Office Applications and Entertainment, Latin Squares

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15.0   Latin Squares (15 x 15)

A Latin Squares of order 15 is a 15 x 15 square filled with 15 different symbols, each occurring only once in each row and only once in each column.

15.1   Latin Diagonal Squares (15 x 15)

Latin Diagonal Squares are Latin Squares for which the 15 different symbols occur also only once in each of the main diagonals.

15.2   Magic Squares, Natural Numbers

15.2.1 Pan Magic Squares (1)

Pan Magic Square M of order 15 with the integers 1 ... 225 can be written as M = A + 15 * B + 1 where the squares A and B contain only the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ... 14.

However, as 15 is a multiple of three, the Cyclic Method to construct order 15 Pan Magic Squares can only be based on pairs of Orthogonal Latin Squares (A, B).

The required Orthogonal Latin Squares (A, B) for Pan Magic Squares can be constructed as follows:

  1. Fill the first row of square A and square B with the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ... 14,
    such that:

    a(1) + a(4) + a(7) + a(10) + a(13) = 35
    a(2) + a(5) + a(8) + a(11) + a(14) = 35
    a(3) + a(6) + a(9) + a(12) + a(15) = 35

    with a(1) ... a(15) the variables of the first row.

  2. Complete square A and B by copying the first row into the following rows of the applicable square,
    according to one of the following two schemes:

    A/B L2 R2 L7 R7
    L2 - y - -
    L7 - - - y

    Ln = shift n columns to the left  (n = 2, 7)
    Rn = shift n columns to the right (n = 2, 7)

An example of such a pair (A, B) and the resulting Pan Magic Square M, based on the first scheme, is shown below:

A(L2)
0 2 4 1 3 5 7 8 6 13 10 9 14 12 11
4 1 3 5 7 8 6 13 10 9 14 12 11 0 2
3 5 7 8 6 13 10 9 14 12 11 0 2 4 1
7 8 6 13 10 9 14 12 11 0 2 4 1 3 5
6 13 10 9 14 12 11 0 2 4 1 3 5 7 8
10 9 14 12 11 0 2 4 1 3 5 7 8 6 13
14 12 11 0 2 4 1 3 5 7 8 6 13 10 9
11 0 2 4 1 3 5 7 8 6 13 10 9 14 12
2 4 1 3 5 7 8 6 13 10 9 14 12 11 0
1 3 5 7 8 6 13 10 9 14 12 11 0 2 4
5 7 8 6 13 10 9 14 12 11 0 2 4 1 3
8 6 13 10 9 14 12 11 0 2 4 1 3 5 7
13 10 9 14 12 11 0 2 4 1 3 5 7 8 6
9 14 12 11 0 2 4 1 3 5 7 8 6 13 10
12 11 0 2 4 1 3 5 7 8 6 13 10 9 14
B(R2)
0 2 4 1 3 5 7 8 6 13 10 9 14 12 11
12 11 0 2 4 1 3 5 7 8 6 13 10 9 14
9 14 12 11 0 2 4 1 3 5 7 8 6 13 10
13 10 9 14 12 11 0 2 4 1 3 5 7 8 6
8 6 13 10 9 14 12 11 0 2 4 1 3 5 7
5 7 8 6 13 10 9 14 12 11 0 2 4 1 3
1 3 5 7 8 6 13 10 9 14 12 11 0 2 4
2 4 1 3 5 7 8 6 13 10 9 14 12 11 0
11 0 2 4 1 3 5 7 8 6 13 10 9 14 12
14 12 11 0 2 4 1 3 5 7 8 6 13 10 9
10 9 14 12 11 0 2 4 1 3 5 7 8 6 13
6 13 10 9 14 12 11 0 2 4 1 3 5 7 8
7 8 6 13 10 9 14 12 11 0 2 4 1 3 5
3 5 7 8 6 13 10 9 14 12 11 0 2 4 1
4 1 3 5 7 8 6 13 10 9 14 12 11 0 2
M = A + 15 * B + 1
1 33 65 17 49 81 113 129 97 209 161 145 225 193 177
185 167 4 36 68 24 52 89 116 130 105 208 162 136 213
139 216 188 174 7 44 71 25 60 88 117 121 93 200 152
203 159 142 224 191 175 15 43 72 16 48 80 107 124 96
127 104 206 160 150 223 192 166 3 35 62 19 51 83 114
86 115 135 103 207 151 138 215 182 169 6 38 69 22 59
30 58 87 106 123 95 197 154 141 218 189 172 14 41 70
42 61 18 50 77 109 126 98 204 157 149 221 190 180 13
168 5 32 64 21 53 84 112 134 101 205 165 148 222 181
212 184 171 8 39 67 29 56 85 120 133 102 196 153 140
156 143 219 187 179 11 40 75 28 57 76 108 125 92 199
99 202 164 146 220 195 178 12 31 63 20 47 79 111 128
119 131 100 210 163 147 211 183 170 2 34 66 23 54 82
55 90 118 132 91 198 155 137 214 186 173 9 37 74 26
73 27 46 78 110 122 94 201 158 144 217 194 176 10 45

Attachment 15.2.1 shows a pair of Orthogonal Latin Squares (A, B) and the resulting Pan Magic Square M, based on the second scheme.

Suitable top rows for the integers {0 ... 14} can be constructed by means of following procedure:

  • Generate Magic Series for the Magic Sum s5 = 35 within the range {0 ... 14}.

  • Construct Generators with three Magic Rows, based on the Magic Series obtained above.
    The Generators can be transformed to the required top rows (s15 = 105) within the same procedure.

Attachment 15.2.31 shows the 141 possible Magic Series for s5 = 35 (ref. MgcLns5)

Attachment 15.2.32 contains the resulting 305 suitable top rows (ref. CnstrGen3)

Each top row corresponds with 3! * (5!)3 = 10.368.000 potential top rows, which can be obtained by permutation of and within the three series of five integers.

15.2.2 Pan Magic Squares (2)

A more controllable collection of top rows can be found under following more strict conditions:

a(1) + a(4) + a(7) + a(10) + a(13) = 35
a(2) + a(5) + a(8) + a(11) + a(14) = 35
a(3) + a(6) + a(9) + a(12) + a(15) = 35

a(1) + a( 6) + a(11) = 21
a(2) + a( 7) + a(12) = 21
a(3) + a( 8) + a(13) = 21
a(4) + a( 9) + a(14) = 21
a(5) + a(10) + a(15) = 21

which can be reduced till following defining equations:

a(1) =  7 - a( 8) + a( 9) - a(11) + a(12) - a(13) + a(15)
a(2) = 14 - a( 8) + a(10) - a(11) - a(14) + a(15)
a(3) = 21 - a( 8) - a(13)
a(4) = 21 - a( 9) - a(14)
a(5) = 21 - a(10) - a(15)
a(6) = 14 + a( 8) - a( 9) - a(12) + a(13) - a(15)
a(7) =  7 + a( 8) - a(10) + a(11) - a(12) + a(14) - a(15)

which have been incorporated in a guessing routine (TopRws15), which returned 15 * 1872 = 28080 potential top rows.

Based on this type top rows, the squares A and B can be completed according to one of the following three schemes:

A/B L2 R2 L4 R4 L7 R7
L2 - y - - - -
L4 - - - y - -
L7 - - - - - y

Ln = shift n columns to the left  (n = 2, 4, 7)
Rn = shift n columns to the right (n = 2, 4, 7)

Attachment 15.2.2 shows a pair of Orthogonal Latin Squares (A, B) and the resulting Pan Magic Square M, based on the first occurring top row of this collection, for each of the three valid combinations.

Alternatively a Pan Magic Square M can be constructed based on foloowing pair of Orthogonal Latin Squares (A, B):

A(L4)
10 12 14 9 13 7 6 5 11 8 4 3 2 1 0
13 7 6 5 11 8 4 3 2 1 0 10 12 14 9
11 8 4 3 2 1 0 10 12 14 9 13 7 6 5
2 1 0 10 12 14 9 13 7 6 5 11 8 4 3
12 14 9 13 7 6 5 11 8 4 3 2 1 0 10
7 6 5 11 8 4 3 2 1 0 10 12 14 9 13
8 4 3 2 1 0 10 12 14 9 13 7 6 5 11
1 0 10 12 14 9 13 7 6 5 11 8 4 3 2
14 9 13 7 6 5 11 8 4 3 2 1 0 10 12
6 5 11 8 4 3 2 1 0 10 12 14 9 13 7
4 3 2 1 0 10 12 14 9 13 7 6 5 11 8
0 10 12 14 9 13 7 6 5 11 8 4 3 2 1
9 13 7 6 5 11 8 4 3 2 1 0 10 12 14
5 11 8 4 3 2 1 0 10 12 14 9 13 7 6
3 2 1 0 10 12 14 9 13 7 6 5 11 8 4
B = R(A)
0 9 5 3 10 13 11 2 12 7 8 1 14 6 4
1 14 6 4 0 9 5 3 10 13 11 2 12 7 8
2 12 7 8 1 14 6 4 0 9 5 3 10 13 11
3 10 13 11 2 12 7 8 1 14 6 4 0 9 5
4 0 9 5 3 10 13 11 2 12 7 8 1 14 6
8 1 14 6 4 0 9 5 3 10 13 11 2 12 7
11 2 12 7 8 1 14 6 4 0 9 5 3 10 13
5 3 10 13 11 2 12 7 8 1 14 6 4 0 9
6 4 0 9 5 3 10 13 11 2 12 7 8 1 14
7 8 1 14 6 4 0 9 5 3 10 13 11 2 12
13 11 2 12 7 8 1 14 6 4 0 9 5 3 10
9 5 3 10 13 11 2 12 7 8 1 14 6 4 0
14 6 4 0 9 5 3 10 13 11 2 12 7 8 1
12 7 8 1 14 6 4 0 9 5 3 10 13 11 2
10 13 11 2 12 7 8 1 14 6 4 0 9 5 3
M = 15 * A + B + 1
151 190 216 139 206 119 102 78 178 128 69 47 45 22 5
197 120 97 80 166 130 66 49 41 29 12 153 193 218 144
168 133 68 54 32 30 7 155 181 220 141 199 116 104 87
34 26 14 162 183 223 143 204 107 105 82 170 121 70 51
185 211 145 201 109 101 89 177 123 73 53 39 17 15 157
114 92 90 172 125 61 55 36 19 11 164 192 213 148 203
132 63 58 38 24 2 165 187 215 136 205 111 94 86 179
21 4 161 194 222 138 208 113 99 77 180 127 65 46 40
217 140 196 115 96 79 176 134 72 48 43 23 9 152 195
98 84 167 135 67 50 31 25 6 154 191 224 147 198 118
74 57 33 28 8 159 182 225 142 200 106 100 81 169 131
10 156 184 221 149 207 108 103 83 174 122 75 52 35 16
150 202 110 91 85 171 124 71 59 42 18 13 158 189 212
88 173 129 62 60 37 20 1 160 186 214 146 209 117 93
56 44 27 3 163 188 219 137 210 112 95 76 175 126 64

The Latin Square B is the rotated aspect of square A (counter clockwise).

15.2.3 Ultra Magic Squares

The collection of 28080 top rows type (2) contains 16 symmetric series which are shown in Attachment 15.2.21, and can be used for the construction of Ultra Magic Squares, if applied as centre rows.

An example of the construction of an order 15 Ultra Magic Square M based on pairs of Orthogonal Latin Squares (A, B), for the untegers {0, 1, ... 14} is shown below:

A(L4)
11 9 14 8 4 7 10 6 0 5 3 2 1 13 12
4 7 10 6 0 5 3 2 1 13 12 11 9 14 8
0 5 3 2 1 13 12 11 9 14 8 4 7 10 6
1 13 12 11 9 14 8 4 7 10 6 0 5 3 2
9 14 8 4 7 10 6 0 5 3 2 1 13 12 11
7 10 6 0 5 3 2 1 13 12 11 9 14 8 4
5 3 2 1 13 12 11 9 14 8 4 7 10 6 0
13 12 11 9 14 8 4 7 10 6 0 5 3 2 1
14 8 4 7 10 6 0 5 3 2 1 13 12 11 9
10 6 0 5 3 2 1 13 12 11 9 14 8 4 7
3 2 1 13 12 11 9 14 8 4 7 10 6 0 5
12 11 9 14 8 4 7 10 6 0 5 3 2 1 13
8 4 7 10 6 0 5 3 2 1 13 12 11 9 14
6 0 5 3 2 1 13 12 11 9 14 8 4 7 10
2 1 13 12 11 9 14 8 4 7 10 6 0 5 3
B = R(A)
12 8 6 2 11 4 0 1 9 7 5 13 14 10 3
13 14 10 3 12 8 6 2 11 4 0 1 9 7 5
1 9 7 5 13 14 10 3 12 8 6 2 11 4 0
2 11 4 0 1 9 7 5 13 14 10 3 12 8 6
3 12 8 6 2 11 4 0 1 9 7 5 13 14 10
5 13 14 10 3 12 8 6 2 11 4 0 1 9 7
0 1 9 7 5 13 14 10 3 12 8 6 2 11 4
6 2 11 4 0 1 9 7 5 13 14 10 3 12 8
10 3 12 8 6 2 11 4 0 1 9 7 5 13 14
7 5 13 14 10 3 12 8 6 2 11 4 0 1 9
4 0 1 9 7 5 13 14 10 3 12 8 6 2 11
8 6 2 11 4 0 1 9 7 5 13 14 10 3 12
14 10 3 12 8 6 2 11 4 0 1 9 7 5 13
9 7 5 13 14 10 3 12 8 6 2 11 4 0 1
11 4 0 1 9 7 5 13 14 10 3 12 8 6 2
M = 15 * A + B + 1
178 144 217 123 72 110 151 92 10 83 51 44 30 206 184
74 120 161 94 13 84 52 33 27 200 181 167 145 218 126
2 85 53 36 29 210 191 169 148 219 127 63 117 155 91
18 207 185 166 137 220 128 66 119 165 101 4 88 54 37
139 223 129 67 108 162 95 1 77 55 38 21 209 195 176
111 164 105 11 79 58 39 22 198 192 170 136 212 130 68
76 47 40 23 201 194 180 146 214 133 69 112 153 102 5
202 183 177 140 211 122 70 113 156 104 15 86 49 43 24
221 124 73 114 157 93 12 80 46 32 25 203 186 179 150
158 96 14 90 56 34 28 204 187 168 147 215 121 62 115
50 31 17 205 188 171 149 225 131 64 118 159 97 3 87
189 172 138 222 125 61 107 160 98 6 89 60 41 19 208
135 71 109 163 99 7 78 57 35 16 197 190 173 141 224
100 8 81 59 45 26 199 193 174 142 213 132 65 106 152
42 20 196 182 175 143 216 134 75 116 154 103 9 82 48

The Latin Square B is the rotated aspect of square A (counter clockwise).

Attachment 15.2.23 shows a pair of Orthogonal Latin Squares (A, B) and the resulting Ultra Magic Square M, based on the same centre row.

15.2.4 Associated Magic Squares (1)

A numerical example of the construction of a classical order 15 Associated Magic Square M, based on pairs of Orthogonal Latin Squares (A, B), is shown below (La Hire):

A(L7)
11 12 13 14 0 1 2 3 4 5 6 7 8 9 10
3 4 5 6 7 8 9 10 11 12 13 14 0 1 2
10 11 12 13 14 0 1 2 3 4 5 6 7 8 9
2 3 4 5 6 7 8 9 10 11 12 13 14 0 1
9 10 11 12 13 14 0 1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 0
8 9 10 11 12 13 14 0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
7 8 9 10 11 12 13 14 0 1 2 3 4 5 6
14 0 1 2 3 4 5 6 7 8 9 10 11 12 13
6 7 8 9 10 11 12 13 14 0 1 2 3 4 5
13 14 0 1 2 3 4 5 6 7 8 9 10 11 12
5 6 7 8 9 10 11 12 13 14 0 1 2 3 4
12 13 14 0 1 2 3 4 5 6 7 8 9 10 11
4 5 6 7 8 9 10 11 12 13 14 0 1 2 3
B(R7)
4 5 6 7 8 9 10 11 12 13 14 0 1 2 3
12 13 14 0 1 2 3 4 5 6 7 8 9 10 11
5 6 7 8 9 10 11 12 13 14 0 1 2 3 4
13 14 0 1 2 3 4 5 6 7 8 9 10 11 12
6 7 8 9 10 11 12 13 14 0 1 2 3 4 5
14 0 1 2 3 4 5 6 7 8 9 10 11 12 13
7 8 9 10 11 12 13 14 0 1 2 3 4 5 6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
8 9 10 11 12 13 14 0 1 2 3 4 5 6 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 0
9 10 11 12 13 14 0 1 2 3 4 5 6 7 8
2 3 4 5 6 7 8 9 10 11 12 13 14 0 1
10 11 12 13 14 0 1 2 3 4 5 6 7 8 9
3 4 5 6 7 8 9 10 11 12 13 14 0 1 2
11 12 13 14 0 1 2 3 4 5 6 7 8 9 10
M = A + 15 * B + 1
72 88 104 120 121 137 153 169 185 201 217 8 24 40 56
184 200 216 7 23 39 55 71 87 103 119 135 136 152 168
86 102 118 134 150 151 167 183 199 215 6 22 38 54 70
198 214 5 21 37 53 69 85 101 117 133 149 165 166 182
100 116 132 148 164 180 181 197 213 4 20 36 52 68 84
212 3 19 35 51 67 83 99 115 131 147 163 179 195 196
114 130 146 162 178 194 210 211 2 18 34 50 66 82 98
1 17 33 49 65 81 97 113 129 145 161 177 193 209 225
128 144 160 176 192 208 224 15 16 32 48 64 80 96 112
30 31 47 63 79 95 111 127 143 159 175 191 207 223 14
142 158 174 190 206 222 13 29 45 46 62 78 94 110 126
44 60 61 77 93 109 125 141 157 173 189 205 221 12 28
156 172 188 204 220 11 27 43 59 75 76 92 108 124 140
58 74 90 91 107 123 139 155 171 187 203 219 10 26 42
170 186 202 218 9 25 41 57 73 89 105 106 122 138 154

When the centre row is replaced by one of the centre rows used in Section 15.2.3 above, also this classical method will return an Ultra Magic Square as illustrated in Attachment 15.2.41.

Attachment 15.2.43 shows the resulting Ultra Magic Squares for each of the Symmetric Centre Rows Type (2), as listed in Attachment 15.2.21.

The defining equations for Symmetric Centre Rows Type (1) can - after deduction -be written as:

a(9) = 7 + a(10) - a(12) + a(13) - a(15)

a(1) = 14 - a(15)
a(2) = 14 - a(14)
a(3) = 14 - a(13)
a(4) = 14 - a(12)

a(5) = 14 - a(11)
a(6) = 14 - a(10)
a(7) = 14 - a(9)
a(8) = 7

which have been incorporated in a guessing routine (CntrRws15), which returned 17280 potential centre rows.

Each of these Symmetric Centre Rows Type (1) will return an Ulta Magic Square if applied in the construction method described above.

15.2.5 Associated Magic Squares (2)

The construction of order 15 Associated Lozenge Squares based on Latin Squares have been discussed in Section 18.8.1.

Attachment 15.2.42 shows the construction of an Associated Magic Square M based on a Self Orthogonal Non Cyclic Associated Latin Square A (Harry White).

15.2.6 Composed Magic Squares

Overlapping Sub Squares

Order 15 Magic Squares, containing order 8 Overlapping Sub Squares with identical Magic Sum, based on Latin Sub Squares, have been discussed in Section 25.6.

15.2.7 Composed Magic Squares

Order 5 Latin Diagonal Sub Squares

Order 15 (Pan) Magic Squares M composed of order 5 (Pan) Magic Sub Squares can be constructed based on pairs of Orthogonal Composed Semi-Latin Squares (A, B).

A numerical example, based on order 5 Latin Diagonal Pan Magic Squares - as discussed in Section 5.2.2 - for the untegers {0, 1, ... 14} is shown below:

A
2 12 10 8 3 0 14 13 7 1 4 11 9 6 5
8 3 2 12 10 7 1 0 14 13 6 5 4 11 9
12 10 8 3 2 14 13 7 1 0 11 9 6 5 4
3 2 12 10 8 1 0 14 13 7 5 4 11 9 6
10 8 3 2 12 13 7 1 0 14 9 6 5 4 11
2 12 10 8 3 0 14 13 7 1 4 11 9 6 5
8 3 2 12 10 7 1 0 14 13 6 5 4 11 9
12 10 8 3 2 14 13 7 1 0 11 9 6 5 4
3 2 12 10 8 1 0 14 13 7 5 4 11 9 6
10 8 3 2 12 13 7 1 0 14 9 6 5 4 11
2 12 10 8 3 0 14 13 7 1 4 11 9 6 5
8 3 2 12 10 7 1 0 14 13 6 5 4 11 9
12 10 8 3 2 14 13 7 1 0 11 9 6 5 4
3 2 12 10 8 1 0 14 13 7 5 4 11 9 6
10 8 3 2 12 13 7 1 0 14 9 6 5 4 11
B = T(A)
2 8 12 3 10 2 8 12 3 10 2 8 12 3 10
12 3 10 2 8 12 3 10 2 8 12 3 10 2 8
10 2 8 12 3 10 2 8 12 3 10 2 8 12 3
8 12 3 10 2 8 12 3 10 2 8 12 3 10 2
3 10 2 8 12 3 10 2 8 12 3 10 2 8 12
0 7 14 1 13 0 7 14 1 13 0 7 14 1 13
14 1 13 0 7 14 1 13 0 7 14 1 13 0 7
13 0 7 14 1 13 0 7 14 1 13 0 7 14 1
7 14 1 13 0 7 14 1 13 0 7 14 1 13 0
1 13 0 7 14 1 13 0 7 14 1 13 0 7 14
4 6 11 5 9 4 6 11 5 9 4 6 11 5 9
11 5 9 4 6 11 5 9 4 6 11 5 9 4 6
9 4 6 11 5 9 4 6 11 5 9 4 6 11 5
6 11 5 9 4 6 11 5 9 4 6 11 5 9 4
5 9 4 6 11 5 9 4 6 11 5 9 4 6 11
C= 15 * A + B + 1
33 189 163 124 56 3 219 208 109 26 63 174 148 94 86
133 49 41 183 159 118 19 11 213 204 103 79 71 168 144
191 153 129 58 34 221 198 114 28 4 176 138 99 88 64
54 43 184 161 123 24 13 214 206 108 84 73 169 146 93
154 131 48 39 193 199 116 18 9 223 139 101 78 69 178
31 188 165 122 59 1 218 210 107 29 61 173 150 92 89
135 47 44 181 158 120 17 14 211 203 105 77 74 166 143
194 151 128 60 32 224 196 113 30 2 179 136 98 90 62
53 45 182 164 121 23 15 212 209 106 83 75 167 149 91
152 134 46 38 195 197 119 16 8 225 137 104 76 68 180
35 187 162 126 55 5 217 207 111 25 65 172 147 96 85
132 51 40 185 157 117 21 10 215 202 102 81 70 170 142
190 155 127 57 36 220 200 112 27 6 175 140 97 87 66
52 42 186 160 125 22 12 216 205 110 82 72 171 145 95
156 130 50 37 192 201 115 20 7 222 141 100 80 67 177

The series {0, 1, 2, 3, 4 ... 14} have been split into three sub series, each summing to s5 = 35:

     {0, 1, 7, 13, 14}, {2, 3, 8, 10, 12} and {4, 5, 6, 9, 11}

which have been used for the construction of the nine Pan Magic Sub Squares.

Each of the 305 Generators, as deducted in Section 15.2.1 above, can be used for the construction method illustrated above.

Each constructed square corresponds with 9! * 288009 = 4,95 * 1045 Composed Pan Magic Squares, which can be obtained by permutation of the sub squares and (or) selecting other aspects (of the sub squares).

15.2.8 Concentric Magic Squares

A numerical example of the construction of an order 15 Concentric Magic Square M, based on pairs of Orthogonal Concentric Semi-Latin Squares (A, B), is shown below:

A
7 1 2 3 4 5 6 8 9 10 11 12 13 14 0
0 7 2 3 4 5 6 8 9 10 11 12 13 1 14
14 1 7 3 4 5 6 8 9 10 11 12 2 13 0
14 1 2 11 4 5 3 6 8 9 10 7 12 13 0
14 13 12 3 10 4 5 6 8 9 7 11 2 1 0
0 13 12 3 10 9 5 6 8 7 4 11 2 1 14
14 1 2 11 4 5 8 6 7 9 10 3 12 13 0
14 1 2 11 4 9 6 7 8 5 10 3 12 13 0
0 1 2 11 10 5 7 8 6 9 4 3 12 13 14
0 13 2 3 4 7 9 8 6 5 10 11 12 1 14
0 13 12 3 7 10 9 8 6 5 4 11 2 1 14
0 1 12 7 10 9 11 8 6 5 4 3 2 13 14
0 13 12 11 10 9 8 6 5 4 3 2 7 1 14
14 13 12 11 10 9 8 6 5 4 3 2 1 7 0
14 13 12 11 10 9 8 6 5 4 3 2 1 0 7
B
0 0 0 14 14 14 0 0 14 14 14 0 0 14 7
1 1 1 1 13 13 13 13 13 13 1 1 1 7 13
6 2 2 2 2 12 12 12 12 2 2 12 7 12 8
2 8 12 7 11 11 3 3 3 11 11 3 2 6 12
8 3 11 10 7 10 4 10 10 4 4 4 3 11 6
3 9 10 9 9 7 5 9 9 5 5 5 4 5 11
5 4 9 8 8 6 7 6 8 8 6 6 5 10 9
9 6 8 6 6 5 8 7 6 9 8 8 6 8 5
4 10 6 3 5 8 6 8 7 6 9 11 8 4 10
10 5 5 5 4 9 9 5 5 7 10 9 9 9 4
13 11 4 4 10 4 10 4 4 10 7 10 10 3 1
11 12 3 11 3 3 11 11 11 3 3 7 11 2 3
14 13 7 12 12 2 2 2 2 12 12 2 12 1 0
12 7 13 13 1 1 1 1 1 1 13 13 13 13 2
7 14 14 0 0 0 14 14 0 0 0 14 14 0 14
M = A + 15 * B + 1
8 2 3 214 215 216 7 9 220 221 222 13 14 225 106
16 23 18 19 200 201 202 204 205 206 27 28 29 107 210
105 32 38 34 35 186 187 189 190 41 42 193 108 194 121
45 122 183 117 170 171 49 52 54 175 176 53 43 104 181
135 59 178 154 116 155 66 157 159 70 68 72 48 167 91
46 149 163 139 146 115 81 142 144 83 80 87 63 77 180
90 62 138 132 125 96 114 97 128 130 101 94 88 164 136
150 92 123 102 95 85 127 113 99 141 131 124 103 134 76
61 152 93 57 86 126 98 129 112 100 140 169 133 74 165
151 89 78 79 65 143 145 84 82 111 161 147 148 137 75
196 179 73 64 158 71 160 69 67 156 110 162 153 47 30
166 182 58 173 56 55 177 174 172 51 50 109 168 44 60
211 209 118 192 191 40 39 37 36 185 184 33 188 17 15
195 119 208 207 26 25 24 22 21 20 199 198 197 203 31
120 224 223 12 11 10 219 217 6 5 4 213 212 1 218

A pair of order 15 Orthogonal Semi-Latin Borders can be constructed for each pair of order 13 Orthogonal Concentric Semi-Latin Squares (A13, B13), as found in Section 13.2.5

Each pair of order 15 Orthogonal Semi-Latin Borders corresponds with numerous pairs, which can be obtained by permutation of the border pairs.

15.2.9 Bordered Magic Squares

Composed Magic Sub Squares

Order 15 Bordered Magic Squares M can be constructed based on pairs of Orthogonal Bordered Semi-Latin Squares (A, B). for miscellaneous types of Centre Squares.

The example shown below is based on Order 13 Magic Centre Squares, containing order 7 Overlapping Sub Squares with identical Magic Sum, based on Semi-Latin Squares, as discussed in Section 25.5.

A
7 1 2 3 4 5 6 8 9 10 11 12 13 14 0
0 11 3 13 12 7 2 1 10 5 6 8 9 4 14
14 13 12 7 2 1 11 3 4 9 6 8 5 10 0
14 7 2 1 11 3 13 12 4 5 8 6 9 10 0
14 1 11 3 13 12 7 2 10 5 8 6 9 4 0
0 3 13 12 7 2 1 11 4 9 8 6 5 10 14
14 12 7 2 1 11 3 13 10 9 6 8 5 4 0
14 2 1 11 3 13 12 7 9 10 6 8 4 5 0
0 13 2 3 11 12 1 10 6 8 4 5 7 9 14
0 1 12 3 11 2 13 8 4 5 7 9 10 6 14
0 1 2 11 3 12 13 5 7 9 10 6 8 4 14
0 13 2 11 3 12 1 9 10 6 8 4 5 7 14
0 1 12 11 3 2 13 6 8 4 5 7 9 10 14
14 13 12 3 11 2 1 4 5 7 9 10 6 8 0
14 13 12 11 10 9 8 6 5 4 3 2 1 0 7
B
0 0 0 14 14 14 0 0 14 14 14 0 0 14 7
1 11 13 7 1 3 12 2 13 1 1 13 1 13 13
6 3 12 2 11 13 7 1 2 12 2 2 12 12 8
2 13 7 1 3 12 2 11 3 3 11 11 11 3 12
8 12 2 11 13 7 1 3 11 11 3 3 3 11 6
3 7 1 3 12 2 11 13 12 2 12 12 2 2 11
5 2 11 13 7 1 3 12 1 13 13 1 13 1 9
9 1 3 12 2 11 13 7 10 8 5 9 6 4 5
4 10 4 4 10 4 10 9 6 4 7 10 8 5 10
10 5 9 5 5 9 9 10 8 5 9 6 4 7 4
13 6 6 8 8 8 6 6 4 7 10 8 5 9 1
11 8 8 6 6 6 8 8 5 9 6 4 7 10 3
14 9 5 9 9 5 5 4 7 10 8 5 9 6 0
12 4 10 10 4 10 4 5 9 6 4 7 10 8 2
7 14 14 0 0 0 14 14 0 0 0 14 14 0 14
M = A + 15 * B + 1
8 2 3 214 215 216 7 9 220 221 222 13 14 225 106
16 177 199 119 28 53 183 32 206 21 22 204 25 200 210
105 59 193 38 168 197 117 19 35 190 37 39 186 191 121
45 203 108 17 57 184 44 178 50 51 174 172 175 56 181
135 182 42 169 209 118 23 48 176 171 54 52 55 170 91
46 109 29 58 188 33 167 207 185 40 189 187 36 41 180
90 43 173 198 107 27 49 194 26 205 202 24 201 20 136
150 18 47 192 34 179 208 113 160 131 82 144 95 66 76
61 164 63 64 162 73 152 146 97 69 110 156 128 85 165
151 77 148 79 87 138 149 159 125 81 143 100 71 112 75
196 92 93 132 124 133 104 96 68 115 161 127 84 140 30
166 134 123 102 94 103 122 130 86 142 99 65 111 158 60
211 137 88 147 139 78 89 67 114 155 126 83 145 101 15
195 74 163 154 72 153 62 80 141 98 70 116 157 129 31
120 224 223 12 11 10 219 217 6 5 4 213 212 1 218

A pair of order 15 Orthogonal Semi-Latin Borders can be constructed for each pair of order 13 Orthogonal Composed Semi-Latin Squares (A13, B13), as found in Section 25.5.

Each pair of order 15 Orthogonal Semi-Latin Borders corresponds with numerous pairs, which can be obtained by permutation of the border pairs.

15.2.10 Bordered Magic Squares

Diamond Inlays Order 5 and 6

The example shown below is based on Order 13 Bordered Magic Centre Squares, containing order 5 and 6 Diamond Inlays, based on Semi-Latin Squares, as discussed in Section 13.2.6.

A
7 1 2 3 4 5 6 8 9 10 11 12 13 14 0
0 7 2 3 4 5 6 8 9 10 11 12 13 1 14
14 1 10 12 6 5 9 3 7 2 4 8 11 13 0
14 1 6 3 2 9 5 12 10 4 7 8 11 13 0
14 13 7 3 12 2 6 10 11 9 4 5 8 1 0
0 13 8 12 9 5 10 3 4 2 6 11 7 1 14
14 1 4 11 2 7 8 10 5 9 6 12 3 13 0
14 1 12 10 3 5 6 7 8 9 11 4 2 13 0
0 1 11 2 8 5 9 4 6 7 12 3 10 13 14
0 13 7 3 8 12 10 11 4 9 5 2 6 1 14
0 13 6 9 10 5 3 4 8 12 2 11 7 1 14
0 1 3 6 7 10 4 2 9 5 12 11 8 13 14
0 13 3 6 10 12 7 11 5 9 8 2 4 1 14
14 13 12 11 10 9 8 6 5 4 3 2 1 7 0
14 13 12 11 10 9 8 6 5 4 3 2 1 0 7
B
0 0 0 14 14 14 0 0 14 14 14 0 0 14 7
1 1 1 1 13 13 13 13 13 13 1 1 1 7 13
6 2 10 6 7 8 4 12 11 7 6 3 3 12 8
2 8 12 3 3 12 11 10 2 3 9 6 6 6 12
8 3 6 2 12 9 2 3 8 8 10 7 10 11 6
3 9 5 9 2 5 7 5 5 12 5 10 12 5 11
5 4 9 5 6 10 8 6 9 10 3 4 7 10 9
9 6 3 12 10 3 10 7 4 11 4 2 11 8 5
4 10 7 10 11 4 5 8 6 4 8 9 5 4 10
10 5 2 4 9 2 9 9 7 9 12 5 9 9 4
13 11 4 7 4 6 6 11 12 5 2 12 8 3 1
11 12 8 8 5 11 12 4 3 2 11 11 2 2 3
14 13 11 11 8 7 3 2 10 6 7 8 4 1 0
12 7 13 13 1 1 1 1 1 1 13 13 13 13 2
7 14 14 0 0 0 14 14 0 0 0 14 14 0 14
M = A + 15 * B + 1
8 2 3 214 215 216 7 9 220 221 222 13 14 225 106
16 23 18 19 200 201 202 204 205 206 27 28 29 107 210
105 32 161 103 112 126 70 184 173 108 95 54 57 194 121
45 122 187 49 48 190 171 163 41 50 143 99 102 104 181
135 59 98 34 193 138 37 56 132 130 155 111 159 167 91
46 149 84 148 40 81 116 79 80 183 82 162 188 77 180
90 62 140 87 93 158 129 101 141 160 52 73 109 164 136
150 92 58 191 154 51 157 113 69 175 72 35 168 134 76
61 152 117 153 174 66 85 125 97 68 133 139 86 74 165
151 89 38 64 144 43 146 147 110 145 186 78 142 137 75
196 179 67 115 71 96 94 170 189 88 33 192 128 47 30
166 182 124 127 83 176 185 63 55 36 178 177 39 44 60
211 209 169 172 131 118 53 42 156 100 114 123 65 17 15
195 119 208 207 26 25 24 22 21 20 199 198 197 203 31
120 224 223 12 11 10 219 217 6 5 4 213 212 1 218

A pair of order 15 Orthogonal Semi-Latin Borders can be constructed for each pair of order 13 Orthogonal Inlaid Bordered Semi-Latin Squares (A13, B13), as found in Section 13.2.6.

Each pair of order 15 Orthogonal Semi-Latin Borders corresponds with numerous pairs, which can be obtained by permutation of the border pairs.

15.2.11 Bordered Magic Squares

Diamond Inlays Order 6 and 7

The example shown below is based on Order 13 Inlaid Magic Centre Squares, containing order 6 and 7 Diamond Inlays, based on Semi-Latin Squares, as discussed in Section 13.2.7.

A
7 1 2 3 4 5 6 8 9 10 11 12 13 14 0
0 6 1 8 13 3 4 9 12 7 2 10 11 5 14
14 6 12 10 7 1 5 3 9 11 2 8 4 13 0
14 5 9 10 6 13 11 7 2 4 8 3 12 1 0
14 11 4 2 3 10 8 7 12 5 6 13 1 9 0
0 9 5 4 6 1 10 2 8 13 11 3 7 12 14
14 9 13 8 3 12 5 7 2 4 10 11 6 1 0
14 2 4 13 8 11 9 7 5 3 6 1 10 12 0
0 13 8 3 4 10 12 7 9 2 11 6 1 5 14
0 2 7 11 3 1 6 12 4 13 8 10 9 5 14
0 5 13 1 8 9 2 7 6 4 11 12 10 3 14
0 13 2 11 6 10 12 7 3 1 8 4 5 9 14
0 1 10 6 12 3 5 11 9 13 7 4 2 8 14
14 9 3 4 12 7 2 5 10 11 1 6 13 8 0
14 13 12 11 10 9 8 6 5 4 3 2 1 0 7
B
0 0 0 14 14 14 0 0 14 14 14 0 0 14 7
1 6 6 5 11 9 9 2 13 2 5 13 1 9 13
6 1 12 9 4 5 13 4 8 7 13 2 10 3 8
2 8 10 10 2 4 8 13 3 11 1 11 6 4 12
8 13 7 6 3 6 3 8 4 3 8 6 12 12 6
3 3 1 13 10 1 12 11 10 1 9 10 3 7 11
5 4 5 11 8 10 5 9 12 6 2 12 5 2 9
9 9 3 7 7 2 7 7 7 12 7 7 11 5 5
4 12 9 2 12 8 2 5 9 4 6 3 9 10 10
10 7 11 4 5 13 4 3 2 13 4 1 13 11 4
13 2 2 8 6 11 10 6 11 8 11 8 7 1 1
11 10 8 3 13 3 11 1 6 10 12 4 4 6 3
14 11 4 12 1 7 6 10 1 9 10 5 2 13 0
12 5 13 1 9 12 1 12 5 5 3 9 8 8 2
7 14 14 0 0 0 14 14 0 0 0 14 14 0 14
M = A + 15 * B + 1
8 2 3 214 215 216 7 9 220 221 222 13 14 225 106
16 97 92 84 179 139 140 40 208 38 78 206 27 141 210
105 22 193 146 68 77 201 64 130 117 198 39 155 59 121
45 126 160 161 37 74 132 203 48 170 24 169 103 62 181
135 207 110 93 49 101 54 128 73 51 127 104 182 190 91
46 55 21 200 157 17 191 168 159 29 147 154 53 118 180
90 70 89 174 124 163 81 143 183 95 41 192 82 32 136
150 138 50 119 114 42 115 113 111 184 112 107 176 88 76
61 194 144 34 185 131 43 83 145 63 102 52 137 156 165
151 108 173 72 79 197 67 58 35 209 69 26 205 171 75
196 36 44 122 99 175 153 98 172 125 177 133 116 19 30
166 164 123 57 202 56 178 23 94 152 189 65 66 100 60
211 167 71 187 28 109 96 162 25 149 158 80 33 204 15
195 85 199 20 148 188 18 186 86 87 47 142 134 129 31
120 224 223 12 11 10 219 217 6 5 4 213 212 1 218

A pair of order 15 Orthogonal Semi-Latin Borders can be constructed for each pair of order 13 Orthogonal Inlaid Semi-Latin Squares (A13, B13), as found in Section 13.2.7.

Each pair of order 15 Orthogonal Semi-Latin Borders corresponds with numerous pairs, which can be obtained by permutation of the border pairs.

15.3   Summary

The obtained results regarding the order 15 (Semi) Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:

Attachment

Subject

Subroutine

-

-

-

Attachment 15.2.1

Construction of Pan   Magic Squares, Type 1

-

Attachment 15.2.2

Construction of Pan   Magic Squares, Type 2

-

Attachment 15.2.23

Construction of Ultra Magic Squares, Type 2

-

-

-

-

Attachment 15.2.41

Ultra Magic Square, Construction

-

Attachment 15.2.43

Ultra Magic Squares

-

Attachment 15.2.42

Associated Magic Square, Self Orthogonal

-

-

-

-

Attachment 15.2.31

Magic Series, s5 = 35

MgcLns5

Attachment 15.2.32

Potential Top Rows Type (1), s15 = 105

CnstrGen3

Attachment 15.2.21

Symmetric Top Rows Type (2), s15 = 105

-

-

-

-

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.

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