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Exhibit P38
Typical for P38, P39, P45, P51, P134, P135, P138 and P139

Analysis Quadrant P38 Bordered Magic Squares

The Quadrant P38 Property of Bordered Magic Squares, with Ultra Magic Centre Squares, is defined by the inner border variables a(i), i = 19, 25, 27, 33, 121, 135 155, 169 257, 263, 265 and 271 as illustrated below:

P38
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 a170
a171 a172 a173 a174 a175 a176 a177 a178 a179 a180 a181 a182 a183 a184 a185 a186 a187
a188 a189 a190 a191 a192 a193 a194 a195 a196 a197 a198 a199 a200 a201 a202 a203 a204
a205 a206 a207 a208 a209 a210 a211 a212 a213 a214 a215 a216 a217 a218 a219 a220 a221
a222 a223 a224 a225 a226 a227 a228 a229 a230 a231 a232 a233 a234 a235 a236 a237 a238
a239 a240 a241 a242 a243 a244 a245 a246 a247 a248 a249 a250 a251 a252 a253 a254 a255
a256 a257 a258 a259 a260 a261 a262 a263 a264 a265 a266 a267 a268 a269 a270 a271 a272
a273 a274 a275 a276 a277 a278 a279 a280 a281 a282 a283 a284 a285 a286 a287 a288 a289

Defining Equations

a(169) = s1 - a(265) - a(271) - s(4)
a(155) = s1 - a(257) - a(263) - s(3)
a( 27) = s1 - a( 33) - a(135) - s(2)
a( 19) = s1 - a( 25) - a(121) - s(1)

a( 19) = Pr15 - a(271)    a( 25) = Pr15 - a(263)
a( 27) = Pr15 - a(265)    a( 33) = Pr15 - a(257)
a(121) = Pr15 - a(135)    a(155) = Pr15 - a(169)

Reduced Equations

a(135) = -11*s1/17 - a(263) - a(271) + s(1)
a(169) =     s1    - a(265) - a(271) - s(4)
a(257) = -24*s1/17 - a(263) - a(265) - a(271) + s(1) + s(2)
s(1)   =  56*s1/17 - s(2)   - s(3)   - s(4)

a( 19) = Pr15 - a(271)    a( 25) = Pr15 - a(263)
a( 27) = Pr15 - a(265)    a( 33) = Pr15 - a(257)
a(121) = Pr15 - a(135)    a(155) = Pr15 - a(169)

with s1 the Magic Sum and s(1), s(2), s(3) and s(4) the sums of the pattern elements within the Ultra Magic Centre Square.

The reduced equations can be incorporated in a guessing routine (ChkPtrn17b) which

  • reads the Order 13 Ultra Magic Center Squares and determines the partial sums s(1), s(2), s(3) and s(4)
  • completes the patterns based on the independent variables a(263), a(265) and a(271)

The applied range (order 15 border) results from the construction method based on Semi Latin Squares:

{19 ... 33,  36,  50,  53,  67,  70,  84,  87, 101, 104, 118, 121, 135, 138
       152, 155, 169, 172, 186, 189, 203, 206, 220, 223, 237, 240, 254, 257 ... 271}

as illustrated in Section 17.2.4.

Numerous solutions - suitable for the construction of Quadrant Bordered Magic Squares - can be obtained of which a few examples are shown in Attachment 12.8.72.

Construction Quadrant P38 Bordered Magic Squares

While starting with a preselected Ultra Magic Centre Square (ref. Section 13.2.2):

Ultra Magic
155 67 114 86 63 96 26 46 27 9 159 136 121
19 43 38 2 166 138 128 148 78 111 79 61 94
126 146 71 108 90 54 101 21 50 31 13 163 131
98 14 48 29 6 160 142 119 153 73 115 83 65
135 130 150 66 113 81 58 95 25 41 36 8 167
60 102 18 52 33 1 165 133 123 147 77 106 88
158 140 125 154 70 117 85 53 100 16 45 30 12
82 64 93 23 47 37 5 169 137 118 152 68 110
3 162 134 129 145 75 112 89 57 104 20 40 35
105 87 55 97 17 51 28 10 164 141 122 156 72
39 7 157 139 120 149 69 116 80 62 99 24 44
76 109 91 59 92 22 42 32 4 168 132 127 151
49 34 11 161 143 124 144 74 107 84 56 103 15
A13
11 1 9 7 10 4 12 6 0 8 2 5 3
5 3 11 1 9 7 10 4 12 6 0 8 2
8 2 5 3 11 1 9 7 10 4 12 6 0
6 0 8 2 5 3 11 1 9 7 10 4 12
4 12 6 0 8 2 5 3 11 1 9 7 10
7 10 4 12 6 0 8 2 5 3 11 1 9
1 9 7 10 4 12 6 0 8 2 5 3 11
3 11 1 9 7 10 4 12 6 0 8 2 5
2 5 3 11 1 9 7 10 4 12 6 0 8
0 8 2 5 3 11 1 9 7 10 4 12 6
12 6 0 8 2 5 3 11 1 9 7 10 4
10 4 12 6 0 8 2 5 3 11 1 9 7
9 7 10 4 12 6 0 8 2 5 3 11 1
B13
11 5 8 6 4 7 1 3 2 0 12 10 9
1 3 2 0 12 10 9 11 5 8 6 4 7
9 11 5 8 6 4 7 1 3 2 0 12 10
7 1 3 2 0 12 10 9 11 5 8 6 4
10 9 11 5 8 6 4 7 1 3 2 0 12
4 7 1 3 2 0 12 10 9 11 5 8 6
12 10 9 11 5 8 6 4 7 1 3 2 0
6 4 7 1 3 2 0 12 10 9 11 5 8
0 12 10 9 11 5 8 6 4 7 1 3 2
8 6 4 7 1 3 2 0 12 10 9 11 5
2 0 12 10 9 11 5 8 6 4 7 1 3
5 8 6 4 7 1 3 2 0 12 10 9 11
3 2 0 12 10 9 11 5 8 6 4 7 1

a suitable Ultra Magic Centre Square can be constructed:

A13'
13 3 11 9 12 6 14 8 2 10 4 7 5
7 5 13 3 11 9 12 6 14 8 2 10 4
10 4 7 5 13 3 11 9 12 6 14 8 2
8 2 10 4 7 5 13 3 11 9 12 6 14
6 14 8 2 10 4 7 5 13 3 11 9 12
9 12 6 14 8 2 10 4 7 5 13 3 11
3 11 9 12 6 14 8 2 10 4 7 5 13
5 13 3 11 9 12 6 14 8 2 10 4 7
4 7 5 13 3 11 9 12 6 14 8 2 10
2 10 4 7 5 13 3 11 9 12 6 14 8
14 8 2 10 4 7 5 13 3 11 9 12 6
12 6 14 8 2 10 4 7 5 13 3 11 9
11 9 12 6 14 8 2 10 4 7 5 13 3
B13'
13 7 10 8 6 9 3 5 4 2 14 12 11
3 5 4 2 14 12 11 13 7 10 8 6 9
11 13 7 10 8 6 9 3 5 4 2 14 12
9 3 5 4 2 14 12 11 13 7 10 8 6
12 11 13 7 10 8 6 9 3 5 4 2 14
6 9 3 5 4 2 14 12 11 13 7 10 8
14 12 11 13 7 10 8 6 9 3 5 4 2
8 6 9 3 5 4 2 14 12 11 13 7 10
2 14 12 11 13 7 10 8 6 9 3 5 4
10 8 6 9 3 5 4 2 14 12 11 13 7
4 2 14 12 11 13 7 10 8 6 9 3 5
7 10 8 6 9 3 5 4 2 14 12 11 13
5 4 2 14 12 11 13 7 10 8 6 9 3
M13' = A13' + 17 * B13' + 1
235 123 182 146 115 160 66 94 71 45 243 212 193
59 91 82 38 250 214 200 228 134 179 139 113 158
198 226 127 176 150 106 165 61 98 75 49 247 207
162 54 96 73 42 244 218 191 233 129 183 143 117
211 202 230 122 181 141 110 159 65 89 80 44 251
112 166 58 100 77 37 249 209 195 227 133 174 148
242 216 197 234 126 185 145 105 164 56 93 74 48
142 116 157 63 95 81 41 253 213 190 232 124 178
39 246 210 201 225 131 180 149 109 168 60 88 79
173 147 107 161 57 99 72 46 248 217 194 236 128
83 43 241 215 192 229 125 184 140 114 163 64 92
132 177 151 111 156 62 90 76 40 252 208 199 231
97 78 47 245 219 196 224 130 175 144 108 167 55

Based on this centre square and the applicable key variables, a partly completed order 15 Bordered Magic Square can be constructed, which contains the four P38 Patterns:

Cntr Sqr (15 x 15)
270 138 254 26
235 123 182 146 115 160 66 94 71 45 243 212 193
59 91 82 38 250 214 200 228 134 179 139 113 158
198 226 127 176 150 106 165 61 98 75 49 247 207
162 54 96 73 42 244 218 191 233 129 183 143 117
211 202 230 122 181 141 110 159 65 89 80 44 251
135 112 166 58 100 77 37 249 209 195 227 133 174 148 155
242 216 197 234 126 185 145 105 164 56 93 74 48
19 142 116 157 63 95 81 41 253 213 190 232 124 178 271
39 246 210 201 225 131 180 149 109 168 60 88 79
173 147 107 161 57 99 72 46 248 217 194 236 128
83 43 241 215 192 229 125 184 140 114 163 64 92
132 177 151 111 156 62 90 76 40 252 208 199 231
97 78 47 245 219 196 224 130 175 144 108 167 55
264 152 36 20
Key Var's
270 138 254 26
135 0 0 155
19 0 07 271
264 152 36 20

which can be completed with a guessing routine (Priem15b) based on the remaining integers of the applicable range:

Cntr Sqr (15 x 15)
270 259 260 261 262 263 138 67 254 25 24 23 22 21 26
172 235 123 182 146 115 160 66 94 71 45 243 212 193 118
186 59 91 82 38 250 214 200 228 134 179 139 113 158 104
203 198 226 127 176 150 106 165 61 98 75 49 247 207 87
220 162 54 96 73 42 244 218 191 233 129 183 143 117 70
237 211 202 230 122 181 141 110 159 65 89 80 44 251 53
135 112 166 58 100 77 37 249 209 195 227 133 174 148 155
169 242 216 197 234 126 185 145 105 164 56 93 74 48 121
19 142 116 157 63 95 81 41 253 213 190 232 124 178 271
101 39 246 210 201 225 131 180 149 109 168 60 88 79 189
84 173 147 107 161 57 99 72 46 248 217 194 236 128 206
50 83 43 241 215 192 229 125 184 140 114 163 64 92 240
33 132 177 151 111 156 62 90 76 40 252 208 199 231 257
32 97 78 47 245 219 196 224 130 175 144 108 167 55 258
264 31 30 29 28 27 152 223 36 265 266 267 268 269 20

which can be completed with an exterior border as constructed in Section 17.2.4.

The application of centre squares containing already complete patterns e.g. P61 will result in Quadrant (P38, P61) Bordered Magic Squares:

P38
9 2 3 4 277 278 279 280 282 283 284 285 14 15 16 17 137
18 270 259 260 261 262 263 138 67 254 25 24 23 22 21 26 272
35 172 235 123 182 146 115 160 66 94 71 45 243 212 193 118 255
52 186 59 91 82 38 250 214 200 228 134 179 139 113 158 104 238
69 203 198 226 127 176 150 106 165 61 98 75 49 247 207 87 221
102 220 162 54 96 73 42 244 218 191 233 129 183 143 117 70 188
119 237 211 202 230 122 181 141 110 159 65 89 80 44 251 53 171
136 135 112 166 58 100 77 37 249 209 195 227 133 174 148 155 154
170 169 242 216 197 234 126 185 145 105 164 56 93 74 48 121 120
187 19 142 116 157 63 95 81 41 253 213 190 232 124 178 271 103
204 101 39 246 210 201 225 131 180 149 109 168 60 88 79 189 86
205 84 173 147 107 161 57 99 72 46 248 217 194 236 128 206 85
222 50 83 43 241 215 192 229 125 184 140 114 163 64 92 240 68
239 33 132 177 151 111 156 62 90 76 40 252 208 199 231 257 51
256 32 97 78 47 245 219 196 224 130 175 144 108 167 55 258 34
289 264 31 30 29 28 27 152 223 36 265 266 267 268 269 20 1
153 288 287 286 13 12 11 10 8 7 6 5 276 275 274 273 281
P61
9 2 3 4 277 278 279 280 282 283 284 285 14 15 16 17 137
18 270 259 260 261 262 263 138 67 254 25 24 23 22 21 26 272
35 172 235 123 182 146 115 160 66 94 71 45 243 212 193 118 255
52 186 59 91 82 38 250 214 200 228 134 179 139 113 158 104 238
69 203 198 226 127 176 150 106 165 61 98 75 49 247 207 87 221
102 220 162 54 96 73 42 244 218 191 233 129 183 143 117 70 188
119 237 211 202 230 122 181 141 110 159 65 89 80 44 251 53 171
136 135 112 166 58 100 77 37 249 209 195 227 133 174 148 155 154
170 169 242 216 197 234 126 185 145 105 164 56 93 74 48 121 120
187 19 142 116 157 63 95 81 41 253 213 190 232 124 178 271 103
204 101 39 246 210 201 225 131 180 149 109 168 60 88 79 189 86
205 84 173 147 107 161 57 99 72 46 248 217 194 236 128 206 85
222 50 83 43 241 215 192 229 125 184 140 114 163 64 92 240 68
239 33 132 177 151 111 156 62 90 76 40 252 208 199 231 257 51
256 32 97 78 47 245 219 196 224 130 175 144 108 167 55 258 34
289 264 31 30 29 28 27 152 223 36 265 266 267 268 269 20 1
153 288 287 286 13 12 11 10 8 7 6 5 276 275 274 273 281

The square shown above corresponds with 8 * (15!)2 * (1!)2 * n13 Quadrant (P38, P61) Bordered Magic Squares (n13 = all suitable order 13 centre squares).

Attachment 12.8.73 shows miscellaneous Quadrant Bordered Magic Squares, as found with routine Priem15b, based on the sub collection of 0rder 13 Ultra Magic Squares described above.

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