$p, q, r$ are natural numbers greater than 1. There are $pq$ balls placed on a circle, and one number among $0, 1, 2, \cdots , pr-1$ is written on each ball, satisfying following conditions. (1) If $i$ and $j$ is written on two adjacent balls, $|i-j|=1$ or $|i-j|=pr-1$. (2) $i$ is written on a ball $A$. If we skip $q-1$ balls clockwise from $A$ and see $q^{th}$ ball, $i+r$ or $i-(p-1)r$ is written on it. (This condition is satisfied for every ball.) If $p$ is even, prove that the number of pairs of two adjacent balls with $1$ and $2$ written on it is odd.