Sup­pose you go out to eat lunch with a group of friends every day. Say for simplicity’s sake you have N sets of lunch­mates, and you eat with each group every N days on a reg­u­lar cycle.

On a day i of this peri­odic sched­ule, the lunch is shared by n_i of your friends, plus you. You have a total of F lunch­mate friends, and some of those friends may occur in more than one lunch group. Every­body who is a mem­ber of each group always shows up for lunch.

The rule for pay­ing for each lunch is that every mem­ber of a group pays in turn. So if you and Bob and Sally go to lunch on Mon­days, every third Mon­day one of you will buy everybody’s lunch. This is true in every group, regard­less of its size or composition.

For every lunch group with n_i friends, you will end up pay­ing for everyone’s lunch every n_i+1 days.

So: Is there any arrange­ment of your F friends into N sub­sets, such that you end up pay­ing less (over a long time, on aver­age) than any of your friends? Does it mat­ter if some friends are mem­bers of more than one lunch group? You are, after all, mem­ber of all the lunch groups.

There will be a follow-​​on ques­tion, when or if some­body deigns to respond