Given four congruent balls A,B,C,D in Rd 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≥7. We also make a conjecture that would imply that
n≥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