Office Applications and Entertainment, Latin Squares | ||
 |
Index | About the Author |
|
A Latin Square of order 13 is a 13 x 13 square filled with 13 different symbols, each occurring only once in each row and only once in each column.
13.1 Latin Diagonal Squares (13 x 13)
Latin Diagonal Squares
are Latin Squares for which the 13 different symbols occur also only once in each of the main diagonals.
13.2 Magic Squares, Natural Numbers
Pan Magic Square M of order 13 with the integers 1 ... 169 can be written as
M = A + 13 * B + 1
where the squares A and B contain only the integers
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12.
Attachment 13.2.1 shows the ten types Latin Diagonal Squares based on the construction method described above.
Each type Latin Diagonal Square described above, corresponds with 12! = 479.001.600 Latin Diagonal Squares.
The Latin Squares discussed and applied in previous section are referred to as Cyclic Pan Diagonal Latin Squares.
An example of the construction of an order 13 Ultra Magic Square M based on pairs of Orthogonal Latin Diagonal Squares (A, B), is shown below for the symbols {ai, i = 1 ... 13} and {bj, j = 1 ... 13}.
The Latin Diagonal Square B is the transposed square of A (rows and columns exchanged).
The Latin Diagonal Squares A can be determined based on the defining equations of the top and bottom row, as provided below for Latin Diagonal Squares type R2:
|
|
a(168) = 2 * s1 / 13 - a(169) |
a(8) = 3 * s1 / 13 - a(168) - a(169) |
a(13) = a(167) |
|
The solutions can be obtained by guessing the 7 parameters:
Overlapping Sub Squares
Order 13 Magic Squares, containing order 7 Overlapping Sub Squares with identical Magic Sum,
based on Latin Diagonal Sub Squares, have been discussed in Section 25.5.
13.2.5 Concentric Magic Squares
Order 13 Concentric Magic Squares M can be constructed based on pairs of Orthogonal Concentric Semi-Latin Squares (A, B), as shown below for the symbols {ai, i = 1 ... 13} and {bj, j = 1 ... 13).
All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
A pair of order 13 Orthogonal Semi-Latin Borders can be constructed
for each pair of order 11 Orthogonal Concentric Semi-Latin Squares
(A11, B11),
as found in Section 11.2.3.
Diamond Inlays Order 4 and 5
Order 13 Bordered Magic Squares M can be constructed based on pairs of Orthogonal Bordered
Semi-Latin Squares
(A, B)
for miscellaneous types of Center Squares.
All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
A pair of order 13 Orthogonal Semi-Latin Borders can be constructed
for each pair of order 11 Orthogonal Bordered Semi-Latin Squares
(A11, B11),
as found in Section 11.2.4.
The example shown below is based on Center Squares with order 5 and 6 Diamond Inlays - as discussed in Section 11.2.6 - and the symbols {ai, i = 1 ... 13} and {bj, j = 1 ... 13).
All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
A pair of order 13 Orthogonal Semi-Latin Borders can be constructed
for each pair of order 11 Orthogonal Inlaid Semi-Latin Squares
(A11, B11),
as found in Section 11.2.6.
13.2.7 Associated Magic Squares
Diamond Inlays order 6 and 7
Order 13 Associated Magic Squares M, with order 6 and 7 Diamond Inlays, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B), as shown below for the integers {0, 1, ... 12}. |
A
5 0 7 12 2 3 8 11 6 1 9 10 4 5 11 9 6 0 4 2 8 10 1 7 3 12 4 8 9 5 12 10 6 1 3 7 2 11 0 10 3 1 2 9 7 6 11 4 5 12 0 8 8 4 3 5 0 9 1 7 12 10 2 6 11 8 12 7 2 11 4 6 1 3 9 10 5 0 1 3 12 7 10 8 6 4 2 5 0 9 11 12 7 2 3 9 11 6 8 1 10 5 0 4 1 6 10 2 0 5 11 3 12 7 9 8 4 4 12 0 7 8 1 6 5 3 10 11 9 2 12 1 10 5 9 11 6 2 0 7 3 4 8 0 9 5 11 2 4 10 8 12 6 3 1 7 8 2 3 11 6 1 4 9 10 0 5 12 7 B = T(A)
5 5 4 10 8 8 1 12 1 4 12 0 8 0 11 8 3 4 12 3 7 6 12 1 9 2 7 9 9 1 3 7 12 2 10 0 10 5 3 12 6 5 2 5 2 7 3 2 7 5 11 11 2 0 12 9 0 11 10 9 0 8 9 2 6 3 4 10 7 9 4 8 11 5 1 11 4 1 8 2 6 6 1 6 6 6 11 6 6 10 4 11 8 1 11 7 1 4 8 3 5 2 8 9 6 10 3 4 12 3 2 1 12 3 0 12 10 1 1 7 5 10 9 5 10 7 10 7 6 0 9 7 2 12 2 10 0 5 9 11 3 3 5 10 3 11 0 6 5 9 0 8 9 4 1 12 4 12 0 8 11 0 11 4 4 2 8 7 7 M = 13 * A + B + 1
71 6 96 167 35 48 106 156 80 18 130 131 61 66 155 126 82 5 65 30 112 137 26 93 49 159 60 114 127 67 160 138 91 16 50 92 37 149 4 143 46 19 29 123 94 86 147 55 73 162 12 116 107 53 52 75 1 129 24 101 157 139 36 81 150 108 161 102 34 153 57 87 25 45 119 142 70 2 22 42 163 98 132 111 85 59 38 72 7 128 148 168 100 28 51 125 145 83 113 17 136 68 9 62 20 89 134 31 13 69 146 41 169 95 118 117 63 54 158 8 97 115 23 84 76 47 141 151 124 27 166 21 133 78 120 154 79 32 10 103 43 56 110 11 121 77 144 33 58 140 105 165 88 44 15 104 109 39 40 152 90 14 64 122 135 3 74 164 99
|
The Semi-Latin Square B is the transposed square of A
(rows and columns exchanged).
With procedure SemiLat13
numerous Self-Orthogonal Semi-Latin Associated Squares A
with order 6 and 7 Diamond Inlays can be found,
of which a few are shown in Attachment 13.2.6.
Order 13 Associated Lozenge Squares based on Latin Squares have been discussed in Section 18.7.1.
The obtained results regarding the order 13 (Semi) Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:
Comparable methods as described above, can be used to construct higher order (Semi) Latin - and related (Pan) Magic Squares,
which will be described in following sections.
|
|
|
Index | About the Author |