Once upon a time, a band of colonists, all having distinct birthdays, settled on a remote island. They had brought some mementos from the old country, around which certain mores regarding friendships and birthdays eventually developed:
Making new friends is encouraged, but with care: to ever break off a friendship is unthinkably rude.
Friendships are mutual. To call someone a friend when they would not say the same of you is unthinkably rude.
When someone has a birthday, each of their friends is expected to give them a birthday present of exactly one memento. To not have a memento to give or to give more than one memento is unthinkably rude.
Mementos may be given to someone only on their birthday. To give a memento on any other day is unthinkably rude.
To give a friend the same memento that they gave you on your most recent birthday is tacky. I.e., unthinkably rude.
One fine stormy day, a castaway washes up on the shore with nothing but the clothes on his back and a small portrait of his family, which counts as a memento in the island reckoning. He is brought to the leader who, after making sure that the sailor doesn't share his birthday with any islander, explains the local customs. "You should make at least one friend soon," she says. "To not have any friends would be looked on as unthinkably rude."
"Are there many people here?" he asks. "Many mementos?"
"Yes, very many." she says. "There are $n$ islanders, all perfectly polite, and as many mementos."
The refugee hesitates. He does not want to be rude, and he does not want to put someone else in a tight spot. Can he safely make a friend, or is a scandal inevitable?