This is a very common puzzle, many variants of which one would have. I still like it as the answer is very elegant, and teaches children a methodology. This one is from Mathematical Circles (Russian Experience) by Dmitri Fomin, Sergey Genkin and Ilia Itenberg.
A person has 10 friends. Over several days he invites some of them to a dinner party so that the company never repeats itself i.e. the exactly the same set of people cannot repeat itself on any of the days. He may, however, not invite anyone on one of the days. For how many days can he continue to invite people without repetition?
As always, please send your answers as comments within the blog (preferred), or send an e-mail to email@example.com. Please do share the puzzle with others if you like, and please also send puzzles that you have come across that you think I can share in this blog.
Happy partying…and there will be a lot of it!