Impossible to intresect

Let ${\rm P_1, P_2, P_3}$ and ${\rm Q_1, Q_2, Q_3}$ be points on a sphere. Prove that it is impossible to connect each ${\rm P_i}$ with each ${\rm Q_i}$ with $9$ curves of the sphere that intersect only at the given points. 

Solution

There do not exist points

Prove that there do not exist four points in $\mathbb{R}^2$ whose pairwise distances are all odd integers.

Solution