Finite Projective Planes and the Math of Spot It!

Spot It!Spot It! is a fun game that has some interesting math behind it. I first took notice of Spot It! because its packaging is similar to a game called Name That! that I co-designed, though the games are quite different. To play Spot It!, you turn over two cards and find the matching symbols. Any two cards will have one and only one matching symbol in common. According to the game documentation, there are 55 cards with 8 symbols on each card and “more than 50” different symbols used in the deck.

Spot It! cardsWhen my brother and I sat down to play this game, we immediately started thinking about the math and wondering about the different combinations that one could use to create simpler or more complex versions of this game, and how many cards and symbols would be necessary in the various configurations.

One way to understand the math of Spot It! is to look at the game as a physical expression of a finite projective plane, which is a set of points and lines such that any two points have only one line that passes through them, and any two lines have only one point in common. This describes the math of Spot It! well. Besides the fact that any two cards have exactly one symbol in common, it is also true that for any two symbols, there is one and only one card that contains those two symbols. A simple type of finite projective plane is the Fano Plane shown below. It is a projective plane of order 2. It has seven points and seven “lines,” one of which is circular. As you can see, it meets the requirements discussed: any two points have exactly one line that intersects them both, and any two lines have exactly one point in common. For this plane, there are three lines through every point, and three points on every line. This led Maxime Bourrigan (use Chrome to read his page in English) to create the game shown below, which he calls Mini Dobble. (Dobble appears to be a French version of Spot It!)

Fano Plane and Mini DobbleSo this is the smallest playable version of Spot It! possible: seven cards and seven symbols, with three symbols on each card. (Actually, by some definitions, a finite projective plane of order 1 is a simple triangle. Based on that, you could have a game of Micro Spot It! with only three cards, three symbols, and two symbols on each card, but it would be a very short game!) From here, we can create more complex versions of the game based on finite projective planes of greater orders.

Projective Plane Order 4For a projective plane of order n there will be n2 + n + 1 points and lines, with n+1 points on each line, and n+1 lines on each point. Here is an illustration of a projective plane of order 4, provided by Ed Pegg, Jr. (who also created another game based on the Fano Plane). This plane has 21 points and 21 lines, with five points on each line and five lines on each point. The small grey circles are the points and each different color squiggle is a “line.” By following an individual colored line, you can see that it intersects with exactly five points. Each point has exactly five different color lines passing through it. So from this model, we could create a 21-card game with 21 symbols and five symbols on each card.

The actual game of Spot It! turns out to be based on a projective plane of order 7. That would give us 57 cards and 57 symbols, with eight symbols on each card. (Apparently, the makers of Spot It! decided to only use 55 cards, though the game could have had 57.) Here is an illustration of a projective plane of order 7, courtesy of Wolfram Alpha.

Projective Plane Order 7

If you are interested in creating your own game based on a finite projective plane, you may want to check out this thread or this one on Stack Overflow.

Graeco-Latin Squares

Finite projective planes also have an interesting relationship to Graeco-Latin squares: a Hyper-Graeco-Latin square of order can have n-1 dimensions if and only if there is a finite projective plane of order n.

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