Anti-SET
In which we get bored with SET and reverse the rules
David Clark
Joint work with George Fisk and Nurry Goren
April 2, 2015
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George Fisk & Nurry Goren (center)
Spring 2014 Minnesota Pi Mu Epsilon Conference
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What is SET?
Color: Red, Green, Purple
Number: 1, 2, 3
Filling: Open, Stripe, Solid
Shape: Squiggle, Oval, Diamond
Set: 3 cards, each attribute all same or all different.
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What is SET?
Color: Red, Green, Purple
Number: 1, 2, 3
Filling: Open, Stripe, Solid
Shape: Squiggle, Oval, Diamond
Set: 3 cards, each attribute all same or all different.
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What is SET?
Color: Red, Green, Purple
Number: 1, 2, 3
Filling: Open, Stripe, Solid
Shape: Squiggle, Oval, Diamond
Set: 3 cards, each attribute all same or all different.
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SET as tic-tac-toe
Xavier (Player 1) vs. Olivia (Player 2)
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Anti-SET
Theorem (Pellegrino, 1971)
Every set of 21 SET cards contains a set.
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Anti-SET
Theorem (Pellegrino, 1971)
Every set of 21 SET cards contains a set.
Anti-SET Rules:
Start with all 81 SET cards
2 players alternate taking any available card
First to have a set in their hand loses
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Anti-SET
Theorem (Pellegrino, 1971)
Every set of 21 SET cards contains a set.
Anti-SET Rules:
Start with all 81 SET cards
2 players alternate taking any available card
First to have a set in their hand loses
Anyone up for a game?
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The geometry of SET
Point
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The geometry of SET
Two points
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The geometry of SET
Line (set)
2 points determine a unique line
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The geometry of SET
Intersecting lines
2 lines intersect in 1 point. . .
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The geometry of SET
Plane of 32 = 9 cards
. . . or lines can be parallel
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The geometry of SET
Hyperplane of 33 = 27 cards (“3D space”)
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The geometry of SET
All 34 = 81 cards (“4D space”)
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The SECRET of WINNING Anti-SET
Moves: X0 , O0 , X1 , O1 , . . .
Winning Strategy for Xavier
Pick Xn to complete the line through X0 and On−1 .
Demo!
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Lemma: Xavier can play
Proof: In this situation:
X0
On−1
??
How could Xavier’s move be occupied?
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Lemma: Xavier can’t lose
X1
Proof by picture:
X2
O0
O1
X0
Vector proof
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Lemma: Xavier can’t lose
X1
Proof by picture:
X2
O0
O1
X
O
X0
Vector proof
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Lemma: Xavier can’t lose
X1
Proof by picture:
X2
O0
O1
X
O
X0
Vector proof
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Lemma: Xavier can’t lose
X1
Proof by picture:
X2
O0
O1
X
O
X0
Vector proof
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Lemma: There are no ties
Proof:
Play two full turns.
This gives 2 lines which span a 9-card plane P.
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Lemma: There are no ties
Proof:
Play two full turns.
This gives 2 lines which span a 9-card plane P.
There is no tie in P.
Proof
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Lemma: There are no ties
Proof:
Play two full turns.
This gives 2 lines which span a 9-card plane P.
There is no tie in P. Proof
If Olivia keeps playing in P, she will lose.
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Lemma: There are no ties
If Olivia plays outside of P, Xavier still can’t lose.
Xavier never picks a point in P.
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Lemma: There are no ties
If Olivia plays outside of P, Xavier still can’t lose.
Xavier never picks a point in P.
Therefore, Olivia must either lose outside of P, or
eventually play in P again
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Lemma: There are no ties
If Olivia plays outside of P, Xavier still can’t lose.
Xavier never picks a point in P.
Therefore, Olivia must either lose outside of P, or
eventually play in P again . . . and lose.
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Summary
Theorem: Winning Strategy for Xavier
Pick Xn to complete the line through X0 and On−1 .
Proof: By the previous lemmas.
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Summary
Theorem: Winning Strategy for Xavier
Pick Xn to complete the line through X0 and On−1 .
Proof: By the previous lemmas.
But wait. . . our proofs only needed:
Every pair of points defines a unique line.
There are 3 points on every line.
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Affine geometry
SET is an affine geometry:
A nonempty set of points P and lines L such that:
Every pair of points defines a unique line.
Every line has the same number of points.
Every line is part of a parallel class which
partitions P.
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Affine geometry: Examples
The Euclidean Plane
(Infinite)
The 4-point plane
(2 points per line)
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Affine geometry: Examples
A 9-card SET Plane
(3 points per line)
All 81 SET Cards
(3 points per line)
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Mathematizing SET
We can represent SET cards as points/vectors:
hc, n, f , si
c: Color
n: Number
f: Filling
s: Shape
0
1
2
Red
Green Purple
3 (?!)
1
2
Open
Stripe
Solid
Squiggle Oval Diamond
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Mathematizing SET
c: Color
n: Number
f: Filling
s: Shape
0
Red
3
Open
Squiggle
1
Green
1
Stripe
Oval
2
Purple
2
Solid
Diamond
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Mathematizing SET
c: Color
n: Number
f: Filling
s: Shape
0
Red
3
Open
Squiggle
1
Green
1
Stripe
Oval
2
Purple
2
Solid
Diamond
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Mathematizing SET
c: Color
n: Number
f: Filling
s: Shape
0
Red
3
Open
Squiggle
1
Green
1
Stripe
Oval
2
Purple
2
Solid
Diamond
h0, 0, 0, 0i
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Mathematizing SET
c: Color
n: Number
f: Filling
s: Shape
0
Red
3
Open
Squiggle
1
Green
1
Stripe
Oval
2
Purple
2
Solid
Diamond
h0, 0, 0, 0i
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Mathematizing SET
c: Color
n: Number
f: Filling
s: Shape
0
Red
3
Open
Squiggle
1
Green
1
Stripe
Oval
2
Purple
2
Solid
Diamond
h0, 0, 0, 0i h1, 1, 1, 2i
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Mathematizing SET
c: Color
n: Number
f: Filling
s: Shape
0
Red
3
Open
Squiggle
1
Green
1
Stripe
Oval
2
Purple
2
Solid
Diamond
h0, 0, 0, 0i h1, 1, 1, 2i
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Mathematizing SET
c: Color
n: Number
f: Filling
s: Shape
0
Red
3
Open
Squiggle
1
Green
1
Stripe
Oval
2
Purple
2
Solid
Diamond
h0, 0, 0, 0i h1, 1, 1, 2i h2, 2, 2, 1i
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Mathematizing SET
c: Color
n: Number
f: Filling
s: Shape
0
Red
3
Open
Squiggle
1
Green
1
Stripe
Oval
2
Purple
2
Solid
Diamond
h0, 0, 0, 0i h1, 1, 1, 2i h2, 2, 2, 1i
These three vectors form a subspace in F43 .
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Affine geometry: Algebra
To construct the 9-card SET plane algebraically:
Points: {hp, qi : p, q ∈ {0, 1, 2}}
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Affine geometry: Algebra
To construct the 9-card SET plane algebraically:
Points: {hp, qi : p, q ∈ {0, 1, 2}}
h0, 2i
h1, 2i
h2, 2i
h0, 1i
h1, 1i
h2, 1i
h0, 0i
h1, 0i
h2, 0i
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Affine geometry: Algebra
To construct the 9-card SET plane algebraically:
Points: {hp, qi : p, q ∈ {0, 1, 2}}
h0, 2i
h1, 2i
h2, 2i
h0, 1i
h1, 1i
h2, 1i
h0, 0i
h1, 0i
h2, 0i
~ and ~b,
Lines: For n
fixed m
o
~
~
` = mx + b : x ∈ {0, 1, 2} (mod 3).
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Affine geometry: Algebra
To construct the 9-card SET plane algebraically:
Points: {hp, qi : p, q ∈ {0, 1, 2}}
h0, 2i
h1, 2i
h2, 2i
h1, 0i x + h1, 2i :
h0, 1i
h1, 1i
h2, 1i
h0, 0i
h1, 0i
h2, 0i
~ and ~b,
Lines: For n
fixed m
o
~
~
` = mx + b : x ∈ {0, 1, 2} (mod 3).
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Affine geometry: Algebra
To construct the 9-card SET plane algebraically:
Points: {hp, qi : p, q ∈ {0, 1, 2}}
h0, 2i
h1, 2i
h2, 2i
h1, 0i x + h1, 2i :
h1, 0i (0) + h1, 2i = h1, 2i
h0, 1i
h1, 1i
h2, 1i
h0, 0i
h1, 0i
h2, 0i
~ and ~b,
Lines: For n
fixed m
o
~
~
` = mx + b : x ∈ {0, 1, 2} (mod 3).
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Affine geometry: Algebra
To construct the 9-card SET plane algebraically:
Points: {hp, qi : p, q ∈ {0, 1, 2}}
h0, 2i
h1, 2i
h2, 2i
h1, 0i x + h1, 2i :
h1, 0i (0) + h1, 2i = h1, 2i
h0, 1i
h1, 1i
h2, 1i
h1, 0i (1) + h1, 2i = h2, 2i
h0, 0i
h1, 0i
h2, 0i
~ and ~b,
Lines: For n
fixed m
o
~
~
` = mx + b : x ∈ {0, 1, 2} (mod 3).
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Affine geometry: Algebra
To construct the 9-card SET plane algebraically:
Points: {hp, qi : p, q ∈ {0, 1, 2}}
h0, 2i
h1, 2i
h2, 2i
h1, 0i x + h1, 2i :
h1, 0i (0) + h1, 2i = h1, 2i
h0, 1i
h1, 1i
h2, 1i
h1, 0i (1) + h1, 2i = h2, 2i
h1, 0i (2) + h1, 2i = h0, 2i
h0, 0i
h1, 0i
h2, 0i
~ and ~b,
Lines: For n
fixed m
o
~
~
` = mx + b : x ∈ {0, 1, 2} (mod 3).
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Constructing the 9-card plane
h0, 2i
h1, 2i
h2, 2i
h0, 1i
h1, 1i
h2, 1i
h0, 0i
h1, 0i
h2, 0i
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Constructing the 9-card plane
h0, 2i
h1, 2i
h2, 2i
h0, 1i
h1, 1i
h2, 1i
h0, 0i
h1, 0i
h2, 0i
h1, 0i x + h0, 0i
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Constructing the 9-card plane
h0, 2i
h1, 2i
h2, 2i
h0, 1i
h1, 1i
h2, 1i
+ h0, 1i
+ h0, 1i
+ h0, 1i
h0, 0i
h1, 0i
h2, 0i
h1, 0i x + h0, 1i
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Constructing the 9-card plane
h0, 2i
h1, 2i
h2, 2i
h0, 1i
h1, 1i
h2, 1i
+ h0, 2i
+ h0, 2i
+ h0, 2i
h0, 0i
h1, 0i
h2, 0i
h1, 0i x + h0, 2i
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Affine geometry: Algebra again
This construction also works with larger vectors:
n
o
n
~
~
Points: F3 , n ≥ 1
Lines: mx + b .
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Affine geometry: Algebra again
This construction also works with larger vectors:
n
o
n
~
~
Points: F3 , n ≥ 1
Lines: mx + b .
SET is played with 4-dimensional vectors:
h0, 0, 0, 0i h1, 1, 1, 2i h2, 2, 2, 1i
h1, 1, 1, 2i x + h0, 0, 0, 0i
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Affine geometry: Extending SET
SET: Searching for lines in an affine geometry.
Anti-SET: Avoiding lines in any dimension affine
geometry with 3 points per line.
Theorem
Xavier can win Anti-SET if played on the points and
lines of any affine geometry in Fn3 with n > 1.
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Back to our inspiration
Theorem (Pellegrino, 1971)
Every set of 21 SET cards contains a set.
(The largest cap in a 4-dimensional affine geometry over F3 is 20 points.)
Why doesn’t Xavier lose on turn 21?
We never actually reach 21 cards.
Many smaller sets of cards contain sets too.
. . . and our strategy carefully forces Olivia’s hand
to be one such set.
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How long does the game actually last?
Cap: A set of points that contains no line.
m2 (k ): Size of a maximal cap in an affine geometry of dimension k .
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How long does the game actually last?
Cap: A set of points that contains no line.
m2 (k ): Size of a maximal cap in an affine geometry of dimension k .
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How long does the game actually last?
X0
O0
X1
Start with 1 point
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How long does the game actually last?
X0
O0
X1
−→
X0
O0
O1
O2
X2
Start with 1 point
X1
X3
Cap of size 2
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How long does the game actually last?
X6
X7
X4
X5
O6
O5
O3
O4
X3
X2
O1
X0
O2
O0
X1
Cap of size 4
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How long does the game actually last?
For standard SET:
1 + ( 2 + 4 + 9 ) + 1 = 17
First move
Last move
maximal caps in 1, 2, and 3 dimensions
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Olivia’s cap strategy
Olivia can extend the game by playing within caps.
Theorem
Suppose Xavier and Olivia play Anti-SET in an affine
geometry of dimension n > 1. Then Olivia can force
n−1
X
the game to last at least 2 +
m2 (k ) turns.
k =1
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Extensions
Some extensions:
Can Olivia force the game to 20 rounds?
More: Options per attribute, Players
Extend beyond Affine geometries: Projective
geometries, Steiner Triple Systems
Can you recover from an error (or starting 2nd)?
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Questions?
Happy Egg Roll day!
(First official White House egg roll: April 2, 1877)
Vector loser proof
Ties on 9-card plane
Lines in 9-card plane
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References
More information:
David Clark and George Fisk and Nurry Goren: A variation on the
game SET.
To appear in Involve.
Benjamin Lent Davis and Diane Maclagan: The card game SET.
Mathematical Intelligencer 25 (3) (2003) 33–40.
Maureen T. Carroll and Steven T. Dougherty: Tic-Tac-Toe on a
finite plane.
Mathematics Magazine 77 (4) (2004) 260–274.
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Proof: Xavier can’t lose
~
2a
~
2b
~a
~b
~0
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Proof: Xavier can’t lose
~
2a
~
2b
~a
~b
X?
O?
~0
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Proof: Xavier can’t lose
~
2a
~
2b
~a
~b
X?
O?
~0
If Xavier has a line, then X = ~a + ~b
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Proof: Xavier can’t lose
~
2a
~a + ~b
~
2b
~a
~b
O?
~0
If Xavier has a line, then X = ~a + ~b
By the strategy, X + O + ~0 = ~0
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Proof: Xavier can’t lose
~
2a
~a + ~b
~
2b
~a
~b
2~a + 2~b
~0
If Xavier has a line, then X = ~a + ~b
By the strategy, X + O + ~0 = ~0
So, O = 2~a + 2~b
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Proof: Xavier can’t lose
~
2a
~a + ~b
~
2b
~a
~b
2~a + 2~b
~0
If Xavier has a line, then X = ~a + ~b
By the strategy, X + O + ~0 = ~0
So, O = 2~a + 2~b
~a + ~b + O = ~0, so Ophelia chose a line first.
Back
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Proof: There are no ties
Choose a set S of 5 points:
Back
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Proof: There are no ties
Choose 3 parallel lines not contained in S.
`1
`2
`3
Back
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Proof: There are no ties
One of these meets S in 1 point P.
`1
`2
`3
P
Back
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Proof: There are no ties
Draw lines connecting P to the two points in `1 ∩ S
`1
`2
x
`3
P
y
Back
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Proof: There are no ties
Each line meets `2 at a different point. But `2 has only
1 point not in S. So one of those points must be in S.
`1
`2
x
`3
P
Back
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Proof: There are no ties
This line is contained entirely in S.
`1
`2
x
`3
P
Back
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Counting lines in an affine geometry
Have we found all of the lines?
There are 9 points.
Each pair of points defines a unique line.
There are 92 = 36 pairs.
Each line contains 3 points (so 32 = 3 pairs)
So there must be 36/3 = 12 lines.
Back
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