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WOLFRAM|DEMONSTRATIONS PROJECT

The Set of Sets in SET

​
deck number
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
BitXor gives the bitwise XOR of a set of integers in binary form. This uses the finite field
F
2
={0,1}
. Is there something similar for
F
3
?
The ternary representations of
(31,38,51)
are
(1011,1102,1220)
; transposing,
1
0
1
1
1
1
0
2
1
2
2
0
=
1
1
1
0
1
2
1
0
2
1
2
0
, which gives
(111,012,102,120)
. Similarly,
(45,48,51)
in ternary is
(1200,1210,1220)
with transpose
(111,222,012,000)
. In each transpose, the triples of numbers are either all the same or all different. Under a ternary form of BitXor, the triples have a ternary bitwise sum of zero.
In the game of SET, there are 81 cards, each having four features: number of shapes (one, two or three), shape (diamond, squiggle or oval), shading (solid, striped or open) and color (red, green or purple). A set of three cards is called a "set" if the four features are pairwise identical or completely different. This is the same as a bitwise
F
3
sum of zero. SET can be thought of as projective geometry
PG(3,4)
.
There are 1080 possible sets in SET. This Demonstration shows them all, using 40 decks of SET to show 27 sets at a time. This is also a Steiner system: each pair of numbers from 0 to 80 is represented in a unique set. This is also a
81–(27,3,1)
resolvable design.
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