The Games and Puzzles Journal — Issue 26, March-April 2003

Awani Kumar has constructed the first 12×12 fully magic knight's tours (both diagonals magic). The details of his work leading to this important result are given in full. Some work by the editor on magic tours on 4m × 4n+2 rectangles follows, first using knight and rook moves, second relaxing the magic condition to ‘quasi-magic’.


Back to: GPJ Index Page
Sections on this page: (36) In Search of 12×12 Magic Square Tours. (37) Four Perfect Magic Tours 12×12. (38) A New Type of Magic Tour. (39) Emperor Magic Tours. (40) Quasi-Magic Tours. End
In Search of Perfect Magic Tours of Knight on 12×12 Board.
by Awani Kumar

17 Rana Pratap Marg, Lucknow 226001. INDIA.
(This article was sent by e-mail on 25 March 2003, tours 9 and 27 were added 3 April 2003.)
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Magic squares and knight's tours have been attracting attention of mathematicians for centuries, however, the twain rarely meet. In spite of the very large number of magic squares and knight's tours possible on doubly even boards, until now 16×16 was the smallest board size on which a ‘perfect magic tour’ (with both diagonals adding to the magic constant) were found. The first, by Helga Em. de Vasa, were published in 1962 [See Knight's Tour Notes: 16×16 Magic Tours].

All the 266 magic knight tours found on the 8×8 board are ‘weak’ magic squares, since their diagonals do not add to the magic constant. On the 12×12 board, Murray, Willcocks, Jelliss and Awani Kumar have constructed scores of ‘simple magic tours’. These composers have also constructed ‘almost perfect magic tours’ (with one diagonal adding to the magic constant) but until now a ‘perfect magic tour’ has remained elusive.


Figure 1 by T. H. Willcocks (published in Recreational Mathematics Magazine 1962) is the nearest to perfection previously achieved. A close look at it reveals that the board can be divided into nine ‘quads’, each of size 4×4. The sum of rows and columns of each quad is 290, that is, one-third of the magic constant. The four 2×2 ‘blocks’ within each quad also sum up to 290. The numbers can be divided into four groups: 1 to 36, 37 to 72, 73 to 108 and 109 to 144. Each of the thirty-six blocks has numbers from the four groups.

99 10117 64107 2109 72 35 74 37144
118 63100 9 70111106 3142 39 34 75
11 98 65116 1108 71110 73 36143 38
62119 8101112 69 4105 40141 76 33
97 12115 66103 6113 68 77 32139 42
120 61102 7114 67104 5140 41 78 31
13 96 55126 19 90 49132 25 84 43138
60121 18 91 54127 24 85 48133 30 79
95 14125 56 89 20131 50 83 26137 44
122 59 92 17128 53 86 23134 47 80 29
15 94 57124 21 88 51130 27 82 45136
58123 16 93 52129 22 87 46135 28 81
Fig.1. OD=888 ED=870


The author has found 262 magic tours by maintaining the even distribution of numbers but keeping the sum of rows and columns of each quad between 282 and 298. Fig. 2 and Fig. 3 have the unique property of diagonals differing by 4 and their sum differing by 8 from twice the magic constant. That is, |OD – ED| = 4 and |OD + ED – 1740| = 8. Fig. 4 and Fig.5 are closer to ‘perfection’ since they have |OD – ED| = 2 and |OD + ED – 1740| = 6.
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32  41 140
120  61 102   7 114  67 104   5 138  43  30  79
 13  96  59 122  23  86  51 130  31  78 139  42
 60 121  14  95  52 129  24  85  44 137  80  29
 93  16 123  58  87  22 131  50  25  84  45 136
124  57  94  15 128  53  88  21 134  47  28  81
 17  92  55 126  19  90  49 132  83  26 135  46
 56 125  18  91  54 127  20  89  48 133  82  27
Fig. 2. OD=868 ED=864
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  31  78  41 140
120  61 102   7 114  67 104   5  42 139  32  77
 13  96  59 122  23  86  51 130  79  30 137  44
 60 121  14  95  52 129  24  85 138  43  80  29
 93  16 123  58  87  22 131  50  25  84  45 136
124  57  94  15 128  53  88  21 134  47  28  81
 17  92  55 126  19  90  49 132  83  26 135  46
 56 125  18  91  54 127  20  89  48 133  82  27
Fig. 3. OD=868 ED=864
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  31  78  41 140
120  61 102   7 114  67 104   5  42 139  32  77
 13  96  53 128  19  90  51 130  79  30 137  44
 60 121  18  91  52 129  24  85 138  43  80  29
 95  14 127  54  89  20 131  50  25  84  45 136
122  59  92  17 126  55  86  23 134  47  28  81
 15  94  57 124  21  88  49 132  83  26 135  46
 58 123  16  93  56 125  22  87  48 133  82  27
Fig. 4. OD=868 ED=866
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32  41 140
120  61 102   7 114  67 104   5 138  43  30  79
 13  96  53 128  19  90  51 130  31  78 139  42
 60 121  18  91  52 129  24  85  44 137  80  29
 95  14 127  54  89  20 131  50  25  84  45 136
122  59  92  17 126  55  86  23 134  47  28  81
 15  94  57 124  21  88  49 132  83  26 135  46
 58 123  16  93  56 125  22  87  48 133  82  27
Fig. 5. OD=868 ED=866
[Editor's note: All the magic squares in this article are shown in the same orientation as the original Willcocks tour for purposes of comparison. If the conventions used in the ‘Knight's Tour Notes’ catalogue of 8×8 tours were followed the numbering of the tours would be reversed (to give 1 in the corner instead of 144) and rotated 90 degrees anticlockwise, to place the 1 at the top left corner.]
Figures 6, 7, 8 and 9, also derived from the Willcocks tour, are ‘almost perfect magic tours’ in which the sum of the odd diagonal is equal to the magic constant. [Of course, reversing the numbering gives a tour with even diagonal magic.]
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32  41 140
120  61 102   7 114  67 104   5 138  43  30  79
 13  96  55 126  23  86  53 128  31  78 139  42
 60 121  22  87  54 127  24  85  44 137  80  29
 95  14 125  56  89  20 129  52  25  84  45 136
122  59  88  21 130  51  90  19 134  47  28  81
 15  94  57 124  17  92  49 132  83  26 135  46
 58 123  16  93  50 131  18  91  48 133  82  27
Fig. 6. OD=870 ED=862
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  31  78  41 140
120  61 102   7 114  67 104   5  42 139  32  77
 13  96  55 126  23  86  53 128  79  30 137  44
 60 121  14  95  54 127  24  85 138  43  80  29
 93  16 125  56  87  22 129  52  25  84  45 136
122  59  94  15 130  51  88  21 134  47  28  81
 17  92  57 124  19  90  49 132  83  26 135  46
 58 123  18  91  50 131  20  89  48 133  82  27
Fig. 7. OD=870 ED=866
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  31  78  41 140
120  61 102   7 114  67 104   5  42 139  32  77
 13  96  55 126  23  86  53 128  79  30 137  44
 60 121  22  87  54 127  24  85 138  43  80  29
 95  14 125  56  89  20 129  52  25  84  45 136
122  59  88  21 130  51  90  19 134  47  28  81
 15  94  57 124  17  92  49 132  83  26 135  46
 58 123  16  93  50 131  18  91  48 133  82  27
Fig. 8. OD=870 ED=862
 99  10 117  64 107   2 109  72  35  74  37 144 
118  63 100   9  70 111 106   3 142  39  34  75 
 11  98  65 116   1 108  71 110  73  36 143  38 
 62 119   8 101 112  69   4 105  40 141  76  33 
 97  12 115  66 103   6 113  68  77  32  41 140 
120  61 102   7 114  67 104   5 138  43  30  79 
 13  96  55 126  23  86  53 128  31  78 139  42 
 60 121  14  95  54 127  24  85  44 137  80  29 
 93  16 125  56  87  22 129  52  25  84  45 136 
122  59  94  15 130  51  88  21 134  47  28  81 
 17  92  57 124  19  90  49 132  83  26 135  46 
 58 123  18  91  50 131  20  89  48 133  82  27 
Fig. 9. OD=870 ED=866

The author has enumerated sixty-five ‘almost perfect magic tours’ with the even diagonal equal to magic constant and a few of them are shown from Fig.10 to Fig 21. Others can be derived from these tours by keeping the quads containing the even diagonal intact and rearranging the other quads.
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32 139  42
120  61 102   7 114  67 104   5 140  41  78  31
 13  96  55 126  19  90  53 128  29  80  43 138
 60 121  18  91  54 127  20  89  44 137  30  79
 95  14 125  56  87  52 129  24 133  28  81  46
122  59  92  17 130  21  88  49  84  45 136  27
 15  94  57 124  51  86  23 132  25 134  47  82
 58 123  16  93  22 131  50  85  48  83  26 135
Fig. 10. OD=1000 ED=870
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32 139  42
120  61 102   7 114  67 104   5 140  41  78  31
 13  96  55 126  19  90  53 128  29  80  43 138
 60 121  18  91  54 127  20  89  44 137  30  79
 95  14 125  56 131  52  87  28 129  46  81  26
122  59  92  17  86  21 130  45  88  27 136  47
 15  94  57 124  51 132  23  84  49 134  25  82
 58 123  16  93  22  85  50 133  24  83  48 135
Fig. 11. OD=956 ED=870
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32 139  42
120  61 102   7 114  67 104   5 140  41  30  79
 13  96  55 126  19  90  53 128  31  78  43 138
 60 121  18  91  54 127  20  89  44 137  80  29
 95  14 125  56  87  22 129  52  81  28  45 136
122  59  92  17 130  51  88  21  48 133  82  27
 15  94  57 124  23  86  49 132  25  84 135  46
 58 123  16  93  50 131  24  85 134  47  26  83
Fig. 12. OD=1072 ED=870
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32 139  42
120  61 102   7 114  67 104   5 140  41  30  79
 13  96  55 126  19  90  53 128  31  78  43 138
 60 121  18  91  54 127  20  89  44 137  80  29
 95  14 125  56  87  22 129  52  81  28  45 136
122  59  92  17 130  51  88  21 134  47  82  27
 15  94  57 124  23  86  49 132  25  84 135  46
 58 123  16  93  50 131  24  85  48 133  26  83
Fig. 13. OD=986 ED=870
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32  41 140
120  61 102   7 114  67 104   5 138  43  30  79
 13  96  55 126  19  90  53 128  31  78 139  42
 60 121  18  91  54 127  24  85  44 137  80  29
 95  14 125  56  89  20 129  52  25  84  45 136
122  59  92  17 130  51  86  23  48 133  28  81
 15  94  57 124  21  88  49 132  83  26 135  46
 58 123  16  93  50 131  22  87 134  47  82  27
Fig. 14. OD=956 ED=870
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  31  78  41 140
120  61 102   7 114  67 104   5  42 139  32  77
 13  96  55 126  15  94  53 128  79  30 137  44
 60 121  14  95  54 127  24  85 138  43  80  29
 91  18 125  56  93  16 129  52  25  84  45 136
122  59  92  17 130  51  86  23  48 133  28  81
 19  90  57 124  21  88  49 132  83  26 135  46
 58 123  20  89  50 131  22  87 134  47  82  27
Fig. 15. OD=956 ED=870
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32 139  42
120  61 102   7 114  67 104   5 140  41  78  31
 13  96  55 122  19  94  45 132  29  84  43 138
 60 121  18  95  54 123  28  85  44 133  30  79
 91  14 125  56  93  20 131  46  83  26 137  48
126  59  92  17 124  53  86  27 134  47  80  25
 15  90  57 128  21  88  51 130  23  82  49 136
 58 127  16  89  52 129  22  87  50 135  24  81
Fig. 16. OD=888 ED=870
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32 139  42
120  61 102   7 114  67 104   5 140  41  78  31
 13  96  55 122  19  94  45 136  25  84  43 138
 60 121  18  95  54 123  24  85  44 137  30  79
 91  14 125  56  93  20 135  46  83  26 133  48
126  59  92  17 124  53  86  23 134  47  80  29
 15  90  57 128  21  88  51 130  27  82  49 132
 58 127  16  89  52 129  22  87  50 131  28  81
Fig. 17. OD=888 ED=870
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32 139  42
120  61 102   7 114  67 104   5 140  41  78  31
 13  96  55 122  19  94  45 136  29  80  43 138
 60 121  18  95  54 123  28  81  44 137  30  79
 91  14 125  56  93  20 135  46  83  26 133  48
126  59  92  17 124  53  82  27 134  47  84  25
 15  90  57 128  21  88  51 130  23  86  49 132
 58 127  16  89  52 129  22  87  50 131  24  85
Fig. 18. OD=888 ED=870
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32 139  42
120  61 102   7 114  67 104   5 140  41  78  31
 13  96  59 122  15  94  49 136  25  80  43 138
 60 121  14  95  58 123  24  81  48 137  30  79
 91  18 125  56  93  16 135  50  83  26 133  44
126  55  92  17 124  57  82  23 134  47  84  29
 19  90  53 128  21  88  51 130  27  86  45 132
 54 127  20  89  52 129  22  87  46 131  28  85
Fig. 19. OD=888 ED=870
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32 139  42
120  61 102   7 114  67 104   5 140  41  78  31
 13  96  59 122  19  90  45 132  29  84  43 138
 60 121  18  91  58 123  28  85  44 133  30  79
 95  14 125  56  89  20 131  46  83  26 137  48
126  55  92  17 124  57  86  27 134  47  80  25
 15  94  53 128  21  88  51 130  23  82  49 136
 54 127  16  93  52 129  22  87  50 135  24  81
Fig. 20. OD=888 ED=870
 99  10 117  64 107   2 109  72  35  74  37 144
118  63 100   9  70 111 106   3 142  39  34  75
 11  98  65 116   1 108  71 110  73  36 143  38
 62 119   8 101 112  69   4 105  40 141  76  33
 97  12 115  66 103   6 113  68  77  32 139  42
120  61 102   7 114  67 104   5 140  41  78  31
 13  96  59 126  15  90  49 136  25  80  43 138
 60 121  14  91  58 127  24  81  48 137  30  79
 95  18 125  56  89  16 135  50  83  26 133  44
122  55  92  17 128  57  82  23 134  47  84  29
 19  94  53 124  21  88  51 130  27  86  45 132
 54 123  20  93  52 129  22  87  46 131  28  85
Fig. 21. OD=888 ED=870

Four Perfect Magic Tours.
by A. Kumar
é
Figures 22, 23, 24 and 25 are the ‘perfect magic tours’, with both diagonals magic.

99 10117 64107 2109 72 35 74 37144
118 63100 9 70111106 3142 39 34 75
11 98 65116 1108 71110 73 36143 38
62119 8101112 69 4105 40141 76 33
97 12115 66103 6113 68 77 32 41140
120 61102 7114 67104 5138 43 30 79
13 96 55126 15 94 53128 31 78139 42
60121 14 95 54127 24 85 44137 80 29
91 18125 56 93 16129 52 25 84 45136
122 59 92 17130 51 86 23134 47 28 81
19 90 57124 21 88 49132 83 26135 46
58123 20 89 50131 22 87 48133 82 27
Fig.22. OD=870 ED=870
99 10117 64107 2109 72 35 74 37144
118 63100 9 70111106 3142 39 34 75
11 98 65116 1108 71110 73 36143 38
62119 8101112 69 4105 40141 76 33
97 12115 66103 6113 68 77 32 41140
120 61102 7114 67104 5138 43 30 79
13 96 55126 19 90 53128 31 78139 42
60121 18 91 54127 24 85 44137 80 29
95 14125 56 89 20129 52 25 84 45136
122 59 92 17130 51 86 23134 47 28 81
15 94 57124 21 88 49132 83 26135 46
58123 16 93 50131 22 87 48133 82 27
Fig.23. OD=870 ED=870
99 10117 64107 2109 72 35 74 37144
118 63100 9 70111106 3142 39 34 75
11 98 65116 1108 71110 73 36143 38
62119 8101112 69 4105 40141 76 33
97 12115 66103 6113 68 31 78 41140
120 61102 7114 67104 5 42139 32 77
13 96 55126 15 94 53128 79 30137 44
60121 14 95 54127 24 85138 43 80 29
91 18125 56 93 16129 52 25 84 45136
122 59 92 17130 51 86 23134 47 28 81
19 90 57124 21 88 49132 83 26135 46
58123 20 89 50131 22 87 48133 82 27
Fig.24. OD=870 ED=870
99 10117 64107 2109 72 35 74 37144
118 63100 9 70111106 3142 39 34 75
11 98 65116 1108 71110 73 36143 38
62119 8101112 69 4105 40141 76 33
97 12115 66103 6113 68 31 78 41140
120 61102 7114 67104 5 42139 32 77
13 96 55126 19 90 53128 79 30137 44
60121 18 91 54127 24 85138 43 80 29
95 14125 56 89 20129 52 25 84 45136
122 59 92 17130 51 86 23134 47 28 81
15 94 57124 21 88 49132 83 26135 46
58123 16 93 50131 22 87 48133 82 27
Fig.25. OD=870 ED=870
For diagrams of these tours go to: Knight's Tour Notes: 12×12 Magic Tours.

Dedication: T. H. Willcocks (1968) raised following question in his paper. “Is there a fully magic tour on a 12×12 board? Can any reader find one or a closer approximation than that of Fig.1?” The author has settled this question. Enumeration of magic tours has remained elusive and more effort is required in this direction. (The author dedicates this paper to Mr. T. H. Willcocks whose seminal works on magic tours have attracted, entertained, motivated and enchanted innumerable people. The author is obliged to Mr. G. P. Jelliss for providing photocopies of Willcocks' papers.)


A New Type of Magic Tour.
by A. Kumar
é

Figures 26 and 27 are curious magic tours since the sums of their diagonals are equal but different from the magic constant.

99 10 117 64 107 2 109 72 35 74 37 144
118 63 100 9 70 111 106 3 142 39 34 75
11 98 65 116 1 108 71 110 73 36 143 38
62 119 8 101 112 69 4 105 40 141 76 33
97 12 115 66 103 6 113 68 31 78 41 140
120 61 102 7 114 67 104 5 42 139 32 77
13 96 59 122 19 90 51 130 79 30 137 44
60 121 18 91 52 129 24 85 138 43 80 29
95 14 123 58 89 20 131 50 25 84 45 136
124 57 92 17 128 53 86 23 134 47 28 81
15 94 55 126 21 88 49 132 83 26 135 46
56 125 16 93 54 127 22 87 48 133 82 27
Fig. 26. OD=868 ED=868
99 10 117 64 107 2 109 72 35 74 37 144
118 63 100 9 70 111 106 3 142 39 34 75
11 98 65 116 1 108 71 110 73 36 143 38
62 119 8 101 112 69 4 105 40 141 76 33
97 12 115 66 103 6 113 68 77 32 41 140
120 61 102 7 114 67 104 5 138 43 30 79
13 96 59 122 19 90 51 130 31 78 139 42
60 121 18 91 52 129 24 85 44 137 80 29
95 14 123 58 89 20 131 50 25 84 45 136
124 57 92 17 128 53 86 23 134 47 28 81
15 94 55 126 21 88 49 132 83 26 135 46
56 125 16 93 54 127 22 87 48 133 82 27
Fig. 27. OD=868 ED=868

These are believed to be the only magic tours of the kind so far found on boards of all sizes. Readers are requested to look for more such magic tours.


Emperor Magic Tours
by G. P. Jelliss
é
The tours shown in this section are sometimes known as two-knight tours or four-knight tours since they consist of two or four sequences of knight moves joined by rook moves. The unorthodox chess-piece that combines the moves of knight and rook is known in Variant Chess as an empress and when the rook moves are restricted to single step (wazir) moves the piece (knight + wazir) is called an emperor. Since the moves of the emperor always take it to a cell of opposite colour to that on which it stands, the results of the theorems in Issue 25, regarding even-sided boards apply to it. The question of the possibility or not of magic knight's tours on rectangles 4m by 4n + 2, led me to consider emperor tours on such boards. As expected, they are possible. Examples follow. I am working on a new page on the Knight's Tour Notes website with details of previous work on tours of these types on the 8×8 board.
The following two are the only two-knight emperor magic tours on the 4×6 board. The magic constants are 50 and 75. These tours can be renumbered cyclically from the half-way point (i.e. 12-13 becomes 24-1 and vice versa) and remain magic but are then empress tours since the 12-13 rook-move link is a three-cell move.

14  7 22  3 18 11
23  4 13 12 21  2
 8 15  6 19 10 17
 5 24  9 16  1 20
23  4 13 12 21  2
14  7 22  3 18 11
 5 24  9 16  1 20
 8 15  6 19 10 17


Here are examples 8 by 6. The first was consciously constructed using ‘contiguous contraparallel chains’, but the others were constructed for their visual symmetry. In the first tour if the ranks are divided into three pairs then all these pairs add to 49 (rank total therefore 3×49 = 147. In the files pairs related by reflection in the horizontal median add to 25 or 73 which together equal 98 (twice 49) thus ensuring the file sum is 4×49 = 196. This is the same in the second tour, but in the third tour the constants in the files are 37 and 61.

34 15 46  3 32 17
47  2 33 16  5 44
14 35  4 45 18 31
 1 48 13 36 43  6
24 25 12 37 30 19
11 38 21 28  7 42
26 23 40  9 20 29
39 10 27 22 41  8
15 32 45  4 17 34
44  3 16 33 46  5
31 14  1 48 35 18
 2 43 36 13  6 47
23 30 37 12 19 26
42 11 24 25 38  7
29 22  9 40 27 20
10 41 28 21  8 39
42  5 36 13 44  7
35 22 43  6 27 14
 4 41 12 37  8 45
21 34 23 26 15 28
40  3 38 11 46  9
33 20 25 24 29 16
 2 39 18 31 10 47
19 32  1 48 17 30


Before going on to the 8×10 board, here are some 8×8 examples of similar type.

The following tour was found as an offshoot from a study of maximum number of 3-move knight-lines in tours of squares and diamonds type. The squares and diamonds are joined to make 12 three-unit lines, thus forming four knight-paths. When the ends of the knight paths are joined by wazir moves this gives a magic emperor (wazir + knight) tour. (Not diagonally magic: DO = 288, DE = 240.)

 6 27 64 33 32  1 38 59
63 34  5 28 37 60 31  2
26  7 36 61  4 29 58 39
35 62 25  8 57 40  3 30
14 19 56 41 24  9 46 51
55 42 13 20 45 52 23 10
18 15 44 53 12 21 50 47
43 54 17 16 49 48 11 22

The next tour was formed after a study of the structure of the first magic knight tour published by William Beverley in 1848. The righthand half of his tour is symmetrical (with horizontal axis). My idea was that perhaps Beverley started from a biaxially symmetric pattern (the H ‘crosspatch’ pattern) and fiddled around with the lefthand side until he found the solution using what we now call ‘beverley quartes’. My tour replaces the four knight moves in the braid on c3d3-c6d6 by four wazir links. The result is a magic emperor tour whose right-hand side is the same as the Beverley tour, both arithmetically and geometrically. (DO = 240, DE = 280, ading to 520, twice the magic constant.)

 3 30 45 52  5 28 43 54
46 51  4 29 44 53  6 27
31  2 49 48 25  8 55 42
50 47 32  1 56 41 26  7
15 18 33 64  9 24 39 58
34 63 16 17 40 57 10 23
19 14 61 36 21 12 59 38
62 35 20 13 60 37 22 11

There is no proof that Beverley found his tour by this method. Another, equally possible method based on ‘Contiguous Contraparallel Chains’ was described by H. J. R. Murray. Alternatively it may be that Beverley found his tour by some sort of search method, since his is the first of the regular tours to be encountered when they are arranged in sequence according to the Frénicle method for magic squares (i.e. oriented with the smallest number at the top left corner and the number to its right less than the number below it, and similar tours arranged according to the numbers when read row by row).

A similar magic empress (rook + knight) tour is possible by connecting d4-c4, d5-c5, c3-c6, d3-d6. The righthand side of this has the same geometrical form as the Beverley tour, but not the same arithmetical form. (The numbers on the diagonals are the same as for the previous tour, but permuted.)

51 46 29  4 53 44 27  6
30  3 52 45 28  5 54 43
47 50  1 32 41 56  7 26
 2 31 48 49  8 25 42 55
63 34 17 16 57 40 23 10
18 15 64 33 24  9 58 39
35 62 13 20 37 60 11 22
14 19 36 61 12 21 38 59

The next diagram shows that the same plan as for the first 8x6 tour above works on the 8×8 board, the right-hand side being geometrically like the Beverley tour, but not arithmetically. The horizontal pairs all add to 65, and the vertical pairs to 97 and 33 which together equal 130. (The diagonal sums are DO = 360, DE = 288).

46 19 62  3 44 21  6 59
63  2 45 20  5 60 43 22
18 47  4 61 24 41 58  7
 1 64 17 48 57  8 23 42
32 33 16 49 40 25 10 55
15 50 29 36  9 56 39 26
34 31 52 13 28 37 54 11
51 14 35 30 53 12 27 38


Now we come to the 8×10 example, on the same plan as the previous 8×8 tour: the horizontal pairs all add to 81 and the vertical pairs add to 41 and 121 giving the magic constants: row sum R = 5×81 = 405 and file sum S = 4×81 = 324.

58 23 78  3 26 55 74  7 28 53
79  2 57 24 75  6 27 54 73  8
22 59  4 77 56 25 10 71 52 29
 1 80 21 60  5 76 51 30  9 72
40 41 20 61 36 45 70 11 32 49
19 62 37 44 65 16 31 50 69 12
42 39 64 17 46 35 14 67 48 33
63 18 43 38 15 66 47 34 13 68

Evidently this scheme can be extended to any boards 8×2n (with n > 2).


Quasi-Magic Tours
by G. P. Jelliss
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By a quasi-magic tour I mean a rectangular (or square) tour in which the ranks add to the magic constant but the files add to two different values (or vice versa). Thus a quasi-magic tour is a special type of semi-magic tour. The following quasi-magic tours have been discovered in course of my attempts to construct magic tours on rectangular boards of dimensions 4h by 2k where k is odd, or to prove that such magic tours are impossible.
We begin with the smallest such board, namely 4×6. On this I find there are just two quasi-magic knight tours, and one two-knight emperor tour, summing to 75 in the 6-cell lines. The sums in the ranks are (a) 52,52,46,46,52,52, (b) 58,58,46,46,46,46, (c) 38,38,38,62,62,62. (See the previous section for two fully magic emperor examples.)

14 11 22  1 18  9    14 11 22  7 18  3    14  1 16  9 24 11
21  6 13 10 23  2    23  8 13  4 21  6    17  4 13 12 21  8
12 15  4 19  8 17    12 15 10 19  2 17     2 15  6 19 10 23
 5 20  7 16  3 24     9 24  1 16  5 20     5 18  3 22  7 20

There are no quasi-magic 4×6 knight tours with the 4-cell lines magic. However there are 15 two-knight emperor tours. The 4-cell lines add to 50 (all being composed of pairs adding to 25). The ranks in the first two add to 73 and 77. The ranks in the others all add to 69 and 81 in various sequences.

20 11 22  7 16  1    20  1 22  7 16 11    14  5 22  9 18  1    14  1 22  9 18  5
13  4 19 10 23  8    23  4 19 10 13  8    21 10 13  6 23  8    23 10 13  6 21  8
12 21  6 15  2 17     2 21  6 15 12 17     4 15 12 19  2 17     2 15 12 19  4 17
 5 14  3 18  9 24     5 24  3 18  9 14    11 20  3 16  7 24    11 24  3 16  7 20

20  3 24  7 16 11    24  3 20  7 16 11    16 11 24  7 20  3    20 11 18  1 22  9
23  6 21 10 13  8    21  6 23 10 13  8    13  8 15  4 23  6    13  6 21 10 17  2
 2 19  4 15 12 17     4 19  2 15 12 17    12 17 10 21  2 19    12 19  4 15  8 23
 5 22  1 18  9 14     1 22  5 18  9 14     9 14  1 18  5 22     5 14  7 24  3 16

20  9 18  1 22 11    24  5 22  9 14  7    16 11 18  3 20  1    22  1 14  5 18  9
17  6 21 10 13  2    21  2 13  6 17 10    13  8 15  6 23  4    13  4 23  8 15  6
 8 19  4 15 12 23     4 23 12 19  8 15    12 17 10 19  2 21    12 21  2 17 10 19
 5 16  7 24  3 14     1 20  3 16 11 18     9 14  7 22  5 24     3 24 11 20  7 16

16  7 14  3 24  5    18  3 20 11 16  1    14  7 16  3 24  5
13 10 17  6 21  2    21  6 17  2 13 10    17 10 13  6 21  2
12 15  8 19  4 23     4 19  8 23 12 15     8 15 12 19  4 23
 9 18 11 22  1 20     7 22  5 14  9 24    11 18  9 22  1 20


I have constructed nine 8×10 tours (three typical examples shown here) based on extending the braid in the Beverley type tours on the 8×8 to cover the extra two files. They all add to 324 in the files, as required in a magic tour, but the ranks sum to two alternating values. The first has the sums 417 and 393, the second 445 and 365, the third 415 and 395.

22 43 80 17 76 15 74 13 28 49
79 18 21 44 25 46 27 48 73 12
20 23 42 77 16 75 14 71 50 29
41 78 19 24 45 26 47 30 11 72
62  3 40 57 36 55 34 51 70  9
39 58 61  4 65  6 67 10 31 52
60 63  2 37 56 35 54 33  8 69
 1 38 59 64  5 66  7 68 53 32
22 43 80 17 76 45 26 15 74 47
79 18 21 44 25 16 75 46 27 14
20 23 42 77 50 71 12 29 48 73
41 78 19 24 11 30 49 72 13 28
62  3 40 57 70 51 32  9 68 53
39 58 61  4 31 10 69 52 33  8
60 63  2 37 56 65  6 35 54 67
 1 38 59 64  5 36 55 66  7 34
22 43 80 17 76 15 74 47 28 13
79 18 21 44 25 46 27 14 49 72
20 23 42 77 16 75 48 73 12 29
41 78 19 24 45 26 11 30 71 50
62  3 40 57 36 55 70 51 10 31
39 58 61  4 65  6 33  8 69 52
60 63  2 37 56 35 54 67 32  9
 1 38 59 64  5 66  7 34 53 68

This final example is constructed by the ‘lozenge’ method that I found for 12×12 magic tours, but due to the limitations of this board the result is only quasi-magic. The 10-cell lines add to 405 in five pairs adding to 81. The 8-cell lines add to 364 and 284 (the magic constant would be 324).

65 70 67 60 63 18 21 14 11 16
68 57 64 19 22 59 62 17 24 13
71 66 69 58 61 20 23 12 15 10
56 29  6 73 54 27  8 75 52 25
 5 72 55 28  7 74 53 26  9 76
30 35 32  3 38 41 80 49 46 51
33  4 37 42 79  2 39 44 77 48
36 31 34  1 40 43 79 47 50 45

Here are some 12×6 examples constructed by joining together two of Awani Kumar's 6×6 semimagic tours, suitably chosen. [See Knight's Tour Notes: 6×6 Semi-Magic Tours]. The 6-cell lines add to 219 and the 12-cell lines to 510 and 366 (the magic constant would be 438).

62  1  4 69 12 71    26 37 28 45 48 35
 3 68 61 72  5 10    39 44 25 36 29 46
60 63  2 11 70 13    24 27 38 47 34 49
67 16 65 56  9  6    43 40 53 32 21 30
64 59 18  7 14 57    52 23 42 19 50 33
17 66 15 58 55  8    41 54 51 22 31 20
44 19 22 51 30 53    56 67 58 15 18  5
21 50 43 54 23 28    69 14 55  6 59 16
42 45 20 29 52 31    66 57 68 17  4  7
49 34 47 38 27 24    13 70 11  2 63 60
46 41 36 25 32 39    10 65 72 61  8  3
35 48 33 40 37 26    71 12  9 64  1 62

Similar tours can be constructed by piling the 6×6 tours on top of one another to any length.


The question remains; are quasi-magic knight tours on boards 4m by (4n + 2) such as the 8×10 and 12×6 the best that can be achieved or are magic knight tours possible on at least some boards of these proportions? To settle this question is the aim of this on-going work.
Sections on this page: (36) In Search of 12×12 Magic Square Tours. (37) Four Perfect Magic Tours 12×12. (38) A New Type of Magic Tour. (39) Emperor Magic Tours. (40) Quasi-Magic Tours. Top
Back to: GPJ Index Page

Copyright 2003 G. P. Jelliss and A. Kumar.