Contributed by
Vijay Fafat
A couple from the race of “Amalgam” wanted to carry out one project before choosing to die after a life spanning tens of thousands of years: Establishing contact with the elusive race called “Aloof”. While the Amalgam had established an empire spaning the disk of the galaxy, they could never penetrate the central bulge, which had apparently been sealed off by the Aloof, who had no desire to establish any sort of communication with them.
The story chronicles this long chase. At one point, the couple discovers traces of a signal leakage coming from the Aloof space but to get into position to properly observe it, they have to solve a geometric problem:
(quoted from Riding the Crocodile)
He asked the map, “Are there two outposts of the Amalgam lying on a straight line that intersects the beam?” The map replied in a tone of mild incredulity. “No.” “That was too much to hope for. Are there three lying on a plane that intersects the beam?” The map said, “There are about tentotheeighteen triples that meet that condition.” Leila suddenly realised what it was he had in mind. She laughed and squeezed his arm. “You are completely insane!” Jasim said, “Let me get the numbers right first, then you can mock me.” He rephrased his question to the map. “For how many of those triples would the beam pass between them, intersecting the triangle whose vertices they lie on?” “About tentothesixth.” “How close to us is the closest point of intersection of the beam with any of those triangles — if the distance in each case is measured via the worst of the three outposts, the one that makes the total path longest.”
“Seven thousand four hundred and twentysix light years.” Leila said, “Collision braking. With three components?” “Do you have a better idea?”

(quoted from Riding the Crocodile)
With standard twobody collision braking, the usual solution was to have the first package, shaped like a cylinder, pass right through a hole in the second package. As it emerged from the other side and the two moved apart again, the magnetic fields were switched from repulsive to attractive. Several “bounces” followed, and in the process as much of the kinetic energy as possible was gradually converted into superconducting currents for storage, while the rest was dissipated as electromagnetic radiation. Having three objects meeting at an angle would not only make the timing and positioning more critical, it would destroy the simple, axial symmetry and introduce a greater risk of instability.
It was dawn before they settled on the optimal design, which effectively split the problem in two. First, package one, a sphere, would meet package two, a torus, threading the gap in the middle, then bouncing back and forth through it seventeen times. The plane of the torus would lie at an angle to its direction of flight, allowing the sphere to approach it headon. When the two finally came to rest with respect to each other, they would still have a component of their velocity carrying them straight towards package three, a cylinder with an axial borehole. Because the electromagnetic interactions were the same as the twobody case — selfcentring, intrinsically stable — a small amount of misalignment at each of these encounters would not be fatal. The usual twobody case, though, didn't require the combined package, after all the bouncing and energy dissipation was completed, to be moving on a path so precisely determined that it could pass through
yet another narrow hoop.

