Games and Puzzles Journal #31

The Games and Puzzles Journal — Issue 32, March-April 2004

This issue is devoted to further new results by Awani Kumar on 12 by 12 magic knight tours. This time he concentrates on reentrant tours that are 4-fold cyclic (i.e. remain magic when renumbered cyclically from the quarter points 37, 73, 109), and also have one diagonal magic.

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Sections on this page: (66) Introduction. (67) Even Diagonal Magic. (68) Even Diagonal Magic with Broken Diagonals. (69) Odd Diagonal Magic with Broken Diagonals. (70) Some Open Tours. End.

Studies in Magic Tours of Knight on 12*12 Board

by Awani Kumar

17, Rana Pratap Marg, Lucknow 226001 INDIA

(66) Introduction

Tours of Knight have unfathomable mysteries and continue to throw up new results in spite of voluminous literature produced over centuries by untiring works of professional and amateur mathematicians. The author has shown that the very first magic knight's tour on 12*12 board, constructed by Krishnaraj Wadiar, Raja of Mysore, India, before 1868 is a rich mine of re-entrant magic tours. [See The Games and Puzzles Journal # 24]. Further studies have revealed that not only 'simple magic tours' (only rows and columns magic, not diagonals), but an ocean of re-entrant cyclic, 'almost perfect magic tours' (rows, columns and one diagonal adding to magic constant) can be constructed from it.

The author has now enumerated over 100 single diagonal closed magic tours. All these tours are reentrant 4-fold cyclic magic tours, that is, they remain magic when numbered from the quarter points 37, 73 and 109. All the tours 2 to 28 that follow are of this type. The 12*12 board can be divided into 9 quads, each of 4*4 size, and the to and fro path of the knight in all the new tours follows the general plan shown in Fig.1. [Magic diagonals are coloured in the diagrams below. Yellow for even, Orange for odd.]
Fig.1

(67) Some Tours with Even Diagonal Magic.

Fig.2 shows an 'almost perfect magic tour' with its even diagonal adding to magic constant. Fig.3 to Fig.11 can be obtained by modifying quads not lying on the magic diagonal.

17 56 91 126 19 54 123 94 35 38 107 110

90 127 18 55 124 93 20 53 106 109 36 39

57 16 125 92 51 22 95 122 37 34 111 108

128 89 50 23 96 121 52 21 112 105 40 33

15 58 129 88 25 48 27 44 117 102 113 104

130 87 24 49 120 97 118 101 28 43 32 41

59 14 85 132 47 26 99 116 45 30 103 114

86 131 60 13 98 119 46 29 100 115 42 31

61 12 133 84 65 8 79 140 69 144 73 2

134 83 64 9 80 137 66 5 76 3 70 143

11 62 81 136 7 78 139 68 141 72 1 74

82 135 10 63 138 67 6 77 4 75 142 71

Fig. 2 OD=686 ED=870

19 54 91 126 21 52 123 94 35 38 107 110

90 127 20 53 124 93 22 51 106 109 36 39

55 18 125 92 49 24 95 122 37 34 111 108

128 89 56 17 96 121 50 23 112 105 40 33

15 58 129 88 25 48 27 44 117 102 113 104

130 87 16 57 120 97 118 101 28 43 32 41

59 14 85 132 47 26 99 116 45 30 103 114

86 131 60 13 98 119 46 29 100 115 42 31

61 12 133 84 65 8 79 140 69 144 73 2

134 83 64 9 80 137 66 5 76 3 70 143

11 62 81 136 7 78 139 68 141 72 1 74

82 135 10 63 138 67 6 77 4 75 142 71

Fig.3 OD=682 ED=870

19 54 125 92 21 52 123 94 35 38 107 110

90 127 20 53 124 93 22 51 106 109 36 39

55 18 91 126 49 24 95 122 37 34 111 108

128 89 56 17 96 121 50 23 112 105 40 33

15 58 129 88 25 48 27 44 117 102 113 104

130 87 16 57 120 97 118 101 28 43 32 41

59 14 85 132 47 26 99 116 45 30 103 114

86 131 60 13 98 119 46 29 100 115 42 31

61 12 133 84 65 8 79 140 69 144 73 2

134 83 64 9 80 137 66 5 76 3 70 143

11 62 81 136 7 78 139 68 141 72 1 74

82 135 10 63 138 67 6 77 4 75 142 71

Fig.4 OD=648 ED=870

19 54 125 92 21 52 123 94 35 38 107 110

90 127 20 53 124 93 22 51 106 109 36 39

55 18 91 126 49 24 95 122 37 34 111 108

128 89 56 17 96 121 50 23 112 105 40 33

57 16 129 88 25 48 27 44 117 102 113 104

86 131 14 59 120 97 118 101 28 43 32 41

15 58 87 130 47 26 99 116 45 30 103 114

132 85 60 13 98 119 46 29 100 115 42 31

61 12 133 84 65 8 79 140 69 144 73 2

134 83 64 9 80 137 66 5 76 3 70 143

11 62 81 136 7 78 139 68 141 72 1 74

82 135 10 63 138 67 6 77 4 75 142 71

Fig.5 OD=648 ED=870

19 54 125 92 21 52 123 94 35 38 107 110

126 91 20 53 124 93 22 51 106 109 36 39

55 18 89 128 49 24 95 122 37 34 111 108

90 127 16 57 96 121 50 23 112 105 40 33

17 56 129 88 25 48 27 44 117 102 113 104

130 87 58 15 120 97 118 101 28 43 32 41

59 14 85 132 47 26 99 116 45 30 103 114

86 131 60 13 98 119 46 29 100 115 42 31

61 12 133 84 65 8 79 140 69 144 73 2

134 83 64 9 80 137 66 5 76 3 70 143

11 62 81 136 7 78 139 68 141 72 1 74

82 135 10 63 138 67 6 77 4 75 142 71

Fig.6 OD=650 ED=870

19 54 125 92 21 52 123 94 35 38 107 110

126 91 20 53 124 93 22 51 106 109 36 39

55 18 89 128 49 24 95 122 37 34 111 108

90 127 56 17 96 121 50 23 112 105 40 33

15 58 129 88 25 48 27 44 117 102 113 104

86 131 16 57 120 97 118 101 28 43 32 41

59 14 87 130 47 26 99 116 45 30 103 114

132 85 60 13 98 119 46 29 100 115 42 31

61 12 133 84 65 8 79 140 69 144 73 2

134 83 64 9 80 137 66 5 76 3 70 143

11 62 81 136 7 78 139 68 141 72 1 74

82 135 10 63 138 67 6 77 4 75 142 71

Fig.7 OD=610 ED=870

19 54 91 126 21 52 123 94 35 38 107 110

90 127 20 53 124 93 22 51 106 109 36 39

55 18 125 92 49 24 95 122 37 34 111 108

128 89 56 17 96 121 50 23 112 105 40 33

57 16 129 88 25 48 27 44 117 102 113 104

86 131 14 59 120 97 118 101 28 43 32 41

15 58 87 130 47 26 99 116 45 30 103 114

132 85 60 13 98 119 46 29 100 115 42 31

61 12 133 84 65 8 79 140 69 144 73 2

134 83 64 9 80 137 66 5 76 3 70 143

11 62 81 136 7 78 139 68 141 72 1 74

82 135 10 63 138 67 6 77 4 75 142 71

Fig.8 OD=682 ED=870

55 18 91 126 53 20 93 124 35 38 107 110

90 127 54 19 92 125 52 21 106 109 36 39

17 56 89 128 23 50 123 94 37 34 111 108

88 129 24 49 122 95 22 51 112 105 40 33

57 16 121 96 25 48 27 44 117 102 113 104

130 87 58 15 120 97 118 101 28 43 32 41

59 14 85 132 47 26 99 116 45 30 103 114

86 131 60 13 98 119 46 29 100 115 42 31

61 12 133 84 65 8 79 140 69 144 73 2

134 83 64 9 80 137 66 5 76 3 70 143

11 62 81 136 7 78 139 68 141 72 1 74

82 135 10 63 138 67 6 77 4 75 142 71

Fig.9 OD=714 ED=870

127 92 19 52 125 94 21 50 35 38 107 110

18 53 126 93 20 51 124 95 106 109 36 39

91 128 55 24 89 122 49 22 37 34 111 108

54 17 90 121 56 23 96 123 112 105 40 33

15 58 129 88 25 48 27 44 117 102 113 104

86 131 16 57 120 97 118 101 28 43 32 41

59 14 87 130 47 26 99 116 45 30 103 114

132 85 60 13 98 119 46 29 100 115 42 31

61 12 133 84 65 8 79 140 69 144 73 2

134 83 64 9 80 137 66 5 76 3 70 143

11 62 81 136 7 78 139 68 141 72 1 74

82 135 10 63 138 67 6 77 4 75 142 71

Fig.10 OD=750 ED=870

127 92 19 52 125 94 21 50 35 38 107 110

18 53 126 93 20 51 124 95 106 109 36 39

91 128 55 24 89 122 49 22 37 34 111 108

54 17 90 121 56 23 96 123 112 105 40 33

15 58 129 88 25 48 27 44 117 102 113 104

86 131 16 57 120 97 118 101 28 43 32 41

59 14 87 130 47 26 99 116 45 30 103 114

132 85 60 13 98 119 46 29 100 115 42 31

61 12 133 84 7 78 139 68 141 72 1 74

134 83 64 9 138 67 6 77 4 75 142 71

11 62 81 136 65 8 79 140 69 144 73 2

82 135 10 63 80 137 66 5 76 3 70 143

Fig.11 OD=1038 ED=870

(68) Even Magic Diagonal with some Broken Diagonals Magic.

Fig.12 is also an 'almost perfect magic tour' with its even magic diagonal different from the previous ones. Fig.13 to Fig.15 can be obtained by modifying non magic-diagonal quads. Similarly, Fig.16 has a different arrangement along its magic diagonals. Fig.17 to Fig.20 can be obtained by modifying its non magic-diagonal quads. [Two tours in each set also have an even-numbered broken diagonal which adds to the magic constant.]

Fig.12 OD=1058 ED=870
19	54	91	126	21	52	123	94	35	38	107	110
90	127	20	53	124	93	22	51	106	109	36	39
55	18	125	92	49	24	95	122	37	34	111	108
128	89	56	17	96	121	50	23	112	105	40	33
15	58	129	88	25	48	119	100	29	42	113	104
130	87	16	57	120	97	26	45	116	103	32	41
59	14	131	86	47	118	99	28	101	30	43	114
84	133	60	13	98	27	46	117	44	115	102	31
11	62	85	132	7	78	139	68	141	72	1	74
134	83	12	61	138	67	6	77	4	75	142	71
63	10	81	136	65	8	79	140	69	144	73	2
82	135	64	9	80	137	66	5	76	3	70	143

Fig.13 OD=1058 ED=870
19	54	91	126	21	52	123	94	35	38	107	110
90	127	20	53	124	93	22	51	106	109	36	39
55	18	125	92	49	24	95	122	37	34	111	108
128	89	56	17	96	121	50	23	112	105	40	33
57	16	87	130	25	48	119	100	29	42	113	104
88	129	58	15	120	97	26	45	116	103	32	41
59	14	131	86	47	118	99	28	101	30	43	114
84	133	60	13	98	27	46	117	44	115	102	31
11	62	85	132	7	78	139	68	141	72	1	74
134	83	12	61	138	67	6	77	4	75	142	71
63	10	81	136	65	8	79	140	69	144	73	2
82	135	64	9	80	137	66	5	76	3	70	143

Fig.14 OD=770 ED=870
19	54	91	126	21	52	123	94	35	38	107	110
90	127	20	53	124	93	22	51	106	109	36	39
55	18	125	92	49	24	95	122	37	34	111	108
128	89	56	17	96	121	50	23	112	105	40	33
15	58	129	88	25	48	119	100	29	42	113	104
130	87	16	57	120	97	26	45	116	103	32	41
59	14	131	86	47	118	99	28	101	30	43	114
84	133	64	9	98	27	46	117	44	115	102	31
13	60	85	132	65	8	79	140	69	144	73	2
134	83	10	63	80	137	66	5	76	3	70	143
61	12	81	136	7	78	139	68	141	72	1	74
82	135	62	11	138	67	6	77	4	75	142	71

Fig.15 OD=770 ED=870
19	54	91	126	21	52	123	94	35	38	107	110
90	127	20	53	124	93	22	51	106	109	36	39
55	18	125	92	49	24	95	122	37	34	111	108
128	89	56	17	96	121	50	23	112	105	40	33
57	16	87	130	25	48	119	100	29	42	113	104
88	129	58	15	120	97	26	45	116	103	32	41
59	14	131	86	47	118	99	28	101	30	43	114
84	133	64	9	98	27	46	117	44	115	102	31
13	60	85	132	65	8	79	140	69	144	73	2
134	83	10	63	80	137	66	5	76	3	70	143
61	12	81	136	7	78	139	68	141	72	1	74
82	135	62	11	138	67	6	77	4	75	142	71

Fig.16 OD=1014 ED=870
19	54	125	92	51	22	123	94	35	38	107	110
90	127	20	53	124	93	50	23	106	109	36	39
55	18	91	126	21	52	95	122	37	34	111	108
128	89	16	57	96	121	24	49	112	105	40	33
17	56	129	88	25	48	97	120	29	44	113	104
130	87	58	15	98	119	28	45	102	115	32	41
59	14	131	86	47	26	117	100	43	30	103	114
84	133	60	13	118	99	46	27	116	101	42	31
61	12	85	132	7	78	139	68	141	72	1	74
134	83	10	63	138	67	6	77	4	75	142	71
11	62	81	136	65	8	79	140	69	144	73	2
82	135	64	9	80	137	66	5	76	3	70	143

Fig.17 OD=936 ED=870
19	54	125	92	51	22	123	94	35	38	107	110
126	91	20	53	124	93	50	23	106	109	36	39
55	18	89	128	21	52	95	122	37	34	111	108
90	127	56	17	96	121	24	49	112	105	40	33
15	58	129	88	25	48	97	120	29	44	113	104
130	87	16	57	98	119	28	45	102	115	32	41
59	14	131	86	47	26	117	100	43	30	103	114
84	133	12	61	118	99	46	27	116	101	42	31
13	60	85	132	7	78	139	68	141	72	1	74
134	83	62	11	138	67	6	77	4	75	142	71
63	10	81	136	65	8	79	140	69	144	73	2
82	135	64	9	80	137	66	5	76	3	70	143

Fig.18 OD=1014 ED=870
19	54	125	92	51	22	123	94	35	38	107	110
126	91	20	53	124	93	50	23	106	109	36	39
55	18	127	90	21	52	95	122	37	34	111	108
128	89	16	57	96	121	24	49	112	105	40	33
17	56	87	130	25	48	97	120	29	44	113	104
88	129	58	15	98	119	28	45	102	115	32	41
13	60	131	86	47	26	117	100	43	30	103	114
84	133	14	59	118	99	46	27	116	101	42	31
61	12	85	132	7	78	139	68	141	72	1	74
134	83	62	11	138	67	6	77	4	75	142	71
63	10	81	136	65	8	79	140	69	144	73	2
82	135	64	9	80	137	66	5	76	3	70	143

Fig.19 OD=1046 ED=870
55	18	125	92	51	22	123	94	35	38	107	110
90	127	54	19	124	93	50	23	106	109	36	39
17	56	91	126	21	52	95	122	37	34	111	108
128	89	20	53	96	121	24	49	112	105	40	33
57	16	87	130	25	48	97	120	29	44	113	104
88	129	58	15	98	119	28	45	102	115	32	41
59	14	131	86	47	26	117	100	43	30	103	114
84	133	12	61	118	99	46	27	116	101	42	31
13	60	85	132	7	78	139	68	141	72	1	74
134	83	62	11	138	67	6	77	4	75	142	71
63	10	81	136	65	8	79	140	69	144	73	2
82	135	64	9	80	137	66	5	76	3	70	143

Fig.20 OD=1014 ED=870
19	54	125	92	51	22	123	94	35	38	107	110
126	91	20	53	124	93	50	23	106	109	36	39
55	18	127	90	21	52	95	122	37	34	111	108
128	89	16	57	96	121	24	49	112	105	40	33
17	56	87	130	25	48	97	120	29	44	113	104
88	129	58	15	98	119	28	45	116	101	32	41
13	60	131	86	47	26	117	100	43	30	103	114
84	133	14	59	118	99	46	27	102	115	42	31
61	12	85	132	7	78	139	68	141	72	1	74
134	83	62	11	138	67	6	77	4	75	142	71
63	10	81	136	65	8	79	140	69	144	73	2
82	135	64	9	80	137	66	5	76	3	70	143

Fig.20a

Fig.28 is the nearest to perfection so far achieved. In this the odd diagonal is 44 short of the magic constant. However, its broken odd diagonal, shown in orange, also adds to the magic constant. [Note this tour has been moved out of sequence, but to avoid confusion the editor decided not to renumber all the tours.]

Fig.28 OD=826 ED=870
127	54	19	92	21	94	123	50	35	38	107	110
18	91	126	53	124	51	22	95	106	109	36	39
55	128	89	20	93	24	49	122	37	34	111	108
90	17	56	125	52	121	96	23	112	105	40	33
15	58	129	88	25	48	27	44	117	102	113	104
86	131	16	57	120	97	118	101	28	43	32	41
59	14	87	130	47	26	99	116	45	30	103	114
132	85	60	13	98	119	46	29	100	115	42	31
61	12	133	84	65	8	79	140	69	144	73	2
134	83	64	9	80	137	66	5	76	3	70	143
11	62	81	136	7	78	139	68	141	72	1	74
82	135	10	63	138	67	6	77	4	75	142	71

Figure 28a

(69) Odd Magic Diagonal with some Broken Diagonals Magic.

It is true that the reverse of an even diagonal magic tour gives an odd diagonal magic tour but the tour shown in Fig.21 is not related like that to previous tours. It has its odd diagonal magic. Fig.22 to Fig.27 can be derived from it by modifying non magic-diagonal quads.

Fig.21 OD=870 ED=996
19	54	125	92	21	52	123	94	35	38	107	110
126	91	20	53	124	93	22	51	106	109	36	39
55	18	127	90	49	24	95	122	37	34	111	108
88	129	16	57	96	121	50	23	112	105	40	33
17	56	89	128	119	48	25	100	29	42	113	104
130	87	58	15	26	97	120	45	116	103	32	41
13	60	131	86	47	118	99	28	101	30	43	114
132	85	14	59	98	27	46	117	44	115	102	31
61	12	81	8	65	138	77	140	69	144	73	2
84	133	64	137	80	7	68	5	76	3	70	143
11	62	135	82	9	66	139	78	141	72	1	74
134	83	10	63	136	79	6	67	4	75	142	71

Fig.22 OD=870 ED=1142
19	54	125	92	21	52	123	94	35	38	107	110
126	91	20	53	124	93	22	51	106	109	36	39
55	18	127	90	49	24	95	122	37	34	111	108
88	129	16	57	96	121	50	23	112	105	40	33
17	56	89	128	119	48	25	100	29	42	113	104
130	87	58	15	26	97	120	45	116	103	32	41
13	60	131	86	47	118	99	28	101	30	43	114
132	85	14	59	98	27	46	117	44	115	102	31
61	12	83	136	65	8	77	140	69	144	73	2
84	133	62	9	80	137	68	5	76	3	70	143
11	82	135	64	7	66	139	78	141	72	1	74
134	63	10	81	138	79	6	67	4	75	142	71

Fig. 23 OD=870 ED=1142
19	54	125	92	21	52	123	94	35	38	107	110
126	91	20	53	124	93	22	51	106	109	36	39
55	18	127	90	49	24	95	122	37	34	111	108
88	129	16	57	96	121	50	23	112	105	40	33
17	56	89	128	119	48	25	100	29	42	113	104
130	87	58	15	26	97	120	45	116	103	32	41
13	60	131	86	47	118	99	28	101	30	43	114
132	85	14	59	98	27	46	117	44	115	102	31
61	12	83	136	65	6	79	140	69	144	73	2
84	133	62	9	80	139	66	5	76	3	70	143
11	82	135	64	137	78	7	68	141	72	1	74
134	63	10	81	8	67	138	77	4	75	142	71

Fig.24 OD=870 ED=1142
19	54	125	92	21	52	123	94	35	38	107	110
126	91	20	53	124	93	22	51	106	109	36	39
55	18	127	90	49	24	95	122	37	34	111	108
88	129	16	57	96	121	50	23	112	105	40	33
17	56	89	128	119	48	25	100	29	42	113	104
130	87	58	15	26	97	120	45	116	103	32	41
13	60	131	86	47	118	99	28	101	30	43	114
132	85	14	59	98	27	46	117	44	115	102	31
61	12	83	136	79	138	67	6	69	144	73	2
84	133	62	9	66	7	78	139	76	3	70	143
11	82	135	64	137	80	5	68	141	72	1	74
134	63	10	81	8	65	140	77	4	75	142	71

Fig.25 OD=870 ED=1164
19	54	125	92	21	52	123	94	35	38	107	110
126	91	20	53	124	93	22	51	106	109	36	39
55	18	127	90	49	24	95	122	37	34	111	108
88	129	16	57	96	121	50	23	112	105	40	33
17	56	89	128	119	48	25	100	29	42	113	104
130	87	58	15	26	97	120	45	116	103	32	41
13	60	131	86	47	118	99	28	101	30	43	114
132	85	14	59	98	27	46	117	44	115	102	31
83	12	61	136	65	8	77	140	69	144	73	2
62	133	84	9	80	137	68	5	76	3	70	143
11	82	135	64	7	66	139	78	141	72	1	74
134	63	10	81	138	79	6	67	4	75	142	71

Fig.26 OD=870 ED=1164
19	54	125	92	21	52	123	94	35	38	107	110
126	91	20	53	124	93	22	51	106	109	36	39
55	18	127	90	49	24	95	122	37	34	111	108
88	129	16	57	96	121	50	23	112	105	40	33
17	56	89	128	119	48	25	100	29	42	113	104
130	87	58	15	26	97	120	45	116	103	32	41
13	60	131	86	47	118	99	28	101	30	43	114
132	85	14	59	98	27	46	117	44	115	102	31
83	12	61	136	79	138	67	6	69	144	73	2
62	133	84	9	66	7	78	139	76	3	70	143
11	82	135	64	137	80	5	68	141	72	1	74
134	63	10	81	8	65	140	77	4	75	142	71

Fig.27 OD=870 ED=1164
19	54	125	92	21	52	123	94	35	38	107	110
126	91	20	53	124	93	22	51	106	109	36	39
55	18	127	90	49	24	95	122	37	34	111	108
88	129	16	57	96	121	50	23	112	105	40	33
17	56	89	128	119	48	25	100	29	42	113	104
130	87	58	15	26	97	120	45	116	103	32	41
13	60	131	86	47	118	99	28	101	30	43	114
132	85	14	59	98	27	46	117	44	115	102	31
83	12	61	136	65	6	79	140	69	144	73	2
62	133	84	9	80	139	66	5	76	3	70	143
11	82	135	64	137	78	7	68	141	72	1	74
134	63	10	81	8	67	138	77	4	75	142	71

Figure 27a

Fig.25 and Fig.27 are unique in the sense that two of their broken diagonals are magic. [The editor notes that in the tour in Fig.27 the magic diagonals occur at regular intervals. Note also that Fig.28 has been moved to the end of section (68).]

Historical note. T. H. Willcocks, G. P. Jelliss and Awani Kumar have previously constructed 'almost perfect magic tours' of open type, but only H. J. R. Murray (1947) had previously constructed, a re-entrant 4-fold cyclic 'almost perfect magic tour' (although he may not have noticed that one diagonal was magic). However, his tour is not 'prolific' in producing other such tours. So, such tours, once rare are now dime a dozen. Human ingenuity can do wonders!

(70) Some Open Tours.

T. H. Willcocks has constructed three open, single diagonal magic tours from which the author has derived over 80 such tours along with four diagonally magic tours. [See The Games and Puzzles Journal # 26]. G.P.Jelliss (2003) has also constructed four single diagonal open magic tours from which the author has derived five more such tours as shown in Fig.29 to Fig.33.

Fig.29 OD=870 ED=850
5	64	101	120	3	66	107	114	35	70	111	74
100	121	4	65	106	115	2	67	112	73	34	71
63	6	119	102	61	8	113	108	69	36	75	110
122	99	62	7	116	105	68	1	76	109	72	33
11	58	103	118	9	60	81	140	37	32	77	144
98	123	10	59	104	117	42	27	80	141	38	31
57	12	125	96	43	26	139	82	29	40	143	78
124	97	56	13	138	83	28	41	142	79	30	39
15	54	95	126	25	44	85	136	23	46	87	134
94	127	14	55	84	137	24	45	86	135	22	47
53	16	129	92	51	18	131	90	49	20	133	88
128	93	52	17	130	91	50	19	132	89	48	21

Fig.30 OD=870 ED=850
5	64	105	116	3	66	107	114	35	70	111	74
104	117	4	65	106	115	2	67	112	73	34	71
63	6	119	102	61	8	113	108	69	36	75	110
118	103	62	7	120	101	68	1	76	109	72	33
11	58	99	122	9	60	81	140	37	32	77	144
98	123	10	59	100	121	42	27	80	141	38	31
57	12	129	92	43	26	139	82	29	40	143	78
124	97	44	25	130	91	28	41	142	79	30	39
13	56	93	128	23	46	83	138	21	48	85	136
96	125	24	45	90	131	22	47	84	137	20	49
55	14	127	94	53	16	133	88	51	18	135	86
126	95	54	15	132	89	52	17	134	87	50	19

Fig.31 OD=870 ED=926
5	64	105	116	3	66	107	114	35	70	111	74
104	117	4	65	106	115	2	67	112	73	34	71
63	6	119	102	61	8	113	108	69	36	75	110
118	103	62	7	120	101	68	1	76	109	72	33
11	58	99	122	9	60	81	140	37	32	77	144
124	97	10	59	100	121	42	27	80	141	38	31
57	12	123	98	43	26	139	82	29	40	143	78
96	125	56	13	138	83	28	41	142	79	30	39
55	14	95	126	25	44	85	136	23	46	87	134
94	127	52	17	84	137	24	45	86	135	22	47
15	54	129	92	51	18	131	90	49	20	133	88
128	93	16	53	130	91	50	19	132	89	48	21

Fig.32 OD=870 ED=890
5	64	105	116	3	66	107	114	35	70	111	74
104	117	4	65	106	115	2	67	112	73	34	71
63	6	119	102	61	8	113	108	69	36	75	110
118	103	62	7	120	101	68	1	76	109	72	33
11	58	123	98	9	60	81	140	37	32	77	144
124	97	10	59	100	121	42	27	80	141	38	31
57	12	99	122	43	26	139	82	29	40	143	78
96	125	56	13	138	83	28	41	142	79	30	39
55	14	95	126	25	44	85	136	23	46	87	134
94	127	16	53	84	137	24	45	86	135	22	47
15	54	129	92	51	18	131	90	49	20	133	88
128	93	52	17	130	91	50	19	132	89	48	21

Fig.33 OD=870 ED=926
5	64	105	116	3	66	107	114	35	70	111	74
104	117	4	65	106	115	2	67	112	73	34	71
63	6	119	102	61	8	113	108	69	36	75	110
118	103	62	7	120	101	68	1	76	109	72	33
11	58	123	98	9	60	81	140	37	32	77	144
124	97	10	59	100	121	42	27	80	141	38	31
57	12	99	122	43	26	139	82	29	40	143	78
96	125	56	13	138	83	28	41	142	79	30	39
55	14	95	126	25	44	85	136	23	46	87	134
94	127	52	17	84	137	24	45	86	135	22	47
15	54	129	92	51	18	131	90	49	20	133	88
128	93	16	53	130	91	50	19	132	89	48	21

Conclusion: A re-entrant 'perfect magic tour' (both diagonals magic) has remained elusive. However, broken diagonals adding to the magic constant have been shown in several of the figures. The whole world of pan diagonal magic tours has remained unexplored. Readers are requested to look into it.

Using powerful computers and intelligent programming, the international team of Hugues Mackay, JC Meyrignac, and Guenter Stertenbrink et al has enumerated all the 280 magic tours on 8*8 board. So the work started by Beverley (1848) has been finished after 155 years in 2003. With dedicated and sustained effort, let us enumerate all the 12*12 magic tours within the next 5 years.

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