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Exhibit P14
Analysis Quadrant P14 Bordered Magic Squares

The Quadrant P14 Property of Bordered Magic Squares, with Associated Compact Pan Magic Centre Squares, is defined by the border variables a(i), i = 1, 7, 13, 79, 91, 157, 163, 169 as illustrated below:

P14
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( 1) = 255 - a(  7) - a(79)
a( 7) = 255 - a( 13) - a(91)
a(79) = 255 - a(157) - a(163)
a(91) = 255 - a(163) - a(169)
a( 1) = 170 - a(169)
a( 7) = 170 - a(163)
a(13) = 170 - a(157)
a(79) = 170 - a( 91)

Reduced Equations

a(157) =  340 - 2 * a(163) - a(169)
a( 91) =  255 -     a(163) - a(169)
a( 79) = - 85 +     a(163) + a(169)
a( 13) = -170 + 2 * a(163) + a(169)
a(  7) =  170 -     a(163)
a(  1) =  170 -     a(169)

The reduced equations can be incorporated in a guessing routine (Priem3c), which however does not return any solution for the (semi consecutive) integers:

{1, 2 ... 24; 146, 147 ... 169}

which are usually applied for order 13 bordered (or concentric) magic squares.

Based on the exterior border range resulting from the Semi Latin Squares construction method:

{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, two unique solutions can be obtained:

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

which are suitable for the construction of Quadrant P14 Bordered Magic Squares.

Construction Quadrant P14 Bordered Magic Squares

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

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

order 11 Bordered Magic Center Squares can be constructed by means of Semi Latin Borders (ref. Section 11.2.3):

A11
6 2 3 4 5 7 8 9 10 11 1
1 10 7 4 8 5 2 9 6 3 11
11 5 8 2 6 9 3 7 10 4 1
11 6 3 9 7 4 10 5 2 8 1
1 7 4 10 5 2 8 6 3 9 11
1 2 5 8 3 6 9 4 7 10 11
1 3 9 6 4 10 7 2 8 5 11
1 4 10 7 2 8 5 3 9 6 11
11 8 2 5 9 3 6 10 4 7 1
11 9 6 3 10 7 4 8 5 2 1
11 10 9 8 7 5 4 3 2 1 6
B11
1 1 1 11 11 11 11 1 1 11 6
11 7 6 5 10 9 8 4 3 2 1
10 4 2 3 7 5 6 10 8 9 2
9 8 10 9 2 4 3 5 7 6 3
8 5 7 6 8 10 9 2 4 3 4
7 2 3 4 5 6 7 8 9 10 5
5 9 8 10 3 2 4 6 5 7 7
4 6 5 7 9 8 10 3 2 4 8
3 3 4 2 6 7 5 9 10 8 9
2 10 9 8 4 3 2 7 6 5 10
6 11 11 1 1 1 1 11 11 1 11
M11 = A11 + 13 * B11 + 1
20 16 17 148 149 151 152 23 24 155 80
145 102 86 70 139 123 107 62 46 30 25
142 58 35 42 98 75 82 138 115 122 28
129 111 134 127 34 57 50 71 94 87 41
106 73 96 89 110 133 126 33 56 49 64
93 29 45 61 69 85 101 109 125 141 77
67 121 114 137 44 37 60 81 74 97 103
54 83 76 99 120 113 136 43 36 59 116
51 48 55 32 88 95 72 128 135 112 119
38 140 124 108 63 47 31 100 84 68 132
90 154 153 22 21 19 18 147 146 15 150

based on this result and the sets of key variables found above, two partly completed order 13 Bordered Magic Square can be constructed:

P14 (Option 1)
169 o o o o o 7 o o o o o 157
o 20 16 17 148 149 151 152 23 24 155 80 o
o 145 102 86 70 139 123 107 62 46 30 25 o
o 142 58 35 42 98 75 82 138 115 122 28 o
o 129 111 134 127 34 57 50 71 94 87 41 o
o 106 73 96 89 110 133 126 33 56 49 64 o
79 93 29 45 61 69 85 101 109 125 141 77 91
o 67 121 114 137 44 37 60 81 74 97 103 o
o 54 83 76 99 120 113 136 43 36 59 116 o
o 51 48 55 32 88 95 72 128 135 112 119 o
o 38 140 124 108 63 47 31 100 84 68 132 o
o 90 154 153 22 21 19 18 147 146 15 150 o
13 o o o o o 163 o o o o o 1
P14 (Option 2)
168 o o o o o 8 o o o o o 156
o 20 16 17 148 149 151 152 23 24 155 80 o
o 145 102 86 70 139 123 107 62 46 30 25 o
o 142 58 35 42 98 75 82 138 115 122 28 o
o 129 111 134 127 34 57 50 71 94 87 41 o
o 106 73 96 89 110 133 126 33 56 49 64 o
79 93 29 45 61 69 85 101 109 125 141 77 91
o 67 121 114 137 44 37 60 81 74 97 103 o
o 54 83 76 99 120 113 136 43 36 59 116 o
o 51 48 55 32 88 95 72 128 135 112 119 o
o 38 140 124 108 63 47 31 100 84 68 132 o
o 90 154 153 22 21 19 18 147 146 15 150 o
14 o o o o o 162 o o o o o 2

which can be completed with a guessing routine ((Priem13a)) based on the remaining integers:

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

Each of the two (unique) squares shown above correspond with 8 * (10!)2 * n11 * n9 Quadrant P14 Bordered Magic Squares (n11 = all possible order 11 borders, n9 = all suitable order 9 centre squares).

Note: Any order 9 centre square for which the elements of the 3 x 3 corner squares sum to s9 can be used.

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