Story problems in math rarely overlap with story problems in screenwriting, but today I have one that I could use some help figuring out.
Far away on a distant planet, an intelligent but very paranoid species is constructing a series of terrestrial watchtowers to scan the heavens, making sure no space-traveling enemies sneak up on them.
You can think of these watchtowers as observatories, each one watching a 180-degree (half-sphere) swath of the sky. For this simplest version, you can ignore complications like atmospheric distortion or possible moons.
**Question #1: For complete coverage, what is the minimum number of watchtowers they need to build?**
**Question #2: What would be a *prudent* number to build? If you want to introduce features like atmosphere or redundancy, go for it.**
This planet’s new Grand Ga’loo was elected on a promise of putting the watchtowers into orbit. After his inauguration, he’s assembled a team of leading scientists to figure out a plan for doing so. The current thinking is to have the satellites be geo-stationary (staying fixed over one spot on the planet), but the Ga’loo can be persuaded otherwise.
**Question #3: What is the minimum number of satellites needed?**
**Question #4: How does the number change if the orbit radius is increased? If the field of view is increased?**
Your answers could help some paranoid aliens sleep better at night.