A puzzle by V. Dubrovsky, from Quantum, January-February 1992:

In a certain planetary system, no two planets are separated by the same distance. On each planet sits an astronomer who observes the planet closest to hers.

Prove that if the total number of planets is odd, there must be a planet that no one is observing.

(In reply to

re: soln by Kenny M)

The case of "all field planets" doesn't exist. The two planets that have the smallest separation, by definition, form a binary pair. I added this fact to the proof. Thanks.

*Edited on ***May 1, 2023, 11:46 pm**