=r+s \end{cases}$ as follows: $\begin{cases} p=a-q\\ q=a-b+r\\ c=a-b+r+s\\ d=r+s \end{cases}$. And take $r,s$ arbitrary so that $r+s=d$.
Is that what you need?

Answer 2

Take $p = 0$, $q = a$, $r = b$, $s = d - b = c - a$. These are also all $\geq 0$.