More Games Involving Geometry: EvenQuads
By Gary Gordon and Liz McMahon
Liz McMahon (left) & Gary Gordon (right)
In our last episode, we saw how the card game SET® is closely tied to affine geometry. [See https://www.mathvalues.org/masterblog/on-your-mark-get-set-geometry for the details.] SET cards have four attributes – number, color, shading, and shape – and each of those attributes has three possibilities: the number is 1, 2, or 3, the color is red, green, or purple, and so on.
What if we want four possibilities for each attribute? For example, what if we allowed 1, 2, 3, or 4 symbols on a card, and we added a fourth color, and a fourth shape? Could we still construct a meaningful game? One approach might be to simply mimic SET by adopting the all-the-same or all-different rule, but there’s a problem. If two cards have two attributes that differ, then there would be two ways to complete a "set" (1 Red …, 2 Blue … could be completed by 3 Green … and 4 Yellow … or by 3 Yellow … and 4 Green …). In this post, we’ll describe how Lauren Rose and her student Jeff Pereira were able to overcome this difficulty to create such a game. It’s the story of EvenQuads.
Before describing the game and its connections to geometry, we encourage you to get one (or both!) of the decks of cards from the Association for Women in Mathematics, the organization that markets the cards. And, as a bonus, each beautiful AWM card also contains a mini-biography of a woman mathematician.
In the EvenQuads deck, each card has:
Four kinds of symbols,
Four colors,
1, 2, 3, or 4 symbols.
(Important: if you get Deck 1 and Deck 2, then they are different: Deck 1 has colorful symbols on a black background, and Deck 2 has black symbols on a colorful background. We'll see in a bit how these two decks can be merged to make a super-deck.)
If you like counting things, this would be a good time to convince yourself that one deck has a total of 43 = 64 cards. Now define a quad as a collection of four cards in which, for each attribute, you have exactly one of the following:
everything is the same;
everything is different;
there’s a 2-2 split.
The first two rules are identical to the rules for SET. The last rule is the key to making the game with four expressions for each attribute work.
First, we'll look at a few examples to understand the game. Here are some collections of four cards (using both decks, but not yet mixing the decks) – try to figure out which ones are quads. Answers are below, but don’t peek!
A.
B.
C.
D.
Peek here: For the four cards in A, we see each of the four numbers, each of the four colors, and each of the four shapes, so this is an all-different quad. For B, numbers and shapes split 2-2, but there are only three colors, so this isn’t a quad: the 2-1-1 split is not allowed for an attribute. The cards in C also fail to form a quad (color and shape split 2-2, but the 2-1-1 split for number is illegal), while D is a quad because we have a 2-2 split for number and shape with all different colors.
Note that the cards in A and B are from a deck where the cards have colorful shapes on a black background (AWM's Deck 1), and C and D are from Deck 2, where the cards have black shapes on a colorful background. You can combine these two decks to produce a 128-card game playing extravaganza. A quad either has all its cards from a single deck, or a 2-2 split. (With only two decks, it’s not possible for the cards to come from four different decks. That requires four decks, of course, which amount to 256 cards. AWM has plans to produce two more decks. We’re super excited.)
For another quick quiz, here are two more collections of four cards that use cards from both decks. One of these is a quad, but the other one isn’t. Figure out which is which and why.
E.
F.
Answer: E is good, F isn't. We won't tell you why.
Returning (briefly) to SET, the most important property that SET cards have is the Fundamental Theorem of SET:
FToS: Given any 2 cards, there is a unique card that completes a Set.
This property is what allows us to interpret the cards as points and the Sets as 3-point lines, connecting the game to geometry.
What should the corresponding property look like when working with quads? If you guessed
FToQ: Given any 3 cards, there is a unique card that completes a quad,
you would be right! For fun, let’s check this property for these 3 cards:
Working attribute by attribute, we can figure out the unique card that completes the quad.
Number: Since we have two 1’s and one 3, we need another card with three symbols.
Color: The colors are all different, so the symbols on the missing card are red.
Shape: We have two cards with nautilus shells and one card with icosahedra, so the missing card must have icosahedra.
This means the card that completes the quad has three red icosahedra. Note that we used the 2-2 split rule twice, for number and shape. And it’s not too hard to see why this procedure will always work. There are three ways an attribute can appear in three cards: all the same, all different, or there’s a 2-1 split. In each case, there’s a unique way to satisfy the EvenQuad attribute rule. This can help us understand why we need the 2-2 split rule in the definition. If all you care about is all-the-same or all-different, then the three cards above will never be in a quad.
In SET, the cards correspond to points and the Sets are lines in the associated affine geometry AG(4,3). For quads, the cards will again correspond to points, but the quads correspond to planes in the affine geometry AG(6,2). This notation is understood as follows: A (affine), G (geometry), 6 (dimension), 2 (points per line).
Here's another cute parallel to SET: If you take a single card in SET, then the remaining 80 cards in the deck can be partitioned into 40 pairs, where each pair makes a set with the given card. (This uses the FToS: pick any new card, and there's a unique card that completes the set. Put that pair aside, and repeat. You'll never need a previously used card.) In EvenQuads, take two cards. Then the remaining 62 cards also pair up so that each pair completes a quad with the given pair, partitioning the rest of the deck. (This uses the FToQ: pick any new card, that card plus the two original cards are three cards which have a unique fourth that makes a quad. Again, no repeats.)
Here’s one more tantalizing geometric trick.
If ABCD and ABEF are both quads, then so is CDEF.
You can work this out directly: given that ABCD and ABEF are quads, think about what the possibilities are for each attribute in the collection CDEF. For instance, if A and B differ in color, then C and D must either use the same two colors as A and B, or C and D use the other two colors (because ABCD is a quad). The same argument applied to the quad ABEF tells us that E and F either use the same two colors as C and D or the other two colors, so CDEF either have a 2-2 color split or show all four colors. Similar arguments work for the remaining cases. We encourage you to think through the details! (By the way, this property that quads enjoy and the facts about deck partitioning both have interpretations that use matroid theory! Contact us if you want more details.)
How can we visualize quads? The geometry AG(6,2) is 6-dimensional, so visualization can be tricky (to say the least). But we can visualize smaller dimensional sub-structures. The cube AG(3,2) is pictured below. We have placed cards at the corners of a cube so that each face of the cube is a quad. In addition, there are lots of other quads corresponding to planes in this cube, some of which are “twisted” in a certain way. We’ll see exactly how “twisted” below
How many quads are there among these eight cards? We’ll use the FToQ (“three cards uniquely determine a quad”) to count them. We first choose three cards from these eight in 8 choose 3, or 56, ways. But this counts each quad four times since the “final” card in the quad could have been any of the four cards in the quad. This gives 56/4 = 14 quads among the eight cards; the same argument works for the entire deck.
Can we actually see all these quads? Yes! First, there are six faces of the cube, and each is a quad. In addition, there are another six that arise from parallel diagonals on the faces (for example, 1 Red Circle, 3 Yellow Nautili, 2 Green Squares, and 4 Blue Icosahedra in the cube above). These 12 quads are all ordinary Euclidean planes intersected with the corners of the cube.
Where are the last two quads? Here’s a cool fact: there are two regular tetrahedra sitting inside the cube, and the vertices of each tetrahedron form a quad (for example, 1 Red Circle, 3 Yellow Nautili, 2 Blue Squares, and 4 Green Icosahedra). That gives us our 14 quads!
The same counting approach works to count the total number of quads when using all 64 cards – you should find that there are a lot of them. (Can you find a geometric way to describe the quads in that 6-dimensional hypercube?) And, as a rainy day activity, try drawing the full 6-dimensional hypercube placing cards at each of the 64 vertices.
Note that the familiar Euclidean geometry property “three points determine a plane” translates directly into the FToQ, “three cards determine a quad.” Then finding planes in AG(6,2), which sounds hard, is equivalent to finding quads, which we hope is an easier task. So, in addition to using the geometry of AG(6,2) to help understand the game, we can also use the game to understand the geometry AG(6,2). The same back-and-forth happens with SET and the geometry AG(4,3).
As in SET, it’s possible to associate vectors to the cards. In the case of EvenQuads, each vector has length 6, with entries of 0 or 1. Typically, the first two positions (which can be 00, 01, 10, or 11) can encode the number of symbols on a card, the next two encode the color, and the last two encode the shape. It’s a quick step to take those coordinates to produce a cube where four points form a plane precisely when the corresponding vectors sum to the 0 vector. That’s worth highlighting:
Four cards form a quad if and only if the corresponding vectors sum to 0 (mod 2).
There are more fun facts about quads that follow from the geometry. For one, parallel quads make sense. Of course, if you look at the cube above, you can pretty easily spot some parallel quads. But there are also some surprising ones. Can you figure out what makes two quads parallel? (Hint: the two tetrahedra are parallel. That’s a mind-bender!)
Finally, both SET and Even Quads are based on affine geometry. In future posts, we’ll look at games that use projective geometry. Stay tuned!
Editor's Note: A new EvenQuads deck was recently released and is available here.
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.