I saw a card game designed for small children. Each card has a picture of 6 animals on it, and there are 31 cards. When any two cards are compared to each other, they share exactly one animal. The game is for the child to find the animal that is common to both cards. I began to wonder how many animals the company would need to draw in order to produce such a set of cards.
So here's the question. Given N cards with K animals on each card, what is the minimum number of unique animals required such that there can exist at least one collection of animals on cards where each card shares precisely one animal with each other card?