  Given four congruent balls $A, B, C, D$ in $R^{d}$ that have disjoint
interior and admit a line that intersects them in the order $ABCD$, we show
that the distance between the centers of consecutive balls is smaller than the
distance between the centers of $A$ and $D$. This allows us to give a new short
proof that $n$ interior-disjoint congruent balls admit at most three geometric
permutations, two if $n\ge 7$. We also make a conjecture that would imply that
$n\geq 4$ such balls admit at most two geometric permutations, and show that if
the conjecture is false, then there is a counter-example of a highly degenerate
nature.
