Spot It! is a fast-paced matching card game. Each of the 55 cards in the deck features eight symbols, and there is always exactly one matching symbol between any two cards in the deck. Your goal is to be the quickest to find the match between two cards. There are a total of 57 different symbols throughout the deck. While playing the game with several family members, my son questioned how it was possible that any two cards always have exactly one match. I tried to explain using the following examples:
If there were 2 symbols on a card, then the deck would contain 3 cards and a total of 3 different symbols.
Let the symbols be A, B, and C.
Then there would be 3 cards: AB, AC, and BC.
Note that between any two of the cards, there is one and only one matching symbol.
If there were 3 symbols on a card, then the deck would contain 7 cards and a total of 7 different symbols.
Let the symbols be A, B, C, D, E, F, and G.
Then there would be 7 cards: ABC, ADE, AFG, BDF, CDG, CEF, and BEG.
Note that between any two of the cards, there is one and only one matching symbol.
Also, note that each symbol occurs on 3 cards.
If there were 7 symbols on each card, what is the maximum number of cards that could be in the deck so that any two cards always have exactly one and only one matching symbol?
Bonus: If there were n symbols on each card, what is the maximum number of cards that could be in the deck so that any two cards have exactly one match?