Understanding the math behind Dobble

Definitions:

  1. Let there be \(N\in \mathbb N\), \(N>1\) unique symbols descibed by the set \(\Omega\).
  2. Each subset of \(\Omega\) is called a card.
  3. A set of cards \(D\) is called a deck iff all cards have the same number \(n\) of symbols and the intersection for any two distinct cards in the deck has exactly one element, i.e. \[ \forall c_1, c_2 \in D, c_1 \neq c_2\exists n\in \mathbb N: |c_1 \cap c_2| = 1, |c_1| = |c_2| = n \]
  4. A good deck is a deck where all symbols appear equally often. In this sense the distribution of symbols is optimal, hence good.

Examples:

  1. The emtpy set is a good deck.
  2. Any set with one card is a good deck.
  3. Let \(N=6\), \(\Omega = \{a, b, c, d, e, f\}\), \(n=3\).
    • \(\{ \{a, b, c\}, \{a, d, e\} \}\) is a deck.
    • \(\{ \{a, b, c\}, \{a, d, e\}, \{b, d, f\}, \{c, e, f\} \}\) is a good deck, as each symbol appears exactly twice.

Questions:

  1. Given \(N\) and \(n\), does a good deck exist? How many cards does it have?
  2. Can multiple good decks exist that are not isomorphic to each other? When?
  3. The demo deck of Dobble has 31 symbols, 6 symbols per card, 16 cards. Why 16? Is it good?
  4. Is the Dobble deck good?
  5. For given \(N, n\), do all good decks have the same number of cards?

Propositions:

  1. Any pair, triplet, … of symbols may only appear once on a card throughout any deck (else the intersection of two cards was larger than 1).
  2. If \(n \geq N / 2\) there never exists a good deck.
  3. If \(N = 2\), there exists no good deck.