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I was exploring the math behind the popular game SpotIt (no affiliation), and realized that the cards represent a complete graph, where the vertices are the cards and the symbols are the edges.

I wanted to know how many symbols you would need for a certain number of cards, so I started drawing, and quickly realized that the number of symbols for n cards corresponds with the triangle number for a triangle of height n - 1.

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I did some online research and tried puzzling it out on my own (with limited time, unfortunately) and so I humbly turn to the math exchange community to help me understand intuitively why this relationship exists.

Erty Seidohl
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    What have you tried? This is a (fairly) simple combinatorics argument. This question will likely be closed if you don’t show more effort on your part :) – Malady Jun 04 '25 at 14:40
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    Oh wait, I think I just had an insight: Let's take n = 6. You need 5 edges to connect the first node, 4 edges to connect the second, and so on. Summing these gets a triangle number! – Erty Seidohl Jun 04 '25 at 14:40
  • In what sense do you need $\frac{n(n-1)}{2}$ symbols for $n$ cards? The default Spot it! deck has $55$ cards, but many fewer symbols that $\frac{55 \cdot 54}{2} = 1485$. – Misha Lavrov Jun 04 '25 at 15:42
  • Ooh, wait, yeah - good point. I need to rethink my finding here. – Erty Seidohl Jun 04 '25 at 18:51

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After writing this up, I realized the answer:

Let's take n = 6. You need 5 edges to connect the first node to every other node, 4 edges to connect the second (since the first node is already connected), and so on. Summing these gets a triangle number.

Erty Seidohl
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  • In the future, don’t answer duplicates :) very good for figuring it out though! – Malady Jun 04 '25 at 15:19
  • @Malady I don't think you can blame someone for self-answering, even if it's a duplicate. You might say that searching more would have avoided the need to ask the question in the first place, but if someone's already asked a question, they're obviously not aware that it's a duplicate, so there's no reason not to post an answer. – Misha Lavrov Jun 04 '25 at 15:46
  • @MishaLavrov I agree, I was trying to let them know of site policy. I don’t think I was rude or “blamed” them, but if I was or did I apologize :) – Malady Jun 04 '25 at 15:52
  • No worries. I did search before, but I was specifically looking for "triangle number" which I didn't find :) – Erty Seidohl Jun 04 '25 at 18:52