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Exhibit P08
Analysis Quadrant P08 Bordered Magic Squares

The Quadrant P08 Property of Bordered Magic Squares, with Associated Compact Pan Magic Centre Squares, is defined by the variables a(i), i = 15, 19, 21, 25, 67, 71, 73, 77, 93, 97, 99, 103. 145, 149, 151, 155 as illustrated below:

P08
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13
a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26
a27 a28 a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39
a40 a41 a42 a43 a44 a45 a46 a47 a48 a49 a50 a51 a52
a53 a54 a55 a56 a57 a58 a59 a60 a61 a62 a63 a64 a65
a66 a67 a68 a69 a70 a71 a72 a73 a74 a75 a76 a77 a78
a79 a80 a81 a82 a83 a84 a85 a86 a87 a88 a89 a90 a91
a92 a93 a94 a95 a96 a97 a98 a99 a100 a101 a102 a103 a104
a105 a106 a107 a108 a109 a110 a111 a112 a113 a114 a115 a116 a117
a118 a119 a120 a121 a122 a123 a124 a125 a126 a127 a128 a129 a130
a131 a132 a133 a134 a135 a136 a137 a138 a139 a140 a141 a142 a143
a144 a145 a146 a147 a148 a149 a150 a151 a152 a153 a154 a155 a156
a157 a158 a159 a160 a161 a162 a163 a164 a165 a166 a167 a168 a169

Defining Equations

a( 15) = 340 - a( 19) - a( 67) - a(71)
a( 21) = 340 - a( 25) - a( 77) - a(73)
a( 93) = 340 - a(145) - a(149) - a(97)
a(103) = 340 - a(151) - a(155) - a(99)

a( 15) = 170 - a(155)   a( 25) = 170 - a(145)
a( 71) = 170 - a( 99)   a( 73) = 170 - a( 97)
a( 19) = 170 - a(149)   a( 21) = 170 - a(151)
a( 67) = 170 - a( 77)   a( 93) = 170 - a(103)

Reduced Equations

a( 73) = 170 - a( 97)
a( 71) = 170 - a( 99)
a(145) = 170 - a(149) - a(151) - a(155)+a(71)+a(73)
a(103) = 340 - a(151) - a(155) - a( 99)
a( 77) = 170 - a(149) - a(155) + a( 71)

a( 15) = 170 - a(155)   a( 25) = 170 - a(145)
a( 19) = 170 - a(149)   a( 21) = 170 - a(151)   
a( 67) = 170 - a( 77)   a( 93) = 170 - a(103)

The reduced equations can be incorporated in a guessing routine (Priem4b) which

  • reads the variables a(71), a(73), a(97) and a(99) from a file with preselected order 9 Center Squares
    (ref. Attachment 12.7.72)
  • and obtains solutions based on the variables a(149), a(151) and a(155)

The applied range (order 11 border) results from the construction method based on Semi Latin Squares:

{15 ... 25, 28,  38,  41,  51,  54,  64,  67,  77,  80
        90, 93, 103, 106, 116, 119, 129, 132, 142, 145 ... 155}

as deducted in Section 13.2.4.

Numerous solutions - suitable for the construction of Quadrant P08 Bordered Magic Squares - can be obtained of which a few are shown in Attachment 12.7.73.

Construction Quadrant P08 Bordered Magic Squares

While starting with a preselected Associated Compact Pan Magic Square (ref. Section 9.5.4):

Ass Comp PM
81 32 10 26 58 39 52 6 65
40 21 62 69 47 7 14 73 36
2 70 51 28 18 77 57 44 22
66 53 4 11 79 33 37 27 59
34 15 74 63 41 19 8 67 48
23 55 45 49 3 71 78 29 16
60 38 25 5 64 54 31 12 80
46 9 68 75 35 13 20 61 42
17 76 30 43 24 56 72 50 1
A9
8 4 0 7 3 2 6 5 1
3 2 7 5 1 6 4 0 8
1 6 5 0 8 4 2 7 3
2 7 3 1 6 5 0 8 4
6 5 1 8 4 0 7 3 2
4 0 8 3 2 7 5 1 6
5 1 6 4 0 8 3 2 7
0 8 4 2 7 3 1 6 5
7 3 2 6 5 1 8 4 0
B9
8 3 1 2 6 4 5 0 7
4 2 6 7 5 0 1 8 3
0 7 5 3 1 8 6 4 2
7 5 0 1 8 3 4 2 6
3 1 8 6 4 2 0 7 5
2 6 4 5 0 7 8 3 1
6 4 2 0 7 5 3 1 8
5 0 7 8 3 1 2 6 4
1 8 3 4 2 6 7 5 0

a suitable Associated Compact Pan Magic Centre Square can be constructed:

M9' = A9' + 13 * B9' + 1
141 72 42 62 110 83 100 34 121
84 57 114 125 95 35 46 133 76
30 126 99 68 50 137 109 88 58
122 101 32 43 139 73 81 63 111
74 47 134 115 85 55 36 123 96
59 107 89 97 31 127 138 69 48
112 82 61 33 120 102 71 44 140
94 37 124 135 75 45 56 113 86
49 136 70 87 60 108 128 98 29
A9'
10 6 2 9 5 4 8 7 3
5 4 9 7 3 8 6 2 10
3 8 7 2 10 6 4 9 5
4 9 5 3 8 7 2 10 6
8 7 3 10 6 2 9 5 4
6 2 10 5 4 9 7 3 8
7 3 8 6 2 10 5 4 9
2 10 6 4 9 5 3 8 7
9 5 4 8 7 3 10 6 2
B9'
10 5 3 4 8 6 7 2 9
6 4 8 9 7 2 3 10 5
2 9 7 5 3 10 8 6 4
9 7 2 3 10 5 6 4 8
5 3 10 8 6 4 2 9 7
4 8 6 7 2 9 10 5 3
8 6 4 2 9 7 5 3 10
7 2 9 10 5 3 4 8 6
3 10 5 6 4 8 9 7 2

Based on this centre square and the applicable set of key variables, a partly completed order 11 Bordered Magic Square can be constructed:

Cntr Sqr (11 x 11)
153 93 129 19
141 72 42 62 110 83 100 34 121
84 57 114 125 95 35 46 133 76
30 126 99 68 50 137 109 88 58
51 122 101 32 43 139 73 81 63 111 119
74 47 134 115 85 55 36 123 96
15 59 107 89 97 31 127 138 69 48 155
112 82 61 33 120 102 71 44 140
94 37 124 135 75 45 56 113 86
49 136 70 87 60 108 128 98 29
151 77 41 17
Key Var's
153 93 129 19
51 43 73 119
15 97 127 155
151 77 41 17

which can be completed with a guessing routine (Priem11a) based on the remaining integers:

Cntr Sqr (11 x 11)
153 25 22 21 93 20 129 147 152 154 19
54 141 72 42 62 110 83 100 34 121 116
38 84 57 114 125 95 35 46 133 76 132
28 30 126 99 68 50 137 109 88 58 142
51 122 101 32 43 139 73 81 63 111 119
90 74 47 134 115 85 55 36 123 96 80
15 59 107 89 97 31 127 138 69 48 155
103 112 82 61 33 120 102 71 44 140 67
106 94 37 124 135 75 45 56 113 86 64
146 49 136 70 87 60 108 128 98 29 24
151 145 148 149 77 150 41 23 18 16 17

By applying an exterior border as constructed for Quadrant P14 Bordered Magic Squares, the resulting square will be Quadrant (P08, P24) Magic:

P08
169 130 143 156 161 162 7 6 5 4 3 2 157
105 153 25 22 21 93 20 129 147 152 154 19 65
117 54 141 72 42 62 110 83 100 34 121 116 53
118 38 84 57 114 125 95 35 46 133 76 132 52
131 28 30 126 99 68 50 137 109 88 58 142 39
144 51 122 101 32 43 139 73 81 63 111 119 26
79 90 74 47 134 115 85 55 36 123 96 80 91
104 15 59 107 89 97 31 127 138 69 48 155 66
92 103 112 82 61 33 120 102 71 44 140 67 78
12 106 94 37 124 135 75 45 56 113 86 64 158
11 146 49 136 70 87 60 108 128 98 29 24 159
10 151 145 148 149 77 150 41 23 18 16 17 160
13 40 27 14 9 8 163 164 165 166 167 168 1
P14
169 130 143 156 161 162 7 6 5 4 3 2 157
105 153 25 22 21 93 20 129 147 152 154 19 65
117 54 141 72 42 62 110 83 100 34 121 116 53
118 38 84 57 114 125 95 35 46 133 76 132 52
131 28 30 126 99 68 50 137 109 88 58 142 39
144 51 122 101 32 43 139 73 81 63 111 119 26
79 90 74 47 134 115 85 55 36 123 96 80 91
104 15 59 107 89 97 31 127 138 69 48 155 66
92 103 112 82 61 33 120 102 71 44 140 67 78
12 106 94 37 124 135 75 45 56 113 86 64 158
11 146 49 136 70 87 60 108 128 98 29 24 159
10 151 145 148 149 77 150 41 23 18 16 17 160
13 40 27 14 9 8 163 164 165 166 167 168 1

The square shown above corresponds with 8 * (10!)2 * (7!)2 * n9 Quadrant (P08, P14) Bordered Magic Squares (n9 = all suitable order 9 centre squares).

Attachment 12.7.71 shows a few Quadrant (P08, P14) Bordered Magic Squares, as found with routine Priem11a, based on the sub collection of 0rder 9 Associated Compact Pan Magic Squares described above.

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