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.

Symmetrical Finite Projective Plane Order 3
Symmetrical finite projective plane order 3

For 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. Above is a projective plane of order 3, with 13 points and 13 lines. There are four points on each line and four lines on each point. Below we have the same projective plane rearranged a bit. This diagram would likely be more useful to create a game, as the lines are easier to follow. Switching from Maxime’s system, we could say that each point represents a symbol, and each different color line represents a card that would contain the symbols the line intersects with.

Finite Projective Plane Order 3
Finite projective plane order 3

This shape is a nice illustration of one of the principles of projective geometry. The grey line represents the “horizon,” or “line at infinity.” Just as parallel train tracks appear to meet at the horizon line, so each set of parallel lines — the 3 rows, the 3 columns, and each of the 2 sets of 3 curvy diagonals — meets at an “infinity point” on the grey horizon line.

Finite Projective Plane Order 4
Finite Projective Plane Order 4

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. But we would probably want to rearrange it into a diagram that includes a square grid. We will actually do this for a projective plane of order 7, below.

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

To turn this into a diagram that would be easier to follow for actually creating a game, we can rearrange it into a shape that includes a square grid, as we did for the projective plane of order 3 above. This will be a 7×7 grid. When we transformed the projective plane of order 3 using a 3×3 grid, we only had two diagonals at 45 degree angles. With a larger grid, we will have more diagonals at different angles. To understand this, it may help to start with a Graeco-Latin square. Below is a hyper-Graeco-Latin square of order 7 with 6 “dimensions” — the concentric squares within each of the 49 units of the grid. This is also called a complete set of mutually orthogonal latin squares.

Complete set of mutually orthogonal latin squares order 7
Complete set of mutually orthogonal latin squares order 7

As with all Graeco-Latin squares, for each “dimension,” the same color appears only once in each row and each column. For our purposes, take note of the diagonals. If you follow a color along a dimension, you will see that it follows a certain angle. For example, if you start with the upper right blue square, and follow the blue color along the outermost dimension, it follows a 45 degree angle to the lower left square. However, if you follow the same color from the same starting square along the second dimension in from the outermost, you can see that the line goes two squares left for each one square down, and when it gets to the left edge it must “wrap around” and continue on the other side. For the next dimension in from the outermost, the line goes three squares left for each square down, and so on. Each of these diagonals is a line in the finite projective plane that arises from this set of latin squares. To start making cards for a game, we can place 49 symbols in a 7×7 grid. Just as with the projective plane of order 3, we will also need the “points at infinity,” one for where the rows meet, one for where the columns meet, and one for the meeting point of each of the 6 sets of 7 parallel diagonal lines, defined by the 6 dimensions of the Graeco-Latin square.

Animal Grid

Now we can easily see which symbols should appear on each card. Each line represents a single card, and the symbols the line passes through are the symbols for that card. In the example below, the red line and green line follow the second-from-outermost dimension in the Graeco-Latin square above. Because they are parallel within the grid, they have no symbol in common until the corresponding “infinity point” of the spider is added. The blue line crosses the red line and the green line so the blue card will have exactly one symbol in common with the red card and with the green card. The blue card also needs the infinity point for the columns, which is the crab. There are 8 sets of 7 parallel lines, which accounts for 56 cards. The 57th card (purple below) contains the 8 infinity points.

Spot It Animal Grid With Cards

For more help 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

For more on the relationship between Graeco-Latin squares and finite projective planes, see this explanation of the equivalence. 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. One can be constructed from the other.

Latin Square Order 3 Projective Plane Order 3
A complete set of orthogonal latin squares of order 3 exists if and only if a projective plane of order 3 exists.

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