I have a set of $X$ symbols which I have to put in cards of $N$ (where $X > N$) slots such that every card must have exactly $1$ symbol in common with each other. What I want to know is, how many cards can I form? I don't even know how to mathematically formulate this, how do I go about this?
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Do you mean every set of two cards should have exactly one symbol in common? – 雨が好きな人 May 02 '19 at 14:17
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I meant that every card must have a common symbol with every other card. Like, suppose we have a card with symbols "A B C" all other cards must have one of those symbols – Alexey May 02 '19 at 14:21
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1Well, you need to say more than that. If my symbols were, as you say, $A,B,C$ then I could have infinitely many cards each just using the symbol $A$. That satisfies your stated requirements, but I seriously doubt you would allow that configuration. – lulu May 02 '19 at 14:40
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I would assume the cards have to be distinct. – 雨が好きな人 May 02 '19 at 14:42
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2"Every card must have exactly 1 symbol in common..." It sounds like you are referring to the game "Spot It." See the following: What is the math behind the game Spot It?, What are the mathematical computational principles behind this game?, Algorithm to Create all Spot It cards. – JMoravitz May 02 '19 at 14:57
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@JMoravitz exactly what i was looking for, thank you! – Alexey May 02 '19 at 16:16
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See also this answer for discussion of how your problem relates to Steiner systems. – Mike Earnest May 02 '19 at 16:29