Challenge 258: Souper Supper
William arranges a weekly meal for his friends.
(i) He has N ingredients in his cupboard and will use them to cook two soups. Each soup must contain at least one ingredient, otherwise his guests will go home hungry. He knows each potential guest dislikes two of the ingredients. When he invites them, he asks which ingredients they dislike and then plans his soups so that every guest can eat at least one of the soups. How many guests can he invite to be certain that he can make a menu that has something for everyone? Give your answer in terms of N.
(ii) For the next week’s meal, William realises he is bored with the previous week's soups, so he decides that one of the two soups must contain at least two ingredients. How many guests can he invite this week and still be sure of catering for everyone?