Secret Santa schemes

In Secret Santa, rather than each person in a group getting a present for each other person, everyone receives just one present, from one person assigned to choose theirs. How do you decide who gets who in secret santa? Here are some of the things you might want to be true (with what I hope are some suitably fancy-sounding names):

Secrecy

Weak

No-one knows who has them

Strong

No-one knows who anyone has (except themselves)

Mixing

Minimal

No-one is assigned themself

Medium

No pair are both assigned each other

Strong

Everyone in the group is part of a single chain of gifts

Knowledge

In a group where some people don’t know others, you may want to ensure everyone is assigned someone they know.

Here are some comments on these conditions:

• In fact the ‘minimal mixing’ conditions is redundant: if no-one can know who has them, necessarily no-one can be assigned themselves. I include as a separate requirement for emphasis.
• Why wouldn’t we want pairs assigned each other? I suppose it feels against the spirit of the thing, maybe because you get to directly compare who gave the better gift.
• ‘Strong mixing’ can makes for fun present giving sessions, where I give my present first, then the receipient gives their present, and so on, and we never have to start again with someone new.
• The ‘knowledge’ condition can also adapt to impose other types of restriction. For example, you might not want for anyone to be assigned their spouse, or their ex. Let’s focus primarily on groups where everyone knows (and likes!) each other so we can ignore the ‘knowledge’ requirement.

Puzzle:  

How many people do you need before it’s possible to meet both a mixing condition and a secrecy condition? Have a think about it before you scroll down.
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