On Your Mark, Get SET, Geometry!

By Gary Gordon and Liz McMahon

As recent retirees, we have been lucky enough to have the time (and the proximity) to volunteer at MoMath, the National Museum of Mathematics, in New York City. We’ve been running Equilibrium, the grown-up game night, where we curate games to present to attendees. It’s a lot of fun, and we’ve enjoyed meeting people who love both math and games.

One evening, we presented Games that Exist Because of Geometry. We presented four games, two based on affine geometry (SET, EvenQuads) and two based on Projective Geometry (Fire and Ice, SpotIt!). That turned out to be a bit ambitious for one evening, but we managed to introduce all four games and explain some of the underlying geometry. Unfortunately, we didn't have time to explore the connections between the games in depth. Fortunately, the opportunity to write for this blog came along, so we can take a deeper dive into the connections between the games and math. A thorough treatment is a book-length project (want proof? – see the comments below about The Joy of SET), so we decided to write a series of blog posts, to have more time and space to play.

During Equilibrium, we first presented a game, then let the participants play for a while. After 20 minutes or so of play, we explained some of the connections between the game and finite geometry. This format won't work quite as well in a blog, so instead, for each game, we'll introduce the game and give the reader some opportunities to "play." We will then discuss both the geometry and how knowing the geometry can enhance your appreciation of the various games. This goes both ways, though: the games will also enhance your appreciation of geometry!

SET®. We are going to start with the game SET, because we know it very well, and because it's probably the best-known of the games we'll consider. There's another possibly surprising reason we should start with SET: it's the only one of the games we'll look at that wasn't actually invented because of geometry. However, we believe that the connection to geometry is the reason the game is so popular.

SET is played with a special deck of cards. Each card in the deck has symbols, characterized by four attributes.

NUMBER: There are 1, 2, or 3 symbols.
COLOR: The symbols are red, purple, or green.
SHADING: The symbols are empty, striped, or solid.
SHAPE: The shapes are ovals, diamonds, or squiggles.

In this game, a set is three cards where each attribute is (independently) either all the same or all different. That means that if there’s any feature where two attributes are one thing and one isn’t, it’s not a set. Important note: The number of attributes that are the same can vary.

**Figure 1:** Examples of sets. Check that the top set has exactly three attributes the same, the second set has exactly two attributes the same, the third set has exactly one attribute the same, and the last set has no attributes the same.

In the figure below, each of the two collections of three cards is NOT a set. In each case, which attribute (or attributes) keeps it from being a set?

**Figure 2:** Why are these not sets? Raise your hand before answering.

The game of SET is played by as many players as you like. Begin by laying 12 cards face up on the table. Each player looks for sets simultaneously. The first person who finds a set yells “Set!” and removes those three cards from the table. Those cards are then replaced, and everyone continues to look for sets. If at some point the players agree that there aren’t any sets among the cards on the table, three additional cards are added. (This isn’t quite the same thing as saying there is no set among the 12 cards, but that’s another story.) If a set is found among this larger group of cards, it is taken, but three new cards are not added unless everyone agrees that they can’t find any sets in the new layout. Once the deck is played out and no one can find any sets in the remaining cards, the winner is the person who took the most sets. See the figure below for a typical layout of 12 cards.

**Figure 3:** There are three (disjoint) sets among these cards. Find them!

This would be a terrific time to either get yourself a deck of SET cards and start playing, or, if you don't have one, you can go to https://www.setgame.com/set/puzzle, where there is a Daily Puzzle, which always consists of 12 cards that contain exactly six sets. (For the puzzle, cards are not removed when a set is identified.)

There's one more important thing to know.

The Fundamental Theorem of SET: Given any two cards, there is a unique third card that makes a set.

As an example, consider the first two cards in the non-set on the left in Figure 2. Since the first card is red and the second is green, the third card must be purple. Do the same thing for the other three attributes—you should end up with 3 purple shaded squiggles. And it’s clear this will always work; for each attribute, if the two cards are the same in that attribute, the third must be as well. If the two are different, then the third card must be the missing expression of that attribute.

Play time! We recommend that you get your deck (because if you didn't have one before, by now, you've surely gotten one!), pick pairs of cards, and verify that this is true. Then maybe play a game. Or SET Solitaire, where you just play against yourself.

Welcome back! Now that you've played a bit, let's discuss geometry.

Some affine geometry. Finite planar affine geometry has three axioms. (Some sources say four, and what those axioms are can vary, but we'll stick with three.) The first axiom, that two points determine a unique line, is what makes a geometry. The second, that there are at least three non-collinear points, ensures that the geometry is interesting. The last is the big one: the Parallel Postulate says that given a line and a point not on the line, there is a unique line through the point parallel to the given line.

The first two axioms clearly hold for the SET deck, where the cards correspond to the points, and the sets correspond to the lines. The problem with the Parallel Postulate is that it’s tied to points in a plane (whatever that is in this context). Just as the Parallel Postulate doesn't hold in 3-dimensional Euclidean space (think about skew lines), the Parallel Postulate doesn't hold for the full SET deck. Can we use a subset of the cards and have this axiom hold, too?

The answer is yes. (It’s the kind of rhetorical question mathematicians like to ask when they feel like they’re losing their audience. If the answer were no, we wouldn’t have asked the question, would we?)

Here’s how this goes: First, choose three cards from the deck that do NOT form a set. We’ll call the cards C1, C2, and C3.

First three cards:

Now pair the cards up (C1&C2, C1&C3, C2&C3), and for each pair, find the card that completes the set (the Fundamental Theorem in action!). This produces our second collection of three cards:

Second three cards:

Now repeat: take these three cards, form three pairs, and complete three more sets. We get the following three cards.

Third three cards:

It’s tempting to do this again, and giving in to temptation is mathematically rewarding. But something puzzling happens this time! When you pair these last three cards up and complete the sets, you get the first three cards back! (This is also a pretty cool trick to show your friends, if your friends are into this sort of thing.)

Wait … what is the connection to geometry? When we show this trick to people, we usually ask them to rearrange the nine cards so that it's easy to see all the sets. Most of the time, they eventually display the cards like this:

**Figure 4:** These nine SET cards form a plane.

How many sets are there among these nine cards? (Don’t peek at Figure 5 yet. OK, now you can peek.) It's easy to see that the three (horizontal) rows are sets, and so are the three (vertical) columns. But you might also notice that the cards along the diagonals form sets, plus there are some “twisted” diagonals: check that the swoopy lines in Figure 5 correspond to sets. We end up with a grand total of 12 sets.

**Figure 5.** The affine plane AG(2,3) has 9 points and 12 lines, and each line has 3 points.

This collection of nine cards is a model for AG(2,3), the Affine Geometry of dimension 2 (meaning it's a plane) and order 3 (meaning there are three points on every line). In fact, the point-line diagram in Figure 5 is unique; it’s the only way to create an affine geometry with 3 points on each line. You can verify that the three axioms listed above are all satisfied (assuming that "two lines are parallel" means "two lines don't intersect”).

But it gets even better, and the connection is deeper. The entire deck is a model for AG(4,3), the 4-dimensional affine geometry of order 3. In Figure 6, you can see all 81 cards in the deck, nicely organized. First, each of the nine 3 x 3 smaller squares are planes like the plane you (well, it was really us) created in Figure 4. But there's more. For instance, if you take the 9 cards in the upper left of each of the nine 3x3 smaller squares, it’s another plane! In fact, there are lots of planes contained in Figure 6. See if you can find a geometric pattern for the location of the planes and the sets in the entire deck.

**Figure 6:** All the cards in a SET deck.

There's plenty more we could tell you about SET and geometry. In fact, if you get The Joy of SET, you can read much more (there are two chapters about geometry), plus there are projects after each chapter, so there's much more to explore. (Full disclosure: we are two of the authors of this book; the other two authors are our daughters.) Enjoy!!

Liz McMahon enjoys working in algebra, combinatorics, and finite geometry, plus reading, cycling, hiking, and traveling.

Gary Gordon is a combinatorialist who ran Lafayette College's REU program for 11 years; he enjoys baseball, climbing things, and the piano.

Liz and Gary both retired from Lafayette College in 2022; their favorite joint collaboration is the book The Joy of SET, which they coauthored with their two daughters Rebecca Gordon and Hannah Gordon.