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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