Spot It Combinatorics

Spot It is a kids game. Pretty addictive actually. I believe it’s called Dobble in the UK. There are 55 circular cards with 8 images chosen from 57 possible images. Cards are such that each card has exactly one match with each other card (no more and no less). Each person starts with a card and you flip over a third in the center. Whoever “spots” the match with their own card takes center card, and flips a new card.

Very quick game to play. But it turns out the math behind how to design the spot it cards so that each has exactly one match with the other cards is cool too. I found the below story on the web as I thought some of the math folk here would be interested.

https://www.smithsonianmag.com/science-nature/math-card-game-spot-it-180970873/

Also makes me wish USPagerGames was still here as he was a combinatorics guy.

That indeed looks like an interesting math challenge.

As to the game itself, looks fun! Similar to Set. That's a game anyone can play from age 3 to 103. I find kids to be better than adults, in general.


Direct: https://www.youtube.com/watch?v=Kc4WrL7cxeg.

Quote: article

The Mind-Bending Math Behind Spot It!, the Beloved Family Card Game]The basic structure of the game is this: the deck has 55 cards, with eight symbols on each card, culled from a bank of 57 symbols in total.

Before looking the above quote I created a 9*-card deck where any two cards overlap with exactly one symbol (unless I'm mistaken). If anyone can find an error or an additional card that is 36 or less and meets the one overlap requirement, please let me know.

1,2,3,4,5,6,7,8
1,9,10,11,12,13,14,15
2,9,16,17,18,19,20,21
3,10,16,22,23,24,25,26
4,11,17,22,27,28,29,30
5,12,18,23,27,31,32,33
6,13,19,24,28,31,34,35
7,14,20,25,28,32,34,36
8,15,21,26,30,33,35,36

It seems the general case for the "smallest deck," where every possible card is utilized, and n is the number of symbols per card, then:

The least number of cards needed is n+1.
The least number of symbols needed is n*(n+1)/2.

As the quote above shows, if you add more symbols, then you can get more cards. I think a good problem, probably the problem, is to find the maximum size deck.

That's enough for now.

* corrected

You had me confused there for a minute - "if that's an 8-card deck with 8 symbols per card, and each symbol is on exactly 2 cards, then there should be only 32 symbols, so why are there 36?" - before I noticed that it's actually a 9-card deck.

Quote: ThatDonGuy

You had me confused there for a minute - "if that's an 8-card deck with 8 symbols per card, and each symbol is on exactly 2 cards, then there should be only 32 symbols, so why are there 36?" - before I noticed that it's actually a 9-card deck.

You're right -- typo -- I meant 9. Thank you.

MQW4FLOTUS used these cards as a part of rehab after a brain injury. They're really cool. There's lots of options for play, too.

A cool test would be to take the Spot It deck and then ask how many more cards you can add given 57 symbols and 8 symbols per card. That’s the easy part. The hard part would then be to draw the missing cards.

Don't ask me what any of this means because I don't know. Just watched all of this get solved by hand, I am willing to bet the math is right so looks like the makes of Spot It made a mistake.

n : number of images per card
Mk : maximum number for possible cards
S : number of cards per evenly distributed sets of images (Ω)

Mk = n²-n+1

S =Mk-Ω*(n-1)

Spot It does not have all cards with 1 matching symbol. There are 2 cards missing & the math says with 8 images there should be 57 possible cards, not 55 since 50 is the closest number of cards that can be used with 8 images to have each image match just 1 imagine in every card.