The math behind the card game "Spot It"

"Spot It" is an incredibly fun card game where you try to match objects printed on round cards — and the twist is that there's aways one, but only one, matching object on each card. (Previously.)

I've played this game with my family for years, and it's delightful. It's pitched as a game for younger children, because it's so easy to learn and hard to master; but my kids are now teenagers and we still play it. (Part of the fun is how much screaming we do while playing, lol. It's very tense!)

Anyway, I'd always wondered — how did they make the deck? How did they ensure that, in a deck with 55 cards — and 57 possible symbols — there is always one, but only one, match between any two cards?

Over at Smithsonian, Linda Rodriguez McRobbie wrote a fascinating piece on the history of the game and its underlying math.

The historical details are super cool. The story begins in 1850, when the clergyman and amateur mathematician Thomas Penyngton Kirkman created a nifty puzzle that he published in a magazine called The Lady's and Gentleman's Diary

The question read, "Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast."

You can see how this sort of matches the problem behind a Spot-It deck, right? In both cases, you're taking a pile of unique items — schoolgirls, in this case — and arranging them so they each match up pairwise, once, with each other.

Anyway, how did this lead to Spot it? Well, "Kirkman's Schoolgirl Problem", as it's called, became a famous problem in a branch of logic known as combinatorics; a theorem to solve it finally emerged in 1968; which got computer scientists totally interested in it; which led to a French math enthusiast using the theorem in 1976 to create a card-game where you matched insects; which sat in a drawer until 2008 when a distant relative saw some of the cards and … created Spot It.

McRobbie's written a really fun story, worth reading in full! The theory behind the math really crystallizes towards the end of the piece, when McRobbie shows a geometric diagram that illustrates how a simple deck with three objects per card would work. Make sure you read to the end.

(There's also a fascinating book, The Fifteen Schoolgirls, about Kirkman's problem and the math behind it.)