Why is there always one and only one solution to the card game Dobble (Spot it)?

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For a bit more info: Dobble is a card game where you lay a card on the table with 8 different icons/objects. Each player has to call whatever the card in the middle of the table has in common with the card they are holding. For whatever reason, there is only ever one card that matches never more never less. Despite there being like 57 symbols. How is this possible?

In: 7

Basically, they are designed that way using math.

More details: https://youtu.be/VTDKqW_GLkw

Imagine a 2D surface like a (huge) piece of paper. Now draw a bunch of straight lines on paper but don’t make any parallel lines and don’t draw any lines ontop of each other.

Because no lines are exactly the same line and no lines are parallel all lines will have exactly 1 intersection.

Now you can create a unique symbol for each intersection and all symbols which are connected by 1 line are put on the same card. Now we have created playable dobble cards 🙂

But is there always one and exactly one of the same symbols for each card?

Yes, because if there were no matching symbols the lines (which represent the cards) would have no intersection (but they do, cuz they are not parallel) and if there were more than 1 matching symbol on 2 cards we would need more than 1 intersection on the lines but this is also impossible (because no lines are drawn ontop of each other, so we only have 1 unique symbol because we only have 1 intersection).