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Exhibit P03
Analysis Quadrant P03 Bordered Magic Squares

The Quadrant P03 Property of Bordered Magic Squares, with Associated Compact Pan Magic Centre Squares, is defined by the variables a(i), i = 4, 10, 40, 46, 52, 82, 88, 118, 124, 130, 160, 166 as illustrated below:

P03
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(  4) = 340 - a( 40) - a( 46) - a( 82)
a( 10) = 340 - a( 52) - a( 46) - a( 88)
a(118) = 340 - a(160) - a( 82) - a(124)
a(130) = 340 - a(166) - a( 88) - a(124)

a(  4) = 170 - a(160)   a( 10) = 170 - a(166)
a( 40) = 170 - a( 52)   a(118) = 170 - a(130)
a( 46) = 170 - a(124)   a( 82) = 170 - a( 88)

Reduced Equations

a( 52) =       a(166) - a(88) + a(124)
a(130) = 340 - a(166) - a(88) - a(124)
a(160) = 340 - a(166) - 2 * a(124)

a( 40) = 170 - a( 52)   a(118) = 170 - a(130)
a(  4) = 170 - a(160)   a( 10) = 170 - a(166)
a( 46) = 170 - a(124)   a( 82) = 170 - a( 88)

The reduced equations can be incorporated in a guessing routine (Priem5a), which:

  • reads the variables a(46), a(82), a(88) and a(124) from a file with preselected order 9 Center Squares
    (ref. Attachment 12.7.74)
  • and obtains solutions based on the independent variable a(166).

The applied range for the exterior border results from the construction method based on Semi Latin Squares:

{1 ... 14, 26,  27,  39,  40,  52,  53,  65,  66,  78,  79,
       91, 92, 104, 105, 117, 118, 130, 131, 143, 144, 156 ... 169}

as deducted in Section 13.2.4.

Numerous solutions - suitable for the construction of Quadrant P03 Bordered Magic Squares - can be obtained of which a few are shown in Attachment 12.7.75.

To enable the construction of Quadrant (P03, P14) Bordered Magic Squares, following variables should be reserved:

Option 1
169 7 157
79 85 91
13 163 1
or Option 2
168 8 156
79 85 91
14 162 2

and excluded from the range of border variables shown above, while generating Attachment 12.7.75.

Construction Quadrant P03 Bordered Magic Squares

While starting with an order 11 Bordered Center Square suitable for a Quadrant P08 Bordered Magic Square:

Cntr Sqr P08
153 38 28 24 67 23 129 148 152 154 19
51 115 98 42 35 138 82 75 58 122 119
21 84 31 140 124 68 63 47 108 100 149
20 56 126 73 96 49 110 133 89 33 150
77 36 136 83 43 113 99 59 120 76 93
80 125 69 61 141 85 29 109 101 45 90
15 94 50 111 71 57 127 87 34 134 155
106 137 81 37 60 121 74 97 44 114 64
116 70 62 123 107 102 46 30 139 86 54
145 48 112 95 88 32 135 128 72 55 25
151 132 142 146 103 147 41 22 18 16 17

a partly completed Quadrant (P03, P08) Bordered Magic Square can be constructed:

P03, P08
o o o 40 o o o o o 164 o o o
o 153 38 28 24 67 23 129 148 152 154 19 o
o 51 115 98 42 35 138 82 75 58 122 119 o
163 21 84 31 140 124 68 63 47 108 100 149 7
o 20 56 126 73 96 49 110 133 89 33 150 o
o 77 36 136 83 43 113 99 59 120 76 93 o
o 80 125 69 61 141 85 29 109 101 45 90 o
o 15 94 50 111 71 57 127 87 34 134 155 o
o 106 137 81 37 60 121 74 97 44 114 64 o
39 116 70 62 123 107 102 46 30 139 86 54 131
o 145 48 112 95 88 32 135 128 72 55 25 o
o 151 132 142 146 103 147 41 22 18 16 17 o
o o o 130 o o o o o 6 o o o
Key Var's
40 164
163 68 7
69 101
39 102 131
130 6

which can be completed with a guessing routine ((Priem13b)) based on the remaining integers.

By applying an exterior border as constructed for Quadrant P14 Bordered Magic Squares, the resulting square will be Quadrant (P03, P08, P24) Magic:

P03
168 78 157 40 160 161 8 5 4 164 3 1 156
53 153 38 28 24 67 23 129 148 152 154 19 117
65 51 115 98 42 35 138 82 75 58 122 119 105
163 21 84 31 140 124 68 63 47 108 100 149 7
66 20 56 126 73 96 49 110 133 89 33 150 104
118 77 36 136 83 43 113 99 59 120 76 93 52
79 80 125 69 61 141 85 29 109 101 45 90 91
144 15 94 50 111 71 57 127 87 34 134 155 26
158 106 137 81 37 60 121 74 97 44 114 64 12
39 116 70 62 123 107 102 46 30 139 86 54 131
27 145 48 112 95 88 32 135 128 72 55 25 143
11 151 132 142 146 103 147 41 22 18 16 17 159
14 92 13 130 10 9 162 165 166 6 167 169 2
P08
168 78 157 40 160 161 8 5 4 164 3 1 156
53 153 38 28 24 67 23 129 148 152 154 19 117
65 51 115 98 42 35 138 82 75 58 122 119 105
163 21 84 31 140 124 68 63 47 108 100 149 7
66 20 56 126 73 96 49 110 133 89 33 150 104
118 77 36 136 83 43 113 99 59 120 76 93 52
79 80 125 69 61 141 85 29 109 101 45 90 91
144 15 94 50 111 71 57 127 87 34 134 155 26
158 106 137 81 37 60 121 74 97 44 114 64 12
39 116 70 62 123 107 102 46 30 139 86 54 131
27 145 48 112 95 88 32 135 128 72 55 25 143
11 151 132 142 146 103 147 41 22 18 16 17 159
14 92 13 130 10 9 162 165 166 6 167 169 2
P14 (Option 2)
168 78 157 40 160 161 8 5 4 164 3 1 156
53 153 38 28 24 67 23 129 148 152 154 19 117
65 51 115 98 42 35 138 82 75 58 122 119 105
163 21 84 31 140 124 68 63 47 108 100 149 7
66 20 56 126 73 96 49 110 133 89 33 150 104
118 77 36 136 83 43 113 99 59 120 76 93 52
79 80 125 69 61 141 85 29 109 101 45 90 91
144 15 94 50 111 71 57 127 87 34 134 155 26
158 106 137 81 37 60 121 74 97 44 114 64 12
39 116 70 62 123 107 102 46 30 139 86 54 131
27 145 48 112 95 88 32 135 128 72 55 25 143
11 151 132 142 146 103 147 41 22 18 16 17 159
14 92 13 130 10 9 162 165 166 6 167 169 2

The square shown above corresponds with 8 * (8!)2 * n11 Quadrant (P03, P08, P14) Bordered Magic Squares (n11 = all suitable order 11 centre squares).

Attachment 12.7.76 shows a few Quadrant (P03, P08, P14) Bordered Magic Squares, as found with routine Priem13b, based on a sub collection of 0rder 11 Quadrant P08 Bordered Magic Centre Squares.

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