There's Math Behind Dobble
There is this game called Dobble. It consists of 55 cards as shown below:
The game is based on an interesting property of these cards: pick any two cards (each of which will have eight symbols), and they will have exactly one symbol (out of 57 possible ones) in common. Try it out!
It’s clear that a game like this can’t have too many cards. Adding new cards, you would eventually run out of options that wouldn’t match more than one symbol of some other card. The good news, if you lose a card, the game still works.
Having fewer symbols, either per card or in the alphabet, has a similar effect: the size of the deck you may be able to create decreases.
For many choices of symbols per game and card, you can cobble together at least a small deck that technically works, but most of them will be lopsided in that some symbols get used a whole lot more than others. If you go through actual Dobble, you find that symbols occur with almost the same frequency.
It turns out that some combinations of numbers result in a perfectly symmetrical, large deck. These numbers - size of deck, symbols per card, and number of symbols - are tightly coupled, through the absolutely nontrivial concept of finite projective planes.
Others have written about this many times over123, and I wouldn’t do a good job at adding another version of the details, so here it will be broad strokes, with an invitation to dive into the links below or search the internet for more details.
The idea is to think of symbols as points, and of each card as a line. To say that any two cards share a point becomes saying that any two lines intersect. At first glance, this might seem like a straightforward analogy: two different lines either intersect in one point, or not at all (so maybe we’ll avoid those lines somehow), and there aren’t any special points or special lines. But of course lines don’t have exactly eight points.
The real analogy is more complex than just a real plane because of the two gotchas above. Mathematicians, inspired, presumably inspired by how railroad tracks seem to “meet” at infinity, have studied “projective planes”, which have the feature that all lines intersect (and any points can still be connected with each other). They also found, and studied, such planes with only a finite set of points.
One (not I) can show that in this last case, all lines have the same number of points (call that P), and that there are as many lines as points, counting
In Dobble, P=8 (symbols per card) and we expect 57 lines (cards) and points (symbols). This almost matches, but Dobble has only 55 cards - presumably because giving out 57 cards would make people doubt that cards were missing; effectively they ship the game with two cards pre-lost by design.
When P-1 is a prime power (for example 7^13, or, Dobble, 7^1), there is a standard (though very abstract) way to construct such a finite projective plane. Interestingly, with very few exceptions, for other values of P whether a corresponding plane exists is an open problem.
In a sense, Dobble sits right next to a hard unsolved problem. Not bad for a game for toddlers.
Stand-up Maths: How does Dobble work? (YouTube)