A basic question about logic

I am trying to solve the following question:

“After a long war with planet Earth, groups of Cylons have decided to leave their mother planet and move to Earth to join the humans. This unthinkable experiment has so far been going so well that the village of Atlantis-on-Earth is entirely made of married human- Cylon couples, with many new hybrids having been born. Suddenly, the information is leaked that some of the Cylons are spying for the enemy. The law of Atlantis is clear on this: once a Cylon is known to be a spy they shall be killed by their partner on that same night. Mere suspicion is not enough. Once a Cylon is killed everyone will know about it.
Humans are blinded by love and do not know whether their partners are spies. But they also love gossip and always know whether others’ partners are, although they’d never dare telling them directly. In other words, humans have knowledge about the others’ partners being spies or not but not their own. Assume, for the sake of this story, that humans are perfectly rational beings.

Assume that it is common knowledge that some Cylons are spies. Will all spies be killed? and, if so, in how many nights?”

My answer to the question is that it takes one night to kill all spies but don’t know how to explain it in logical form.