I have played the game many times. It still seems like magic.
Out of the 8 symbols on each card in the card deck, there are always two symbols that match when you compare two or more cards. I know it has something to do with math (magic) But how?
In: Mathematics
It’s all to do with an area of maths known as combinatorics. There’s a thorough explanation [here](https://www.smithsonianmag.com/science-nature/math-card-game-spot-it-180970873/) of the story behind it, but you can start small and build up.
Two cards, 1 symbol on each.
Three cards, 1, 2 & 3. Card 1 has A&B, card 2 has B&C, card 3 has A&C.
An easy approach for a deck of N cards would be to make each card have N-1 symbols, one matching each of the other cards. Each symbol would only appear on two cards.
But Dobble manages to be more efficient. It works out how you can use fewer symbols, and still have each pair of cards share one and only one.
The way to achieve this is by using [geometry](https://en.m.wikipedia.org/wiki/Dobble). Instead of thinking of cards and symbols, think of points with lines between them. Each point represents a single card, and each straight or curved line a symbol.
You can arrange 7 points to form a triangle. 3 mark the corners, 3 mark the middle of each edge, and 1 marks the centre. Now you can draw 7 lines to connect them. One line goes between each pair or corners (crossing through the edge points), so 3 lines. Another line goes from each corner to the middle of each edge through the centre, so another 3 lines. The final line is a circle going around (but missing) the centre, where it goes through each edge point. Now each of the 7 points is connected to each of the others exactly once. This would be a deck of 7 cards, with 7 symbols, 3 on each card.
For real Dobble, we use 8 on each card. Crunching the numbers, we get 57 symbols on 57 cards. That is, a Dobble deck has 57 potential cards using the 57 symbols. However, the deck only has 55 cards. There are an extra two cards that never got made, probably because 57 is prime but 55 can be divided by 5 🙂
[Here’s](https://puzzlewocky.com/games/the-math-of-spot-it/) a blogpost about the math behind Spot It/Dobble, although it’s not quite at an ELI5 level.
https://www.petercollingridge.co.uk/blog/mathematics-toys-and-games/dobble/#:~:text=The%20real%20Dobble%20deck%20has,total%20of%201485%20different%20symbols.
Here’s a guy who crunched out a bunch of math and analysis about Dobble. There’s more than one way to make a deck that works, and he says if you have *s* symbols on each card then you can have up to *s*^2 – *s* + 1 cards in the deck and still have one duplicate in every possible pairing, if you lay out the symbols correctly.
With 2 symbols per card your max deck size is 3, with 3 symbols it’s 7, with 4 symbols it’s 13, with 5 symbols it’s 21, with 6 (the kid’s game) it’s 31, with 7 it’s 43, and for 8 symbols the limit is 57. (He calls these max deck sizes *Dobble numbers*; not sure if that’ll catch on.)
Instead of 57, they made the adult game with only 55 cards; guess they thought that sounded better. Or maybe it allowed them to have one less page in each print batch.
This is way too hard to explain in text only, read this blog post:
https://puzzlewocky.com/games/the-math-of-spot-it/
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