# Andy's Math/CS page

## Friday, October 19, 2007

### Space is a Lonely Place

Puzzle time again--this time a dressed-up version of a problem in Claude George's book 'Exercises in Integration'--not as boring as it might sound!

The year is 2100. SETI, having failed to make contact with any extraterrestrials, is facing termination. They haven't lost faith, but everyone else on Earth is pretty tired of them, so they decide to spend their remaining budget by blasting themselves off into space--a reconnaissance mission of no return.

The SETI folks are interested in coming near as many potentially-inhabited star systems as possible. They've watched 'Biodome' a few times, and they're pretty sure they can get their spaceship to support human life indefinitely. But fuel is limited to their blast-off needs, so they have to plan on following a straight-line path at constant velocity thru space (which is 3-dimensional, or 2 if you like, Euclidean, and devoid of gravity or other impedances--stars can be passed thru, e.g.).

A star-system is defined as the billion miles surrounding a star, within which SETI figures they'll have a good shot at communication with any planets in the system. In the first version of the problem, they also assume, somewhat optimistically, that candidate stars do not move and that candidate star-systems occupy a nonzero asymptotic fraction of the universe's volume. E.g., for any N, if we draw a ball of radius N around the Earth, at least .001 of its volume lies in some candidate star system.

Here's SETI making contact with 3 star systems:

Problem 1: Can the crew plot a straight-line course that will put them in contact with infinitely many candidate star-systems?

Problem 2: What if their navigation equipment forces their initial bearing to be a rational angle?

Problem 3: There's all sorts of potential crazy variants: do the stars move? How fast? Adaptively or non-adaptively? But I've already wasted enough time on this problem, so dear readers, I leave it to you.