Points and dimensions

If we are able of finding at least one example where we cannot answer the question we're done. By taking $n=d=2$ we know that $2+1=3=n+1$ is an odd number and we have to check if we can satisfy the other requests. Now if we choose the three distinct points to be the three vertices of an equilateral tringle whoose sides measure $\pi$ we can say that we'll get for sure $\pi$ if we measure the distance between any couple of vertices. We can put this triangle in a bidimensional space (a plane) in order to satisfy the condition $d=2$ . This same reasoning holds even if we consider $n=2$ and $d\ge 2$ so whether we only know that $n+1$ is odd we can't decide if $d$ is even or odd.