There are already few partially duplicate questions about Dobble (SpotIt cards) :
What is the algorithm to generate the cards in the game "Dobble" ( known as "Spot it" in the USA )?h
Dobble card game - mathematical background
How many combinations can be made with these rules? (game of Dobble)
Here is the summary that answers this question :
Assuming that we have cards with $x = q+1$ symbols, the set of Dobble cards is based on Projective planes (see Wikipedia for Projective planes) of $q$th order. I.e. if there is prime number $p$ and positive integer $i$ such that $q = p^i$, then maximal number of cards is $q^2+q+1$ and the same is necessary number of symbols. If $i = 1$, then Incidence matrix for generating Projective plane in Paige-Wexler normal form may easily be generated using Galois Filed $GF(p)$. If $i$ is not 1, then it is a little bit problematic to generate corresponding Galois Field $GF(p^i)$ consisting of polynomials, so it is much harder to generate corresponding projective field and game of Dobble. Available Dobble sets (with $x = 8$ and $x = 6$) have two and one missing card respectively (i.e. 55 instead of 57 and 30 instead of 31).
There are additional interesting properties that are fulfilled by the projective field and would be fulfilled for set of Dobble cards if it had all 57 (or 31) cards:
- Every pair of symbols appears on exactly one card
- Every symbol appears on exactly $x$ cards
- Role of symbols and role of cards may be interchanged so that the incidence between symbols and cards is still maintained (principle of self-duality).
For $q$ not equal to prime power, the projective plane does not exist, so more general Steiner system specifically $S(2,x,y)$ have to be used, which is more difficult to construct generally.