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This web page lists difference triangle sets known to be optimal (ie of smallest size). There is also a plain text version of this file.

M(1,1)=1 -- Exactly 1 distinct example exists.
0 1

M(1,2)=3 -- Exactly 1 distinct example exists.
0 1 3

M(1,3)=6 -- Exactly 1 distinct example exists.
0 1 4 6

M(1,4)=11 -- Exactly 2 distinct examples exist.
0 1 4 9 11
0 3 4 9 11

M(1,5)=17 -- Exactly 4 distinct examples exist.
0 1 4 10 12 17
0 1 4 10 15 17
0 3 5 9 16 17
0 4 6 9 16 17

M(1,6)=25 -- Exactly 5 distinct examples exist.
0 1 4 10 18 23 25
0 2 3 10 16 21 25
0 2 6 9 14 24 25
0 1 7 11 20 23 25
0 3 4 12 18 23 25

M(1,7)=34 -- Exactly 1 distinct example exists.
0 1 4 9 15 22 32 34

M(1,8)=44 -- Exactly 1 distinct example exists.
0 3 9 17 19 32 39 43 44

M(1,9)=55 -- Exactly 1 distinct example exists.
0 1 6 10 23 26 34 41 53 55

M(1,10)=72 -- Exactly 2 distinct examples exist.
0 1 4 13 28 33 47 54 64 70 72
0 1 9 19 24 31 52 56 58 69 72

M(1,11)=85 -- Exactly 1 distinct example exists.
0 2 6 24 29 40 43 55 68 75 76 85

M(1,12)=106 -- Exactly 1 distinct example exists.
0 7 8 17 21 36 47 63 69 81 101 104 106

M(1,13)=127 -- Exactly 1 distinct example exists.
0 5 28 38 41 49 50 68 75 92 107 121 123 127

M(1,14)=151 -- Exactly 1 distinct example exists.
0 6 7 15 28 40 51 75 89 92 94 121 131 147 151

M(1,15)=177 -- Exactly 1 distinct example exists.
0 1 4 11 26 32 56 68 76 115 117 134 150 163 168 177

M(2,1)=2 -- Exactly 1 distinct example exists.
0 2
0 1

M(2,2)=7 -- Exactly 3 distinct examples exist.
0 2 5
0 1 7
0 2 7
0 1 4
0 3 7
0 1 6

M(2,3)=13 -- Exactly 2 distinct examples exist.
0 3 4 12
0 2 7 13
0 4 6 11
0 1 10 13

M(2,4)=22 -- Exactly 1 distinct example exists.
0 3 8 14 18
0 2 9 21 22

M(2,5)=34 -- Exactly 2 distinct examples exist.
0 3 7 23 28 34
0 2 14 15 24 32
0 7 10 19 25 33
0 4 5 21 32 34

M(2,6)=51 -- Exactly 2 distinct examples exist.
0 5 12 16 33 36 51
0 1 10 23 42 48 50
0 6 9 20 24 50 51
0 2 12 19 35 40 48

M(2,7)=70 -- Exactly 1 distinct example exists.
0 6 11 18 28 37 62 70
0 1 4 24 40 54 67 69

M(2,8)=94 -- Exactly 2 distinct examples exist.
0 10 15 29 45 51 77 85 94
0 1 3 24 28 61 74 81 92
0 14 16 22 34 43 62 79 94
0 4 7 37 42 68 81 91 92

M(2,9)=121 -- Exactly 1 distinct example exists.
0 4 13 45 46 69 94 109 116 121
0 3 19 21 29 57 87 101 107 118

M(3,1)=3 -- Exactly 1 distinct example exists.
0 3
0 2
0 1

M(3,2)=10 - Exactly 12 distinct examples exist, one of which is:
0 3 7
0 2 8
0 1 10

M(3,3)=19 - Exactly 9 distinct examples exist.
0 3 14 18
0 2 10 19
0 1 6 13
0 3 14 19
0 2 6 15
0 1 8 18
0 3 15 19
0 2 7 13
0 1 10 18
0 4 13 19
0 2 5 16
0 1 8 18
0 5 11 18
0 2 10 19
0 1 4 16
0 5 12 18
0 2 10 19
0 1 4 15
0 6 8 19
0 4 9 16
0 1 15 18
0 7 8 18
0 4 6 19
0 3 12 17
0 7 8 18
0 4 13 19
0 3 5 17

M(3,4)=32 - Exactly 4 distinct examples exist.
0 3 15 26 31
0 2 10 19 32
0 1 7 21 25
0 8 10 27 30
0 6 11 18 32
0 4 13 28 29
0 8 11 18 32
0 5 9 25 31
0 1 13 28 30
0 9 14 26 32
0 8 10 29 30
0 4 15 28 31

M(3,5)=49 - Exactly 3 distinct examples exist.
0 4 9 35 37 47
0 3 18 25 29 45
0 1 17 30 41 49
0 7 10 30 35 49
0 6 8 24 37 46
0 1 12 27 44 48
0 8 23 25 45 49
0 7 10 19 40 46
0 1 14 32 43 48

M(3,6)=72 - Exactly 4 distinct examples exist.
0 5 19 21 50 57 70
0 4 10 32 43 58 66
0 3 12 30 47 71 72
0 7 22 30 62 66 71
0 3 19 20 48 54 72
0 2 13 27 39 60 70
0 10 15 27 61 70 72
0 7 13 33 49 63 71
0 1 25 29 48 66 69
0 13 16 31 56 67 68
0 7 21 26 48 65 71
0 2 10 30 34 63 72

M(3,7)=100 - Exactly 1 distinct example exists.
0 12 15 31 55 87 88 93
0 7 30 41 51 77 90 99
0 2 29 37 54 82 96 100

M(4,1)=4 -- Exactly 1 distinct example exists.
0 4
0 3
0 2
0 1

M(4,2)=12 -- Exactly 8 distinct examples exist.
0 4 9
0 3 10
0 2 8
0 1 12
0 4 9
0 3 11
0 2 12
0 1 7
0 4 10
0 3 8
0 2 9
0 1 12
0 4 11
0 3 12
0 2 10
0 1 6
0 4 12
0 3 9
0 2 7
0 1 11
0 4 12
0 3 10
0 2 11
0 1 6
0 5 11
0 3 7
0 2 12
0 1 9
0 5 12
0 3 11
0 2 6
0 1 10

M(4,3)=24 -- Exactly 12 distinct examples exist, one of which is:
0 4 13 24
0 3 8 18
0 2 14 21
0 1 17 23

M(4,4)=41 -- Exactly 6 distinct examples exist.
0 8 17 27 40
0 5 11 31 33
0 4 16 34 41
0 1 15 36 39
0 10 12 36 39
0 8 15 31 40
0 6 11 28 41
0 4 18 37 38
0 10 16 38 39
0 5 14 32 40
0 4 11 24 41
0 3 15 34 36
0 11 15 35 41
0 10 17 33 38
0 8 9 27 40
0 2 14 36 39
0 12 13 36 40
0 9 19 34 41
0 6 8 26 37
0 3 17 33 38
0 15 20 37 41
0 6 16 34 35
0 3 14 27 39
0 2 9 32 40

M(4,5)=64 -- Exactly 1 distinct example exists.
0 12 13 31 58 60
0 8 24 34 57 62
0 4 21 41 56 63
0 3 9 39 53 64

M(4,6)=94 -- Exactly 4 distinct examples exist.
0 10 13 43 67 71 93
0 7 25 41 56 73 94
0 6 20 29 65 84 92
0 2 39 44 79 90 91
0 12 14 29 73 86 92
0 8 28 38 60 84 93
0 4 25 41 68 91 94
0 1 35 40 71 82 89
0 12 39 43 79 88 90
0 10 30 38 56 80 93
0 7 29 32 73 89 94
0 1 15 34 69 86 92
0 11 24 44 71 80 90
0 8 26 43 49 91 94
0 5 30 37 58 87 89
0 4 16 38 77 78 92

M(5,1)=5 -- Exactly 1 distinct example exists.
0 5
0 4
0 3
0 2
0 1

M(5,2)=15 -- Exactly 21 distinct examples exist, one of which is:
0 5 11
0 4 12
0 3 13
0 2 9
0 1 15

M(5,3)=30 -- Exactly 174 distinct examples exist, one of which is:
0 5 11 24
0 4 20 30
0 3 21 28
0 2 17 29
0 1 9 23

M(5,4)=51 -- Exactly 92 distinct examples exist, one of which is:
0 6 14 44 45
0 5 21 41 48
0 4 22 33 46
0 3 15 40 50
0 2 19 28 51

M(5,5)=79 -- At least 1 distinct example exists, one of which is:
0 12 20 41 72 75
0 6 25 49 62 79
0 4 15 51 65 74
0 2 18 44 66 71
0 1 33 40 68 78

M(6,1)=6 -- Exactly 1 distinct example exists.
0 6
0 5
0 4
0 3
0 2
0 1

M(6,2)=19 -- Exactly 720 distinct examples exist, one of which is:
0 6 13
0 5 14
0 4 12
0 3 18
0 2 19
0 1 11

M(6,3)=36 -- Exactly 3132 distinct examples exist, one of which is:
0 6 13 28
0 5 24 36
0 4 20 34
0 3 11 32
0 2 25 35
0 1 18 27

M(6,4)=60 -- Exactly 117 distinct examples exist, one of which is.
0 14 26 51 60
0 10 23 52 58
0 7 18 45 50
0 4 28 44 59
0 2 19 41 49
0 1 21 54 57

M(7,1)=7 -- Exactly 1 distinct example exists.
0 7
0 6
0 5
0 4
0 3
0 2
0 1

M(7,2)=22 -- Exactly 4079 distinct examples exist, one of which is:
0 7 15
0 6 16
0 5 14
0 4 17
0 3 21
0 2 22
0 1 12

M(7,3)=42 -- Exactly 71930 distinct examples exist, one of which is:
0 7 15 27
0 6 23 32
0 5 29 42
0 4 22 38
0 3 28 39
0 2 21 35
0 1 31 41

M(7,4)=71 -- About 400000 distinct examples exist, one of which is:
0 16 23 61 65
0 14 32 53 68
0 11 19 62 67
0 10 13 40 60
0 6 28 52 69
0 2 31 57 66
0 1 34 59 71

M(8,1)=8 -- Exactly 1 distinct example exists.
0 8
0 7
0 6
0 5
0 4
0 3
0 2
0 1

M(8,2)=24 -- Exactly 3040 distinct examples exist, one of which is:
0 8 17
0 7 18
0 6 19
0 5 15
0 4 20
0 3 24
0 2 14
0 1 23

M(8,3)=48 -- About 2.7*10**6 distinct examples exist, one of which is:
0 8 17 27
0 7 31 43
0 6 35 48
0 5 30 45
0 4 22 38
0 3 23 44
0 2 28 39
0 1 33 47

M(8,4)=80 -- At least 1 distinct example exists.
0 19 34 73 80
0 18 32 69 75
0 13 30 72 77
0 12 33 74 78
0 11 36 67 76
0 8 35 63 79
0 2 22 60 70
0 1 24 50 53

M(9,1)=9 -- Exactly 1 distinct example exists.
0 9
0 8
0 7
0 6
0 5
0 4
0 3
0 2
0 1

M(9,2)=27 -- Exactly 20505 distinct examples exist, one of which is:
0 9 19
0 8 20
0 7 18
0 6 21
0 5 27
0 4 17
0 3 26
0 2 16
0 1 25

M(9,3)=54 -- About 1.0*10**8 distinct examples exist, one of which is:
0 9 19 30
0 8 23 48
0 7 27 51
0 6 37 53
0 5 38 50
0 4 26 43
0 3 35 49
0 2 36 54
0 1 29 42

M(9,4)=91 -- About 2 billion distinct examples exist, one of which is:
0 26 42 79 90
0 20 25 69 76
0 17 21 82 84
0 13 35 75 87
0 10 33 80 83
0 9 38 66 81
0 8 27 68 86
0 6 45 77 91
0 1 31 55 89

M(10,1)=10 -- Exactly 1 distinct example exists.
0 10
0 9
0 8
0 7
0 6
0 5
0 4
0 3
0 2
0 1

M(10,2)=31 -- Exactly 1669221 distinct examples exist, one of which is:
0 10 21
0 9 22
0 8 20
0 7 23
0 6 24
0 5 19
0 4 29
0 3 31
0 2 17
0 1 27

M(10,3)=60 -- About 9.*10**9 distinct examples exist, one of which is:
0 10 21 33
0 9 22 49
0 8 34 59
0 7 44 60
0 6 38 58
0 5 29 48
0 4 35 50
0 3 39 57
0 2 30 47
0 1 42 56

M(10,4)=100 -- At least 1 distinct example exists.
0 22 42 85 91
0 15 46 76 93
0 14 37 88 95
0 12 40 79 92
0 10 35 83 94
0 9 36 77 96
0 8 34 72 90
0 5 21 71 75
0 3 32 65 89
0 1 45 98 100

M(11,1)=10 -- Exactly 1 distinct example exists.
0 11
0 10
0 9
0 8
0 7
0 6
0 5
0 4
0 3
0 2
0 1

M(11,2)=34 -- Exactly 14708525 distinct examples exist, one of which is:
0 11 23
0 10 24
0 9 22
0 8 25
0 7 26
0 6 27
0 5 20
0 4 32
0 3 34
0 2 18
0 1 30

M(11,3)=66 -- About 4.*10**11 distinct examples exist, one of which is:
0 11 23 36
0 10 24 50
0 9 31 61
0 8 43 64
0 7 45 60
0 6 47 65
0 5 44 63
0 4 46 66
0 3 37 54
0 2 29 57
0 1 33 49

M(12,1)=10 -- Exactly 1 distinct example exists.
0 12
0 11
0 10
0 9
0 8
0 7
0 6
0 5
0 4
0 3
0 2
0 1

M(12,2)=36 -- Exactly 10567748 distinct examples exist, one of which is:
0 12 25
0 11 26
0 10 24
0 9 27
0 8 28
0 7 23
0 6 35
0 5 22
0 4 34
0 3 36
0 2 21
0 1 32

M(12,3)=72 -- About 4*10**12 distinct examples exist, one of which is:
0 17 32 63
0 14 35 65
0 13 24 58
0 9 27 71
0 8 50 69
0 7 40 56
0 6 60 70
0 5 41 53
0 4 47 72
0 3 26 55
0 2 22 59
0 1 39 67

M(13,1)=10 -- Exactly 1 distinct example exists.
0 13
0 12
0 11
0 10
0 9
0 8
0 7
0 6
0 5
0 4
0 3
0 2
0 1

M(13,2)=39 -- About 1.2*10**8 distinct examples exist, one of which is:
0 13 27
0 12 28
0 11 26
0 10 29
0 9 30
0 8 25
0 7 31
0 6 38
0 5 23
0 4 37
0 3 39
0 2 22
0 1 35

M(13,3)=78 -- About 4*10**15 distinct examples exist, one of which is:
0 19 48 68
0 16 23 67
0 14 36 64
0 13 40 66
0 10 45 62
0 9 21 63
0 8 47 77
0 6 65 76
0 5 60 78
0 4 38 75
0 3 46 61
0 2 33 74
0 1 25 57

M(14,1)=10 -- Exactly 1 distinct example exists.
0 14
0 13
0 12
0 11
0 10
0 9
0 8
0 7
0 6
0 5
0 4
0 3
0 2
0 1

M(14,2)=43 -- About 1.7*10**10 distinct examples exist, one of which is:
0 14 29
0 13 30
0 12 28
0 11 31
0 10 32
0 9 27
0 8 33
0 7 26
0 6 40
0 5 42
0 4 43
0 3 41
0 2 23
0 1 36

M(14,3)=84 -- About 2*10**17 distinct examples exist, one of which is:
0 14 50 65
0 13 46 69
0 12 31 49
0 11 27 52
0 10 42 77
0 9 53 70
0 8 38 72
0 7 47 73
0 6 45 74
0 5 48 76
0 4 59 79
0 3 57 81
0 2 60 82
0 1 63 84

M(15,1)=10 -- Eactly 1 distinct example exists.
0 15
0 14
0 13
0 12
0 11
0 10
0 9
0 8
0 7
0 6
0 5
0 4
0 3
0 2
0 1

M(15,2)=46 -- About 1.8*10**11 examples exist, one of which is:
0 15 31
0 14 32
0 13 30
0 12 33
0 11 34
0 10 29
0 9 35
0 8 28
0 7 43
0 6 44
0 5 27
0 4 46
0 3 40
0 2 41
0 1 25

M(15,3)=90 -- About 10**19 distinct examples exist, one of which is:
0 15 55 71
0 14 53 74
0 13 33 50
0 12 44 80
0 11 38 57
0 10 28 75
0 9 52 82
0 8 42 77
0 7 48 79
0 6 51 76
0 5 54 83
0 4 63 85
0 3 61 87
0 2 64 88
0 1 67 90

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