Office Applications and Entertainment, Latin Squares

13.0   Latin Squares (13 x 13)

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

13.2.1 Pan Magic Squares (1)

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.

Consequently order 13 Pan Magic Squares can be based on pairs of Orthogonal Latin Diagonal Squares (A, B).

The required Orthogonal Latin Diagonal 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 and 12.
   While starting with 0 there are 12! = 479.001.600 possible combinations for each square.
   
   
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 45 schemes:
   
   

   A/B	L2	R2	L3	R3	L4	R4	L5	R5	L6	R6
   L2	-	y	y	y	y	y	y	y	y	y
   L3	y	y	-	y	y	y	y	y	y	y
   L4	y	y	y	y	-	y	y	y	y	y
   L5	y	y	y	y	y	y	-	y	y	y
   L6	y	y	y	y	y	y	y	y	-	y

   
   Ln = shift n columns to the left  (n = 2, 3, 4, 5, 6)
   Rn = shift n columns to the right (n = 2, 3, 4, 5, 6)

Attachment 13.2.1 shows the ten types Latin Diagonal Squares based on the construction method described above.

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

A(L2) 0 1 2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 11 12 0 1 4 5 6 7 8 9 10 11 12 0 1 2 3 6 7 8 9 10 11 12 0 1 2 3 4 5 8 9 10 11 12 0 1 2 3 4 5 6 7 10 11 12 0 1 2 3 4 5 6 7 8 9 12 0 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 12 0 3 4 5 6 7 8 9 10 11 12 0 1 2 5 6 7 8 9 10 11 12 0 1 2 3 4 7 8 9 10 11 12 0 1 2 3 4 5 6 9 10 11 12 0 1 2 3 4 5 6 7 8 11 12 0 1 2 3 4 5 6 7 8 9 10

B(R2) 0 1 2 3 4 5 6 7 8 9 10 11 12 11 12 0 1 2 3 4 5 6 7 8 9 10 9 10 11 12 0 1 2 3 4 5 6 7 8 7 8 9 10 11 12 0 1 2 3 4 5 6 5 6 7 8 9 10 11 12 0 1 2 3 4 3 4 5 6 7 8 9 10 11 12 0 1 2 1 2 3 4 5 6 7 8 9 10 11 12 0 12 0 1 2 3 4 5 6 7 8 9 10 11 10 11 12 0 1 2 3 4 5 6 7 8 9 8 9 10 11 12 0 1 2 3 4 5 6 7 6 7 8 9 10 11 12 0 1 2 3 4 5 4 5 6 7 8 9 10 11 12 0 1 2 3 2 3 4 5 6 7 8 9 10 11 12 0 1

M = A + 13 * B + 1 1 15 29 43 57 71 85 99 113 127 141 155 169 146 160 5 19 33 47 61 75 89 103 117 118 132 122 136 150 164 9 23 37 51 65 66 80 94 108 98 112 126 140 154 168 13 14 28 42 56 70 84 74 88 102 116 130 131 145 159 4 18 32 46 60 50 64 78 79 93 107 121 135 149 163 8 22 36 26 27 41 55 69 83 97 111 125 139 153 167 12 158 3 17 31 45 59 73 87 101 115 129 143 144 134 148 162 7 21 35 49 63 77 91 92 106 120 110 124 138 152 166 11 25 39 40 54 68 82 96 86 100 114 128 142 156 157 2 16 30 44 58 72 62 76 90 104 105 119 133 147 161 6 20 34 48 38 52 53 67 81 95 109 123 137 151 165 10 24

Each type Latin Diagonal Square described above, corresponds with 12! = 479.001.600 Latin Diagonal Squares.

The possible combinations of square A and B described above will result in 45 * 479.001.600² / 4 = 2,58123 10¹⁸ unique solutions.

Each unique Pan Magic Square results in a Class C_n and finally in 8 * 169 * 2,58123 10¹⁸ = 3,48982 10²¹ possible Pan Magic Squares of the 13^th order.

Attachment 13.2.2 shows one Pan Magic Square for each valid type combination (A, B) as defined above.

13.2.2 Pan Magic Squares (2)

The Latin Squares discussed and applied in previous section are referred to as Cyclic Pan Diagonal Latin Squares.

It has been proven that for order 5, 7 and 11 all Pan Diagonal Latin Squares are cyclic.

Attachment 13.2.3 shows the construction of a Pan Magic Square M based on a Self Orthogonal Non Cyclic Pan Diagonal Latin Square A (Dabbaghian and Wu).

The square M = A + 13 * T(A) + 1 corresponds with 8 * 13 * 13 = 1352 Pan Magic Squares.

13.2.3 Ultra Magic 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 {a_i, i = 1 ... 13} and {b_j, j = 1 ... 13}.

A a1 a13 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a11 a12 a1 a13 a2 a3 a4 a5 a6 a7 a8 a9 a10 a9 a10 a11 a12 a1 a13 a2 a3 a4 a5 a6 a7 a8 a7 a8 a9 a10 a11 a12 a1 a13 a2 a3 a4 a5 a6 a5 a6 a7 a8 a9 a10 a11 a12 a1 a13 a2 a3 a4 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a1 a13 a2 a13 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a1 a12 a1 a13 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a10 a11 a12 a1 a13 a2 a3 a4 a5 a6 a7 a8 a9 a8 a9 a10 a11 a12 a1 a13 a2 a3 a4 a5 a6 a7 a6 a7 a8 a9 a10 a11 a12 a1 a13 a2 a3 a4 a5 a4 a5 a6 a7 a8 a9 a10 a11 a12 a1 a13 a2 a3 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a1 a13

B = T(A) b1 b11 b9 b7 b5 b3 b13 b12 b10 b8 b6 b4 b2 b13 b12 b10 b8 b6 b4 b2 b1 b11 b9 b7 b5 b3 b2 b1 b11 b9 b7 b5 b3 b13 b12 b10 b8 b6 b4 b3 b13 b12 b10 b8 b6 b4 b2 b1 b11 b9 b7 b5 b4 b2 b1 b11 b9 b7 b5 b3 b13 b12 b10 b8 b6 b5 b3 b13 b12 b10 b8 b6 b4 b2 b1 b11 b9 b7 b6 b4 b2 b1 b11 b9 b7 b5 b3 b13 b12 b10 b8 b7 b5 b3 b13 b12 b10 b8 b6 b4 b2 b1 b11 b9 b8 b6 b4 b2 b1 b11 b9 b7 b5 b3 b13 b12 b10 b9 b7 b5 b3 b13 b12 b10 b8 b6 b4 b2 b1 b11 b10 b8 b6 b4 b2 b1 b11 b9 b7 b5 b3 b13 b12 b11 b9 b7 b5 b3 b13 b12 b10 b8 b6 b4 b2 b1 b12 b10 b8 b6 b4 b2 b1 b11 b9 b7 b5 b3 b13

The Latin Diagonal Square B is the transposed square of A (rows and columns exchanged).

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A 0 12 1 2 3 4 5 6 7 8 9 10 11 10 11 0 12 1 2 3 4 5 6 7 8 9 8 9 10 11 0 12 1 2 3 4 5 6 7 6 7 8 9 10 11 0 12 1 2 3 4 5 4 5 6 7 8 9 10 11 0 12 1 2 3 2 3 4 5 6 7 8 9 10 11 0 12 1 12 1 2 3 4 5 6 7 8 9 10 11 0 11 0 12 1 2 3 4 5 6 7 8 9 10 9 10 11 0 12 1 2 3 4 5 6 7 8 7 8 9 10 11 0 12 1 2 3 4 5 6 5 6 7 8 9 10 11 0 12 1 2 3 4 3 4 5 6 7 8 9 10 11 0 12 1 2 1 2 3 4 5 6 7 8 9 10 11 0 12

B = T(A) 0 10 8 6 4 2 12 11 9 7 5 3 1 12 11 9 7 5 3 1 0 10 8 6 4 2 1 0 10 8 6 4 2 12 11 9 7 5 3 2 12 11 9 7 5 3 1 0 10 8 6 4 3 1 0 10 8 6 4 2 12 11 9 7 5 4 2 12 11 9 7 5 3 1 0 10 8 6 5 3 1 0 10 8 6 4 2 12 11 9 7 6 4 2 12 11 9 7 5 3 1 0 10 8 7 5 3 1 0 10 8 6 4 2 12 11 9 8 6 4 2 12 11 9 7 5 3 1 0 10 9 7 5 3 1 0 10 8 6 4 2 12 11 10 8 6 4 2 12 11 9 7 5 3 1 0 11 9 7 5 3 1 0 10 8 6 4 2 12

M = A + 13 * B + 1 1 143 106 81 56 31 162 150 125 100 75 50 25 167 155 118 104 67 42 17 5 136 111 86 61 36 22 10 141 116 79 65 28 159 147 122 97 72 47 33 164 152 127 102 77 40 26 2 133 108 83 58 44 19 7 138 113 88 63 38 157 156 119 94 69 55 30 161 149 124 99 74 49 24 12 131 117 80 78 41 16 4 135 110 85 60 35 166 154 129 92 90 53 39 158 146 121 96 71 46 21 9 140 115 101 76 51 14 13 132 107 82 57 32 163 151 126 112 87 62 37 168 144 130 93 68 43 18 6 137 123 98 73 48 23 11 142 105 91 54 29 160 148 134 109 84 59 34 165 153 128 103 66 52 15 3 145 120 95 70 45 20 8 139 114 89 64 27 169

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(162) = 3 * s1 / 13 - a(168) - a(169)
a(161) = 2 * s1 / 13 - a(163)
a(160) = 2 * s1 / 13 - a(164)
a(159) = 2 * s1 / 13 - a(165)
a(158) = 2 * s1 / 13 - a(166)
a(157) = 2 * s1 / 13 - a(167)

a(8) = 3 * s1 / 13 - a(168) - a(169)
a(7) = 2 * s1 / 13 - a(163)
a(6) = 2 * s1 / 13 - a(164)
a(5) = 2 * s1 / 13 - a(165)
a(4) = 2 * s1 / 13 - a(166)
a(3) = 2 * s1 / 13 - a(167)

a(13) = a(167)
a(12) = a(166)
a(11) = a(165)
a(10) = a(164)
a( 9) = a(163)
a( 2) = a(169)
a( 1) = a(168)

The solutions can be obtained by guessing the 7 parameters:

      a(i) for i = 163, 164, 165, 166, 167, 168 and 169

filling out these guesses in the abovementioned equations, and completing the square by copying the first row into the following rows, while shifting 2 columns to the right.

With an appropriate guessing routine (UltraLat13) 46080 Ultra Magic Squares can be constructed based on Diagonal Latin Squares type R2.

Comparable results can be obtained for Diagonal Latin Squares type L2, L3, R3, L4, R4, L5, R5, L6 and R6.

13.2.4 Composed Magic Squares

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.

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

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 {a_i, i = 1 ... 13} and {b_j, j = 1 ... 13).

A a7 a2 a3 a4 a5 a6 a8 a9 a10 a11 a12 a13 a1 a1 a7 a3 a4 a5 a6 a8 a9 a10 a11 a12 a2 a13 a1 a2 a11 a4 a5 a3 a6 a8 a9 a10 a7 a12 a13 a13 a12 a3 a10 a4 a5 a6 a8 a9 a7 a11 a2 a1 a13 a12 a3 a10 a9 a5 a6 a8 a7 a4 a11 a2 a1 a1 a2 a11 a4 a5 a8 a6 a7 a9 a10 a3 a12 a13 a1 a2 a11 a4 a9 a6 a7 a8 a5 a10 a3 a12 a13 a1 a2 a11 a10 a5 a7 a8 a6 a9 a4 a3 a12 a13 a13 a2 a3 a4 a7 a9 a8 a6 a5 a10 a11 a12 a1 a13 a12 a3 a7 a10 a9 a8 a6 a5 a4 a11 a2 a1 a1 a12 a7 a10 a9 a11 a8 a6 a5 a4 a3 a2 a13 a13 a12 a11 a10 a9 a8 a6 a5 a4 a3 a2 a7 a1 a13 a12 a11 a10 a9 a8 a6 a5 a4 a3 a2 a1 a7

B b1 b1 b1 b13 b13 b13 b13 b13 b13 b1 b1 b1 b7 b2 b2 b2 b2 b12 b12 b12 b12 b2 b2 b12 b7 b12 b8 b12 b7 b11 b11 b3 b3 b3 b11 b11 b3 b2 b6 b3 b11 b10 b7 b10 b4 b10 b10 b4 b4 b4 b3 b11 b9 b10 b9 b9 b7 b5 b9 b9 b5 b5 b5 b4 b5 b4 b9 b8 b8 b6 b7 b6 b8 b8 b6 b6 b5 b10 b6 b8 b6 b6 b5 b8 b7 b6 b9 b8 b8 b6 b8 b10 b6 b3 b5 b8 b6 b8 b7 b6 b9 b11 b8 b4 b5 b5 b5 b4 b9 b9 b5 b5 b7 b10 b9 b9 b9 b11 b4 b4 b10 b4 b10 b4 b4 b10 b7 b10 b10 b3 b12 b3 b11 b3 b3 b11 b11 b11 b3 b3 b7 b11 b2 b13 b7 b12 b12 b2 b2 b2 b2 b12 b12 b2 b12 b1 b7 b13 b13 b1 b1 b1 b1 b1 b1 b13 b13 b13 b13

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A 6 1 2 3 4 5 7 8 9 10 11 12 0 0 6 2 3 4 5 7 8 9 10 11 1 12 0 1 10 3 4 2 5 7 8 9 6 11 12 12 11 2 9 3 4 5 7 8 6 10 1 0 12 11 2 9 8 4 5 7 6 3 10 1 0 0 1 10 3 4 7 5 6 8 9 2 11 12 0 1 10 3 8 5 6 7 4 9 2 11 12 0 1 10 9 4 6 7 5 8 3 2 11 12 12 1 2 3 6 8 7 5 4 9 10 11 0 12 11 2 6 9 8 7 5 4 3 10 1 0 0 11 6 9 8 10 7 5 4 3 2 1 12 12 11 10 9 8 7 5 4 3 2 1 6 0 12 11 10 9 8 7 5 4 3 2 1 0 6

B 0 0 0 12 12 12 12 12 12 0 0 0 6 1 1 1 1 11 11 11 11 1 1 11 6 11 7 11 6 10 10 2 2 2 10 10 2 1 5 2 10 9 6 9 3 9 9 3 3 3 2 10 8 9 8 8 6 4 8 8 4 4 4 3 4 3 8 7 7 5 6 5 7 7 5 5 4 9 5 7 5 5 4 7 6 5 8 7 7 5 7 9 5 2 4 7 5 7 6 5 8 10 7 3 4 4 4 3 8 8 4 4 6 9 8 8 8 10 3 3 9 3 9 3 3 9 6 9 9 2 11 2 10 2 2 10 10 10 2 2 6 10 1 12 6 11 11 1 1 1 1 11 11 1 11 0 6 12 12 0 0 0 0 0 0 12 12 12 12

M = A + 13 * B + 1 7 2 3 160 161 162 164 165 166 11 12 13 79 14 20 16 17 148 149 151 152 23 24 155 80 156 92 145 89 134 135 29 32 34 139 140 33 25 78 39 142 120 88 121 44 123 125 48 46 50 28 131 117 129 107 114 87 57 110 112 59 56 63 41 53 40 106 102 95 70 86 71 98 100 75 68 64 130 66 93 76 69 61 97 85 73 109 101 94 77 104 118 67 37 62 96 72 99 84 74 108 133 103 52 65 54 55 43 111 113 60 58 83 127 115 116 105 143 51 42 124 49 126 47 45 122 82 128 119 27 144 38 137 36 35 141 138 136 31 30 81 132 26 169 90 154 153 22 21 19 18 147 146 15 150 1 91 168 167 10 9 8 6 5 4 159 158 157 163

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.

Each pair of order 13 Orthogonal Semi-Latin Borders corresponds with 8 * (11!)² = 1,27468 10¹⁶ pairs.

Consequently 3,62137 10⁴⁴ Concentric Magic Squares can be constructed based on the method described above.

13.2.6 Bordered Magic Squares

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.

The example shown below is based on Center Squares with order 4 and 5 Diamond Inlays - as discussed in Section 11.2.4 - and the symbols {a_i, i = 1 ... 13} and {b_j, j = 1 ... 13).

A a7 a2 a3 a4 a5 a6 a8 a9 a10 a11 a12 a13 a1 a1 a7 a3 a4 a5 a6 a8 a9 a10 a11 a12 a2 a13 a1 a2 a4 a7 a9 a3 a10 a8 a6 a5 a11 a12 a13 a13 a12 a11 a9 a7 a10 a4 a3 a5 a6 a8 a2 a1 a13 a12 a8 a7 a3 a5 a6 a9 a10 a11 a4 a2 a1 a1 a2 a4 a3 a9 a8 a10 a5 a11 a7 a6 a12 a13 a1 a2 a9 a10 a11 a6 a7 a8 a3 a4 a5 a12 a13 a1 a2 a8 a7 a3 a9 a4 a6 a5 a11 a10 a12 a13 a13 a2 a10 a3 a4 a5 a8 a9 a11 a7 a6 a12 a1 a13 a12 a6 a8 a9 a11 a10 a4 a7 a5 a3 a2 a1 a1 a12 a3 a9 a8 a6 a4 a11 a5 a7 a10 a2 a13 a13 a12 a11 a10 a9 a8 a6 a5 a4 a3 a2 a7 a1 a13 a12 a11 a10 a9 a8 a6 a5 a4 a3 a2 a1 a7

B b1 b1 b1 b13 b13 b13 b13 b13 b13 b1 b1 b1 b7 b2 b2 b2 b2 b12 b12 b12 b12 b2 b2 b12 b7 b12 b8 b12 b3 b6 b10 b8 b9 b4 b8 b11 b4 b2 b6 b3 b11 b9 b8 b3 b7 b10 b3 b7 b9 b7 b3 b11 b9 b10 b8 b9 b4 b3 b11 b9 b3 b7 b9 b4 b5 b4 b9 b6 b11 b5 b9 b6 b8 b5 b10 b3 b5 b10 b6 b8 b4 b10 b8 b4 b7 b10 b6 b4 b10 b6 b8 b10 b6 b11 b4 b9 b6 b8 b5 b9 b3 b8 b8 b4 b5 b5 b5 b7 b11 b5 b3 b11 b10 b5 b6 b9 b9 b11 b4 b7 b5 b7 b11 b4 b7 b11 b6 b5 b10 b3 b12 b3 b10 b3 b6 b10 b5 b6 b4 b8 b11 b11 b2 b13 b7 b12 b12 b2 b2 b2 b2 b12 b12 b2 b12 b1 b7 b13 b13 b1 b1 b1 b1 b1 b1 b13 b13 b13 b13

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A 6 1 2 3 4 5 7 8 9 10 11 12 0 0 6 2 3 4 5 7 8 9 10 11 1 12 0 1 3 6 8 2 9 7 5 4 10 11 12 12 11 10 8 6 9 3 2 4 5 7 1 0 12 11 7 6 2 4 5 8 9 10 3 1 0 0 1 3 2 8 7 9 4 10 6 5 11 12 0 1 8 9 10 5 6 7 2 3 4 11 12 0 1 7 6 2 8 3 5 4 10 9 11 12 12 1 9 2 3 4 7 8 10 6 5 11 0 12 11 5 7 8 10 9 3 6 4 2 1 0 0 11 2 8 7 5 3 10 4 6 9 1 12 12 11 10 9 8 7 5 4 3 2 1 6 0 12 11 10 9 8 7 5 4 3 2 1 0 6

B 0 0 0 12 12 12 12 12 12 0 0 0 6 1 1 1 1 11 11 11 11 1 1 11 6 11 7 11 2 5 9 7 8 3 7 10 3 1 5 2 10 8 7 2 6 9 2 6 8 6 2 10 8 9 7 8 3 2 10 8 2 6 8 3 4 3 8 5 10 4 8 5 7 4 9 2 4 9 5 7 3 9 7 3 6 9 5 3 9 5 7 9 5 10 3 8 5 7 4 8 2 7 7 3 4 4 4 6 10 4 2 10 9 4 5 8 8 10 3 6 4 6 10 3 6 10 5 4 9 2 11 2 9 2 5 9 4 5 3 7 10 10 1 12 6 11 11 1 1 1 1 11 11 1 11 0 6 12 12 0 0 0 0 0 0 12 12 12 12

M = A + 13 * B + 1 7 2 3 160 161 162 164 165 166 11 12 13 79 14 20 16 17 148 149 151 152 23 24 155 80 156 92 145 30 72 126 94 114 47 97 135 50 25 78 39 142 115 100 33 88 121 29 83 110 86 28 131 117 129 99 111 42 31 136 113 36 89 108 41 53 40 106 69 133 61 112 75 96 63 124 32 64 130 66 93 48 127 102 45 85 125 68 43 122 77 104 118 67 138 46 107 74 95 58 109 37 101 103 52 65 54 62 81 134 57 34 139 128 59 71 116 105 143 51 84 60 87 141 49 82 137 70 55 119 27 144 38 120 35 73 123 56 76 44 98 140 132 26 169 90 154 153 22 21 19 18 147 146 15 150 1 91 168 167 10 9 8 6 5 4 159 158 157 163

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.

Each pair of order 13 Orthogonal Semi-Latin Borders corresponds with 8 * (11!)² = 1,27468 10¹⁶ pairs.

Consequently 2,92735 10³¹ Bordered Magic Squares can be constructed based on the method described above.

Diamond Inlays Order 5 and 6

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 {a_i, i = 1 ... 13} and {b_j, j = 1 ... 13).

A a7 a2 a3 a4 a5 a6 a8 a9 a10 a11 a12 a13 a1 a1 a10 a12 a6 a5 a9 a3 a7 a2 a4 a8 a11 a13 a1 a6 a3 a2 a9 a5 a12 a10 a4 a7 a8 a11 a13 a13 a7 a3 a12 a2 a6 a10 a11 a9 a4 a5 a8 a1 a13 a8 a12 a9 a5 a10 a3 a4 a2 a6 a11 a7 a1 a1 a4 a11 a2 a7 a8 a10 a5 a9 a6 a12 a3 a13 a1 a12 a10 a3 a5 a6 a7 a8 a9 a11 a4 a2 a13 a1 a11 a2 a8 a5 a9 a4 a6 a7 a12 a3 a10 a13 a13 a7 a3 a8 a12 a10 a11 a4 a9 a5 a2 a6 a1 a13 a6 a9 a10 a5 a3 a4 a8 a12 a2 a11 a7 a1 a1 a3 a6 a7 a10 a4 a2 a9 a5 a12 a11 a8 a13 a13 a3 a6 a10 a12 a7 a11 a5 a9 a8 a2 a4 a1 a13 a12 a11 a10 a9 a8 a6 a5 a4 a3 a2 a1 a7

B b1 b1 b1 b13 b13 b13 b13 b13 b13 b1 b1 b1 b7 b2 b10 b6 b7 b8 b4 b12 b11 b7 b6 b3 b3 b12 b8 b12 b3 b3 b12 b11 b10 b2 b3 b9 b6 b6 b6 b3 b6 b2 b12 b9 b2 b3 b8 b8 b10 b7 b10 b11 b9 b5 b9 b2 b5 b7 b5 b5 b12 b5 b10 b12 b5 b4 b9 b5 b6 b10 b8 b6 b9 b10 b3 b4 b7 b10 b6 b3 b12 b10 b3 b10 b7 b4 b11 b4 b2 b11 b8 b10 b7 b10 b11 b4 b5 b8 b6 b4 b8 b9 b5 b4 b5 b2 b4 b9 b2 b9 b9 b7 b9 b12 b5 b9 b9 b11 b4 b7 b4 b6 b6 b11 b12 b5 b2 b12 b8 b3 b12 b8 b8 b5 b11 b12 b4 b3 b2 b11 b11 b2 b2 b13 b11 b11 b8 b7 b3 b2 b10 b6 b7 b8 b4 b1 b7 b13 b13 b1 b1 b1 b1 b1 b1 b13 b13 b13 b13

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A 6 1 2 3 4 5 7 8 9 10 11 12 0 0 9 11 5 4 8 2 6 1 3 7 10 12 0 5 2 1 8 4 11 9 3 6 7 10 12 12 6 2 11 1 5 9 10 8 3 4 7 0 12 7 11 8 4 9 2 3 1 5 10 6 0 0 3 10 1 6 7 9 4 8 5 11 2 12 0 11 9 2 4 5 6 7 8 10 3 1 12 0 10 1 7 4 8 3 5 6 11 2 9 12 12 6 2 7 11 9 10 3 8 4 1 5 0 12 5 8 9 4 2 3 7 11 1 10 6 0 0 2 5 6 9 3 1 8 4 11 10 7 12 12 2 5 9 11 6 10 4 8 7 1 3 0 12 11 10 9 8 7 5 4 3 2 1 0 6

B 0 0 0 12 12 12 12 12 12 0 0 0 6 1 9 5 6 7 3 11 10 6 5 2 2 11 7 11 2 2 11 10 9 1 2 8 5 5 5 2 5 1 11 8 1 2 7 7 9 6 9 10 8 4 8 1 4 6 4 4 11 4 9 11 4 3 8 4 5 9 7 5 8 9 2 3 6 9 5 2 11 9 2 9 6 3 10 3 1 10 7 9 6 9 10 3 4 7 5 3 7 8 4 3 4 1 3 8 1 8 8 6 8 11 4 8 8 10 3 6 3 5 5 10 11 4 1 11 7 2 11 7 7 4 10 11 3 2 1 10 10 1 1 12 10 10 7 6 2 1 9 5 6 7 3 0 6 12 12 0 0 0 0 0 0 12 12 12 12

M = A + 13 * B + 1 7 2 3 160 161 162 164 165 166 11 12 13 79 14 127 77 84 96 48 146 137 80 69 34 37 156 92 149 29 28 152 135 129 23 30 111 73 76 78 39 72 16 155 106 19 36 102 100 121 83 125 131 117 60 116 22 57 88 55 56 145 58 128 150 53 40 108 63 67 124 99 75 109 126 32 51 81 130 66 38 153 120 31 123 85 47 139 50 17 132 104 118 89 119 138 44 61 95 71 46 103 107 62 52 65 20 42 112 25 114 115 82 113 148 54 110 105 143 45 87 49 70 68 134 151 64 15 154 98 27 144 94 97 59 140 147 41 35 18 142 141 21 26 169 133 136 101 90 33 24 122 74 86 93 43 1 91 168 167 10 9 8 6 5 4 159 158 157 163