Office Applications and Entertainment, Magic Squares

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20.0   Magic Squares (14 x 14)

This Chapter 20 regarding order 14 Magic Squares, has been placed behind Chapter 10 because of the similarity with order 10 Magic Squares (single even order). Order 11 Magic Squares will be discussed in Chapter 11.

20.1   Medjig Solutions (14 x 14)

20.1.1 General

As described in Section 6.8, for any integer n, a Magic Square C of order 2n can be constructed from any n x n Medjig-Square A and any n x n Magic Square B, by application of the equations:

cj = bi + n2 aj with i = 1, 2, ... n2, j = 1, 2, ... 4n2 and sa = 3n.

The Medjig method of constructing a Magic Square of order 14 is as follows:

  • Construct a 7 x 7 Medjig-Square A
    (ignoring the original game's limit on the number of times that a given sequence is used)
  • Construct a 7 x 7 (Pan) Magic Square B
    (Already 304.819.200 possibilities for Pan Magic Squares as constructed in Section 7.1)
  • Construct a 14 x 14 Magic Square C by applying the equations mentioned above.

The rows, columns and main diagonals of Square C sum to 2 times the corresponding sum of Magic Square B plus 49 times the corresponding sum of Medjig square A which results in s1 = 2 * 175 + 49 * 21 = 1379.

As b(i) ≠ b(j) for i ≠ j with i, j = 1, 2, ... 49 it is obvious that also c(m) ≠ c(n) for n ≠ m with n, m = 1, 2, ... 196.

A numerical example is shown below:

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

Attachment 14.1.1 contains 24 Medjig Squares (23/2 * 3!), which have been obtained by (Medjig) Row and Column Permutations applied on the Medjig Square shown in the numerical example above.

Attachment 14.1.2 contains the resulting Simple Magic Squares, based on the 7th order Pan Magic Square B shown in the numerical example above.

Notes:

  1. For convenience the 7 x 7 Medjig Square A shown above is composed out of a 5 x 5 Bordered Medjig Square with a border around it.
  2. As the Magic Square B is not bordered, the resulting Simple Magic Square C will not be bordered.
  3. The (Bordered) Medjig Square A shown above correponds with 4 * (3!)2 * 4 * (5!)2 * 1.740.800 = 1,444 1013
    possible Medjig Squares.
  4. A full enumeration as executed for 3 x 3 Medjig Squares in Section 6.6.2 is beyond the scope of this section.
  5. It should be noted that not all possible Magic Squares of the 14th order can be found by means of the Medjig method.

20.1.2 Concentric Magic Squares

The Medjig method of constructing a Concentric Magic Square of order 14 is as follows:

  • Construct a  7 x  7 Concentric Medjig Square A
  • Construct a  7 x  7 Concentric Magic  Square B (ref. Attachment 7.5.1)
  • Construct a 14 x 14 Concentric Magic  Square C by applying the method as described in Section 20.1.1.

A numerical example is shown below:

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

The Concentric Magic Squares resulting from the Medjig Square A shown above and 24 of the possible 7th order Concentric Magic Squares, are shown in Attachment 14.2.1.

20.1.3 Eccentric Magic Squares

The Medjig method of constructing an Eccentric Magic Square of order 14 is as follows:

  • Construct a  7 x  7 Eccentric Medjig Square A
  • Construct a  7 x  7 Eccentric Magic  Square B (ref. Attachment 7.5.3)
  • Construct a 14 x 14 Eccentric Magic  Square C by applying the method as described in Section 20.1.1.

A numerical example is shown below:

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

The Eccentric Magic Squares resulting from the Medjig Square A shown above and 24 of the possible 7th order Eccentric Magic Squares, are shown in Attachment 14.3.1.

20.1.4 Almost Associated Magic Squares

The Medjig method of constructing an Almost Associated Magic Square of order 14 is as follows:

  • Construct a 7 x 7 Almost Associated Medjig Square A composed of:
    An Associated Medjig Border
    An Almost Associated Medjig Center Square (ref. Section 10.1.5)
  • Construct a 7 x 7 Associated Magic Square B (ref. Attachment 7.4.4)
  • Construct a 14 x 14 Almost Associated Magic Square C by applying the method as described in Section 20.1.1.

A numerical example is shown below:

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

Attachment 14.4.2 shows a few examples of Almost Associated Magic Squares resulting from the Associated Magic Square B shown above and miscellaneous Almost Associated Medjig Squares.

The Almost Associated Medjig Squares are based on the border of Square A above and the 120 Almost Associated Medjig Center Squares shown in Attachment 14.4.1.

It should be noted that although the Almost Associated Medjig Square A is bordered, the resulting Square C will be only Almost Associated as the 7 x 7 Square B is Associated.

20.1.5 Composed Magic Squares

The Medjig method of constructing a Composed Magic Square of order 14 is as follows:

  • Construct a  7 x  7 Composed Medjig Square A composed of:
    One 3 x 3 Medjig Corner Square (top/left)
    One 4 x 4 Medjig Corner Square (bottom/right)
    Two 3 x 4 Medjig Rectangles
  • Construct a  7 x  7 Composed Magic  Square B (ref. Attachment 7.8.8)
  • Construct a 14 x 14 Composed Magic  Square C by applying the method as described in Section 20.1.1.

A numerical example is shown below:

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

The resulting 14 x 14 Magic Square C is composed out of:

One 6 x 6 Almost Associated Corner Square (top/left)
One 8 x 8 Associated Corner Square (bottom/right)
Two 6 x 8 Associated Rectangles

The Composed Magic Squares C resulting from the Medjig Square A shown above and 24 of the possible 7th order Composed Magic Squares, are shown in Attachment 14.5.1.

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