The geometry of the card game SET...The rules of SET To start the game, twelve SET cards are dealt face up. All players look for SETs simultaneously. If a player ﬁnds a SET, they

The geometry of the card game SET

Loyola Marymount UniversityMath Club

Robert WonUniversity of Washington

September 19, 2019 1 / 20

The card game SET

• SET is played with a special deck of cards. On each card, there aresymbols with four attributes:

Number: 1 2 3Color: red purple green

Pattern: empty striped solidShape: ovals diamonds squiggles

• Q: How many cards are there in the deck?• A: There are 3 · 3 · 3 · 3 = 34 = 81 cards.

September 19, 2019 An introduction to SET 2 / 20

The card game SET




• Q: How many cards are there in the deck?

• A: There are 3 · 3 · 3 · 3 = 34 = 81 cards.


The card game SET




• Q: How many cards are there in the deck?• A: There are 3 · 3 · 3 · 3 = 34 = 81 cards.


The card game SET

A SET is a collection three cards such that, for each attribute, the cardsare either all the same or all different.

The number of attributes that are the same can vary.

Shape and pattern are the same,color and number are different.

All attributes are different.


The card game SET






The card game SET






The card game SET






The rules of SET

• To start the game, twelve SET cards are dealt face up.

• All players look for SETs simultaneously.• If a player finds a SET, they take it and three new cards are dealt.• If there are no SETs, three more cards are dealt. The three cards are

not replaced on the next SET, reducing the number back to twelve.• The player who finds the most SETs is the winner.

SET is a popular game among mathematicians (and amongnon-mathematicians). So popular that...


The rules of SET

• To start the game, twelve SET cards are dealt face up.• All players look for SETs simultaneously.

• If a player finds a SET, they take it and three new cards are dealt.• If there are no SETs, three more cards are dealt. The three cards are




The rules of SET

• To start the game, twelve SET cards are dealt face up.• All players look for SETs simultaneously.• If a player finds a SET, they take it and three new cards are dealt.

• If there are no SETs, three more cards are dealt. The three cards arenot replaced on the next SET, reducing the number back to twelve.

• The player who finds the most SETs is the winner.



The rules of SET

• To start the game, twelve SET cards are dealt face up.• All players look for SETs simultaneously.• If a player finds a SET, they take it and three new cards are dealt.• If there are no SETs, three more cards are dealt. The three cards are

not replaced on the next SET, reducing the number back to twelve.




The rules of SET





The rules of SET








Now time to play!

Get into groups and play some games!

• To start the game, twelve SET cards are dealt face up.• All players look for SETs simultaneously.• If a player finds a SET, they say “SET” and show the other players

the SET. Once everyone has agreed that this is a SET, the playertakes the three cards and three new cards are dealt.

• If there are no SETs, three more cards are dealt. The three cards arenot replaced on the next SET, reducing the number back to twelve.



The rules of SET• To start the game, twelve SET cards are dealt face up.• All players look for SETs simultaneously.• If a player finds a SET, they take it and three new cards are dealt.• If there are no SETs, three more cards are dealt.• The player who finds the most SETs is the winner.

QuestionHow many cards are needed to guarantee existence of a SET?

Question (Rephrased)How big is the largest collection of cards containing no SETs?

• For SET, the answer 20 cards. This was proved in 1971, before thegame was invented! Why was this question being studied?

















Finite affine geometry

• A deck of SET cards forms a finite geometry.• Key observation: Any two SET cards determine a unique SET:

• Geometry axiom: Any two points determine a unique line.• So the cards are the points and the SETs are the lines.

September 19, 2019 Finite affine geometry 8 / 20













Finite affine geometry• Any two points determine a unique line. Any three non-collinear

points determine a unique plane. What is a plane of SET cards?

• This plane is called F23 or AG(2, 3). (2 for the dimension and 3

points on each line).• How many lines does this plane contain? (How many SETs?)








































points on each line).

• How many lines does this plane contain? (How many SETs?)








Every pair of points determines a line, and all of these lines arecompletely contained in the plane. But we’ve overcounted by threetimes. So the answer is

(92

)/3 = 12.



Every pair of points determines a line, and all of these lines arecompletely contained in the plane.

But we’ve overcounted by threetimes. So the answer is

(92

)/3 = 12.




(92

)/3 = 12.




(92

)/3 = 12.


A 3-dimensional flat

Select any remaining card. Along with the three initial cards, thesefour cards should determine a 3-dimensional flat.



Select any remaining card. Along with the three initial cards, thesefour cards should determine a 3-dimensional flat.



A line in this 3-dimensional flat, F33:

AG(3, 3) is represented by three side-by-side 3 ! 3 grids. Again, three points are collinear if their summod 3 is the point (0, 0, 0). In Figure 2, three collinear points are shown.

! !!

Figure 2. AG(3, 3) with one set of collinear points shown.

AG(4, 3) is represented by a 9 ! 9 grid, consisting of nine 3 ! 3 grids. A line will be three points thatappear either in the same subgrid, or in three subgrids that correspond to a line in AG(2, 3). Figure 3 showsa complete cap in AG(4, 3) whose anchor point is in the upper left. You can verify that the cap does consistof 10 pairs of points, where the third point completing the line for each pair is the point in the upper left.

Figure 3. A complete cap in AG(4, 3) whose anchor point is in the upper left.

This grid can also be realized as a way of viewing points with 4 coordinates, where the first two coordinatesgive the subgrid, and the second two give the point within that subgrid. In this case, the cap S picturedabove is the first complete cap lexicographically.

Let S1 be an arbitrary cap with anchor point a. Since the coordinates of a line sum to (0, 0, 0, 0), if thecoordinates for the points in S1 are summed with 10a, the result must be (0, 0, 0, 0), since this is sum of 10lines. Thus, the sum of the points in S1 is "a mod 3.

A computer search shows that any two caps with di!erent anchor points necessarily intersect. This factis interesting; it would be useful to have a geometric proof.

Another computer search verifies that there are 8424 complete caps with anchor a. These are all a"nelyequivalent, and the a"ne transformation that take one to another is actually a linear transformation, sincethe anchor point (0, 0, 0, 0) must go to itself. This reduces the transformations we must consider to GL(4, 3).

It is possible to decompose all of AG(4, 3) into 4 mutually disjoint complete caps with anchor a. Onesuch decomposition, where one of the complete caps is S, pictured above, is shown in Figure 4

Figure 4. A complete cap in AG(4, 3) whose anchor point is in the upper left.

2


A 4-dimensional flatEach card has four attributes: the deck is 4-dimensional, F4

3.


A 4-dimensional flatEach card has four attributes: the deck is 4-dimensional, F4

3.



• And we’re done! But... we’re mathematicians.

• For a fifth dimension, we need a fifth attribute:

• We’ve constructed F53!

• This game is playable (but difficult).



• And we’re done! But... we’re mathematicians.• For a fifth dimension, we need a fifth attribute:



















Caps

• A capset or cap in Fn3 is a collection of points with no three

collinear (containing no lines).

• A complete cap is a cap for which any other point in the spacemakes a line with a subset of points from the complete cap.

• A maximal cap is a cap of maximum size.• In F2

3, all complete caps are maximal and contain four points.

w w ww w ww w w

September 19, 2019 Caps in Fn3 15 / 20

Caps


collinear (containing no lines).• A complete cap is a cap for which any other point in the space

makes a line with a subset of points from the complete cap.

• A maximal cap is a cap of maximum size.• In F2


w w ww w ww w w


Caps



makes a line with a subset of points from the complete cap.• A maximal cap is a cap of maximum size.

• In F23, all complete caps are maximal and contain four points.

w w ww w ww w w


Caps



makes a line with a subset of points from the complete cap.• A maximal cap is a cap of maximum size.• In F2


w w ww w ww w w


Caps





w w ww w ww w w

{ {

{ {September 19, 2019 Caps in Fn

3 15 / 20

Caps





w w ww w ww w w

{ {{ {


Caps

Generalize this construction to a complete cap in F33 of size 8:v v t

t t ttv vtv v tt t ttv vt

t t tt t tt t t

But this isn’t maximal! A (maximal) cap of size 9:v vtt t tv vtt t tt tvt t t

t tvv vtt tv


Caps

Generalize this construction to a complete cap in F33 of size 8:v v t

t t ttv vtv v tt t ttv vt

t t tt t tt t tBut this isn’t maximal! A (maximal) cap of size 9:v vtt t tv vt

t t tt tvt t tt tvv vtt tv


Maximal caps

Denote by r(Fn3) the size of a maximal cap in Fn

3 .

• r(F13) = 2

• r(F23) = 4

• r(F33) = 9

• r(F43) = 20 [Pellegrino (1971)]. This is the answer for SET.

• r(F53) = 45 [Edel, Ferret, Landjev, and Storme (2002)]

• r(F63) = 112 [Potechin (2008)]

Open questionWhen n ≥ 7, what is r(Fn

3)?


Maximal caps


3 .• r(F1

3) = 2

• r(F23) = 4

• r(F33) = 9



• r(F63) = 112 [Potechin (2008)]


3)?


Maximal caps


3 .• r(F1

3) = 2• r(F2

3) = 4

• r(F33) = 9



• r(F63) = 112 [Potechin (2008)]


3)?


Maximal caps


3 .• r(F1

3) = 2• r(F2

3) = 4• r(F3

3) = 9



• r(F63) = 112 [Potechin (2008)]


3)?


Maximal caps


3 .• r(F1

3) = 2• r(F2

3) = 4• r(F3

3) = 9• r(F4

3) = 20 [Pellegrino (1971)]. This is the answer for SET.


• r(F63) = 112 [Potechin (2008)]


3)?


Maximal caps


3 .• r(F1

3) = 2• r(F2

3) = 4• r(F3

3) = 9• r(F4

3) = 20 [Pellegrino (1971)]. This is the answer for SET.• r(F5

3) = 45 [Edel, Ferret, Landjev, and Storme (2002)]

• r(F63) = 112 [Potechin (2008)]


3)?


Maximal caps


3 .• r(F1

3) = 2• r(F2

3) = 4• r(F3

3) = 9• r(F4


3) = 45 [Edel, Ferret, Landjev, and Storme (2002)]• r(F6

3) = 112 [Potechin (2008)]


3)?


Maximal caps


3 .• r(F1

3) = 2• r(F2

3) = 4• r(F3

3) = 9• r(F4


3) = 45 [Edel, Ferret, Landjev, and Storme (2002)]• r(F6

3) = 112 [Potechin (2008)]


3)?


An integer sequence

... let’s check OEIS!


An integer sequence

... let’s check OEIS!


An integer sequence

• Terry Tao’s blog: “Open question: best bounds for cap sets”( http://terrytao.wordpress.com/2007/02/23/open-question-best-bounds-for-cap-sets/)

• “Perhaps my favourite open question is the problem on themaximal size of a cap set—a subset of Fn

3 which contains nolines...”

n = 1 2 3 4 5 6 7 8 9 10r(Fn

3) ≥ 2 4 9 20 45 112 236 496 1064 2240r(Fn

3) ≤ 2 4 9 20 45 112 291 771 2070 5619


https://terrytao.wordpress.com/2007/02/23/open-question-best-bounds-for-cap-sets/

Thank you!

September 19, 2019 20 / 20

Documents

The geometry of the card game SET...The rules of SET To start the game, twelve SET cards are dealt face up. All players look for SETs simultaneously. If a player ﬁnds a SET, they