This issue is devoted entirely to some excellent work by Awani Kumar that was sent to us by e-mail on 3rd August 2002. Where successive tours are very similarly numbered the editor, to assist in recognition, has highlighted in the second tour cells that differ from the numbering in the first tour. The author is at the following postal address: Awani Kumar, B-4, Forest Colony, Vibhuti Khand, Gomti Nagar, Lucknow - 226010, INDIA.

é (26) The Raja of Mysore's Tour and Related Tours.

The very first magic Knight’s Tour on 12x12 board (Fig.1) was constructed by Krishnaraj Wadiar, Raja of Mysore, India, before 1868. The author has observed that it is a rich mine of re-entrant magic tours. Hundreds of 'simple magic tours' (only rows and columns are magic, not diagonals) can be derived from it by modifying and connecting the sectors. Fig.1 can be divided into 9 sectors, each of 4x4 size, as shown in Fig.2. By modifying sectors E and F, another magic tour (Fig.3) can be obtained. Similarly, Fig.4 and Fig.5 can be obtained by modifying sector H of Fig.3. All these are 4-fold cyclic magic tours, that is, they remain magic when numbered from 37, 73 and 109. Fig.6 is unique in the sense that the sum of the diagonals (=1268) is minimum in it. A host of magic tours, with sum of the diagonals being twice the magic constant, can be constructed. Fig.7 and Fig.8 are just two examples. Fig.9 and Fig.10 show the magic tours with one diagonal (= 872), nearest to the magic constant (= 870).

Fig. 2.

A	B	C
D	E	F
G	H	I

71 74 143 2 103 42 113 32 105 36 111 38 142 1 70 73 114 31 104 41 112 39 108 35 75 72 3 144 43 102 29 116 33 106 37 110 4 141 76 69 30 115 44 101 40 109 34 107 77 68 139 6 99 46 117 28 95 50 123 22 140 5 78 67 118 27 100 45 122 23 94 51 65 80 7 138 47 98 25 120 49 96 21 124 8 137 66 79 26 119 48 97 24 121 52 93 81 64 9 136 85 60 129 16 89 56 125 20 10 135 84 61 132 13 88 57 128 17 92 53 63 82 173 12 59 86 15 130 55 90 19 126 134 11 62 83 14 131 58 87 18 127 54 91 Fig. 1. D1=608 D2=980

71 74 143 2 103 42 113 32 105 36 111 38 142 1 70 73 114 31 104 41 112 39 108 35 75 72 3 144 43 102 29 116 33 106 37 110 4 141 76 69 30 115 44 101 40 109 34 107 77 68 139 6 99 28 117 48 121 50 95 22 140 5 78 67 118 45 100 25 96 23 122 51 65 80 7 138 27 98 47 120 49 124 21 94 8 137 66 79 46 119 26 97 24 93 52 123 81 64 9 136 85 60 129 16 89 56 125 20 10 135 84 61 132 13 88 57 128 17 92 53 63 82 133 12 59 86 15 130 55 90 19 126 134 11 62 83 14 131 58 87 18 127 54 91 Fig. 3. D1=648 D2=1020

71 74 143 2 103 42 113 32 105 36 111 38 142 1 70 73 114 31 104 41 112 39 108 35 75 72 3 144 43 102 29 116 33 106 37 110 4 141 76 69 30 115 44 101 40 109 34 107 77 68 139 6 99 28 117 48 121 50 95 22 140 5 78 67 118 45 100 25 96 23 122 51 65 80 7 138 27 98 47 120 49 124 21 94 8 137 66 69 46 119 26 97 24 93 52 123 81 64 9 136 85 60 15 130 89 56 125 20 10 135 84 61 14 131 88 57 128 17 92 53 63 82 133 12 59 86 129 16 55 90 19 126 134 11 62 83 132 13 58 87 18 127 54 91 Fig. 4. D1=648 D2=1020

71 74 143 2 103 42 113 32 105 36 111 38 142 1 70 73 114 31 104 41 112 39 108 35 75 72 3 144 43 102 29 116 33 106 37 110 4 141 76 69 30 115 44 101 40 109 34 107 77 68 139 6 99 28 117 48 121 50 95 22 140 5 78 67 118 45 100 25 96 23 122 51 65 80 7 138 27 98 47 120 49 124 21 94 8 137 66 69 46 119 26 97 24 93 52 123 81 64 9 136 85 60 129 16 89 56 125 20 10 135 84 61 14 131 88 57 128 17 92 53 63 82 133 12 59 86 15 130 55 90 19 126 134 11 62 83 132 13 58 87 18 127 54 91 Fig. 5. D1=648 D2=1020

Fig. 6. D1=824 D2=444

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

19 54 91 126 21 52 123 94 35 38 107 110 90 127 20 53 124 93 50 23 106 109 36 39 55 18 125 92 51 22 95 122 37 34 111 108 128 89 56 17 96 121 24 49 112 105 40 33 57 16 129 88 25 48 97 120 29 44 113 104 86 131 58 15 98 119 28 45 102 115 32 41 13 60 87 130 47 26 117 100 43 30 103 114 132 85 14 59 118 99 46 27 116 101 42 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 Fig. 7. D1=720 D2=1020

19 54 91 126 21 52 123 94 35 38 107 110 90 127 20 53 124 93 50 23 106 109 36 39 55 18 125 92 51 22 95 122 37 34 111 108 128 89 56 17 96 121 24 49 112 105 40 33 57 16 129 88 25 48 97 120 29 44 113 104 86 131 58 15 98 119 28 45 102 115 32 41 13 60 87 130 47 26 117 100 43 30 103 114 132 85 14 59 118 99 46 27 116 101 42 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. 8. D1=720 D2=1020

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 85 132 47 118 99 28 101 30 43 114 86 131 60 13 98 27 46 117 44 115 102 31 61 12 133 84 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. 9. D1=1058 D2=872

19 54 125 92 21 52 123 94 35 38 107 110 126 91 20 53 124 93 72 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 48 119 100 29 42 113 104 86 131 16 57 120 97 26 45 116 103 32 41 59 14 87 130 47 118 99 28 101 30 43 114 132 85 60 13 98 27 46 117 44 115 102 31 61 12 133 84 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. 10. D1=936 D2=872

é (27) Some New 'Almost Perfect' Magic Tours.

T.H. Willcocks, H.J.R. Murray, E. Lange and G.P. Jelliss have also constructed magic knight tours on 12x12 board but a 'perfect magic tour' (with both the diagonals equal to magic constant) is still elusive. Willcocks has given three 'almost perfect magic tours' (with one diagonal equal to magic constant). The author has constructed 11 new 'almost perfect magic tours' shown in Fig.11 to Fig.21. Their reverse tours are also 'almost perfect magic tours.'

83 26 133 48 85 24 129 52 89 20 127 54 134 47 84 25 132 49 88 21 128 53 90 19 27 82 45 136 23 86 51 130 17 92 55 126 46 135 28 81 50 131 22 87 56 125 18 91 29 80 137 44 99 10 119 62 93 16 123 58 42 139 78 31 118 63 98 11 124 57 94 15 79 30 43 138 9 100 61 120 13 96 59 122 140 41 32 77 64 117 12 97 60 121 14 95 33 76 141 40 101 8 65 116 107 2 67 114 38 143 36 73 110 71 108 1 66 115 106 3 75 34 39 142 7 102 111 70 5 104 113 68 144 37 74 35 72 109 6 103 112 69 4 105 Fig. 11. D1=1016 D2=870

83 26 133 48 85 24 129 52 89 20 127 54 134 47 84 25 132 49 88 21 128 53 90 19 27 82 45 136 23 86 51 130 17 92 55 126 46 135 28 81 50 131 22 87 56 125 18 91 79 30 137 44 99 10 119 62 93 16 123 58 138 43 80 29 118 63 98 11 124 57 94 15 31 78 41 140 9 100 61 120 13 96 59 122 42 139 32 77 64 117 12 97 60 121 14 95 33 76 141 40 101 8 65 116 107 2 67 114 38 143 36 73 110 71 108 1 66 115 106 3 75 34 39 142 7 102 111 70 5 104 113 68 144 37 74 35 72 109 6 103 112 69 4 105 Fig. 12. D1=1016 D2=870

Fig. 13. D1=888 D2=870

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

83 26 133 48 85 24 131 50 87 22 129 52 134 47 84 25 132 49 86 23 130 51 88 21 27 82 45 136 13 96 59 122 15 94 53 128 46 135 28 81 60 121 14 95 58 123 20 89 29 80 137 44 97 12 119 62 93 16 127 54 42 139 78 31 120 61 98 11 124 57 90 19 79 30 43 138 9 100 63 118 17 92 55 126 140 41 32 77 64 117 10 99 56 125 18 91 33 76 141 40 101 8 65 116 107 2 67 114 38 143 36 73 110 71 108 1 66 115 106 3 75 34 39 142 7 102 111 70 5 104 113 68 144 37 74 35 72 109 6 103 112 69 4 105 Fig. 14. D1=1016 D2=870	83 26 133 48 85 24 131 50 87 22 129 52 134 47 84 25 132 49 86 23 130 51 88 21 27 82 45 136 13 96 59 122 15 94 53 128 46 135 28 81 60 121 14 95 58 123 20 89 79 30 137 44 97 12 119 62 93 16 127 54 138 43 80 29 120 61 98 11 124 57 90 19 31 78 41 140 9 100 63 118 17 92 55 126 42 139 32 77 64 117 10 99 56 125 18 91 33 76 141 40 101 8 65 116 107 2 67 114 38 143 36 73 110 71 108 1 66 115 106 3 75 34 39 142 7 102 111 70 5 104 113 68 144 37 74 35 72 109 6 103 112 69 4 105 Fig. 15. D1=1016 D2=870
83 26 133 48 85 24 131 50 93 16 123 58 134 47 84 25 132 49 86 23 124 57 94 15 27 82 45 136 21 88 51 130 17 92 59 122 46 135 28 81 52 129 22 87 56 125 14 95 29 80 137 44 89 20 127 54 91 18 121 60 42 139 78 31 128 53 90 19 126 55 96 13 79 30 43 138 5 104 67 114 7 102 61 120 140 41 32 77 68 113 6 103 66 115 12 97 33 76 141 40 105 4 69 112 101 8 119 62 38 143 36 73 110 71 108 1 116 65 98 11 75 34 39 142 3 106 111 70 9 100 63 118 144 37 74 35 72 109 2 107 64 117 10 99 Fig.16. D1=896 D2=870	85 24 131 50 87 22 129 52 89 20 127 54 132 49 86 23 130 51 88 21 128 53 90 19 25 84 47 134 27 82 57 124 17 92 55 126 48 133 26 83 46 135 16 93 56 125 18 91 79 30 137 44 81 28 123 58 95 14 121 60 138 43 80 29 136 45 94 15 122 59 96 13 31 78 41 140 5 104 67 114 7 102 61 120 42 139 32 77 68 113 6 103 66 115 12 97 33 76 141 40 105 4 69 112 101 8 119 62 38 143 36 73 110 71 108 1 116 65 98 11 75 34 39 142 3 106 111 70 9 100 63 118 144 37 74 35 62 109 2 107 64 117 10 99 Fig. 17. D1=888 D2=870
85 24 131 50 87 22 129 52 91 18 125 56 132 49 86 23 130 51 88 21 126 55 92 17 25 84 47 134 27 82 53 128 19 90 57 124 48 133 26 83 46 135 20 89 54 127 16 93 79 30 137 44 81 28 119 62 97 12 123 58 138 43 80 29 136 45 98 11 120 61 94 15 31 78 41 140 9 100 63 118 13 96 59 122 42 139 32 77 64 117 10 99 60 121 14 95 33 76 141 40 101 8 65 116 107 2 67 114 38 143 36 73 110 71 108 1 66 115 106 3 75 34 39 142 7 102 111 70 5 104 113 68 144 37 74 35 72 109 6 103 112 69 4 105 Fig. 18. D1=992 D2=870	85 24 131 50 87 22 129 52 93 16 123 58 132 49 86 23 130 51 88 21 124 57 94 15 25 84 47 134 27 82 53 128 17 92 59 122 48 133 26 83 46 135 20 89 56 125 14 95 79 30 137 49 81 28 127 54 91 18 121 60 138 43 80 29 136 45 90 19 126 55 96 13 31 78 41 140 5 104 67 114 7 102 61 120 42 139 32 77 68 113 6 103 66 115 12 97 33 76 141 40 105 4 69 112 101 8 119 62 38 143 36 73 110 71 108 1 116 65 98 11 75 34 39 142 3 106 111 70 9 100 63 118 144 37 74 35 72 109 2 107 64 117 10 99 Fig. 19. D1=888 D2=870
87 22 129 52 95 14 121 60 97 12 119 62 130 51 88 21 122 59 96 13 120 61 98 11 23 86 53 128 15 94 57 124 9 100 63 118 50 131 20 89 58 123 16 93 64 117 10 99 85 24 127 54 91 18 125 56 101 8 115 66 132 49 90 19 126 55 92 17 116 65 102 7 25 84 43 138 31 78 41 140 5 104 67 114 48 133 30 79 42 139 32 77 68 113 6 103 83 26 137 44 33 76 141 40 105 4 69 112 134 47 80 29 144 37 74 35 72 109 2 107 27 82 45 136 75 34 39 142 3 106 111 70 46 135 28 81 38 143 36 73 110 71 108 1 Fig. 20. D1=870 D2=844	87 22 129 52 97 12 119 62 99 10 117 64 130 51 88 21 120 61 98 11 118 63 100 9 23 86 53 128 13 96 59 122 15 94 65 116 50 131 20 89 60 121 14 95 58 123 8 101 85 24 127 54 91 18 125 56 93 16 115 66 132 49 90 19 126 55 92 17 124 57 102 7 25 84 43 138 31 18 41 140 5 104 67 114 48 133 30 79 42 139 32 77 68 113 6 103 83 26 137 44 33 76 141 40 105 4 69 112 134 47 80 29 144 37 74 35 72 109 2 107 27 82 43 136 75 34 39 142 3 106 111 70 46 135 28 81 38 143 36 73 110 71 108 1 Fig.21. D1=870 D2=836

é (28) Two Tours Nearest to 'Perfection'.

Fig.22 and Fig.23 have the unique property of diagonals differing by 2 and their sum differing by 18 from twice the magic constant. That is, mod (D1- D2) = 2 and mod (D1+ D2 - 1740) = 18. Their reverse tours also have this property. This is the nearest one has approached to 'perfection'.

83 26 133 48 85 24 131 50 87 22 129 52 134 47 84 25 132 49 86 23 130 51 88 21 27 82 45 136 7 102 65 116 9 100 53 128 46 135 28 81 66 115 8 101 64 117 20 89 29 80 137 44 103 6 63 118 99 10 127 54 42 139 78 31 114 67 12 97 120 61 90 19 79 30 43 138 5 104 119 62 11 98 55 126 140 41 32 77 68 113 96 13 60 121 18 91 33 76 141 40 105 4 69 112 95 14 125 56 38 143 36 73 110 71 108 1 122 59 92 17 75 34 39 142 3 106 111 70 15 94 57 124 144 37 74 35 72 109 2 107 58 123 16 93 Fig. 22. D1=862 D2=860

83 26 133 48 85 24 131 50 87 22 129 52 134 47 84 25 132 49 86 23 130 51 88 21 27 82 45 136 7 102 65 116 9 100 53 128 46 135 28 81 66 115 8 101 64 117 20 89 79 30 137 44 103 6 63 118 99 10 127 54 138 43 80 29 114 67 12 97 120 61 90 19 31 78 41 140 5 104 119 62 11 98 55 126 42 139 32 77 68 113 96 13 60 121 18 91 33 76 141 40 105 4 69 112 95 14 125 56 38 143 36 73 110 71 108 1 122 59 92 17 75 34 39 142 3 106 111 70 15 94 57 124 144 37 74 35 72 109 2 107 58 123 16 93 Fig. 23. D1=862 D2=860

é (29) A Tour by Murray and Related Tours.

H.J.R. Murray has also given an 'almost perfect magic tour' as shown in Fig.24. Fig.25 can be derived from Murray's tour by modifying its sector F and I. Like Murray's tour, it remains 'almost perfect magic tour' if the tour begins from 73. Fig.26 and Fig.27 can also be obtained. However, Wadiar's tour is more amenable to manipulation than Murray's.

19 54 95 122 21 52 93 124 23 50 91 126 96 121 20 53 94 123 22 51 92 125 24 49 55 18 131 86 45 28 129 88 47 26 127 90 120 97 44 29 130 87 46 27 128 89 48 25 17 56 85 132 35 38 81 136 71 138 79 2 98 119 30 43 134 83 34 37 80 3 70 139 57 16 133 84 39 36 135 82 137 72 1 78 118 99 42 31 104 113 40 33 4 77 140 69 15 58 103 114 41 32 105 112 73 68 5 144 100 117 12 61 106 109 10 63 8 141 76 67 59 14 115 102 11 62 111 108 65 74 143 6 116 101 60 13 110 107 64 9 142 7 66 75 Fig. 24. D1=1018 D2=870

19 54 95 122 21 52 93 124 23 50 91 126 96 121 20 53 94 123 22 51 92 125 24 49 55 18 131 86 45 28 129 88 47 26 127 90 120 97 44 29 130 87 46 27 128 89 48 25 17 56 85 132 35 38 81 136 3 138 79 70 98 119 30 43 134 83 34 37 80 71 2 139 57 16 133 84 39 36 135 82 137 4 69 78 118 99 42 31 104 113 40 33 72 77 140 1 15 58 103 114 41 32 105 112 141 68 5 76 100 117 12 61 106 109 10 63 8 73 144 67 59 14 115 102 11 62 111 108 65 142 75 6 116 101 60 13 110 107 64 9 74 7 66 143 Fig. 25. D1=1018 D2=870

19 54 95 122 21 52 93 124 23 50 91 126 96 121 20 53 94 123 22 51 92 125 24 49 55 18 131 86 45 28 129 88 47 26 127 90 120 97 44 29 130 87 46 27 128 89 48 25 17 56 85 132 35 38 81 136 71 138 79 2 98 119 30 43 134 83 34 37 80 3 70 139 57 16 133 84 39 36 135 82 137 72 1 78 118 99 42 31 104 113 40 33 4 77 140 69 15 58 103 114 41 32 105 112 73 68 5 144 100 117 60 13 106 109 10 63 8 141 76 67 59 14 115 102 11 62 111 108 65 74 143 6 116 101 12 61 110 107 64 9 142 7 66 75 Fig. 26. D1=1018 D2=918

19 54 95 122 21 52 93 124 23 50 91 126 96 121 20 53 94 123 22 51 92 125 24 49 55 18 131 86 45 28 129 88 47 26 127 90 120 97 44 29 130 87 46 27 128 89 48 25 17 56 85 132 35 38 81 136 3 138 79 70 98 119 30 43 134 83 34 37 80 71 2 139 57 16 133 84 39 36 135 82 137 4 69 78 118 99 42 31 104 113 40 33 72 77 140 1 15 58 103 114 41 32 105 112 141 68 5 76 100 117 60 13 106 109 10 63 8 73 144 67 59 14 115 102 11 62 111 108 65 142 75 6 116 101 12 61 110 107 64 9 74 7 66 143 Fig. 27. D1=1018 D2=918

é (30) Enumeration of Magic Tours.

Enumeration of magic tours is most difficult and little progress has been made in this direction. Till now, we have enumerated about 1000 'simple magic tours', 32 'almost perfect magic tours' and 'perfect magic tours' are still to be discovered. The author estimates that their total number will be around 100000, 5000 and 10 respectively. So we have a long way to go!