On the existence of "Spot It!" decks that are not projective planes

Abstract: The game "Spot It!" is played with a deck of cards in which every pair of cards has exactly one matching symbol and the aim is to be the fastest at finding the match. It is known that finite projective planes correspond to decks in which every card contains $n$ symbols and every symbol appears on $n$ cards. In this paper we relax the hypothesis on the number of cards on which a symbol appears: we study symmetric decks in which every symbol appears the same number of times and we introduce the concept of maximal deck, providing a sufficient condition for a deck to have this property. We also produce various examples of interesting decks which do not correspond to projective planes.