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12.9   Quadrant Magic Squares (17 x 17)

The concept of Quadrant Magic Squares, as discussed in following sections for order 17 Magic Squares, was introduced by Harvey Heinz (2001/2002).

12.9.1 Definition and Terminology

An order 17 magic square can be divided into four overlapping quadrants of 9 x 9 cells.

Each quadrant might contain symmetric patterns of 17 cells (16 cells + the centre cell) which sum to the Magic Sum s1, as illustrated in following example:

p001 (Plusmagic)
 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

For order 17 magic squares 253 patterns can be recognised (ref.Patterns17), further referred to as Magic Pattern P01 thru P253, which are shown in Attachment 12.8.1.

An order 17 Magic Square is called Quadrant Magic if the same pattern(s) occur in all four quadrants.

The number of patterns which occur in all four quadrants will be referred to as nQ4.

Note:
As only 15 of the possible patterns (ref. Section 12.8.2 below) are archived, the tag numbers of the patterns shown in Attachment 12.8.1 will differ from tag numbers used in other (possible) publications.

12.9.2 Historical Results

The properties of the historical order 17 (Multi) Quadrant Magic Squares, as published by Harvey Heinz, are based on the 15 Quadrant Magic Patterns shown in Attachment 12.8.2 and summarised below:

 nQ4 Type Quadrant Magic Patterns 1 PM p082 2 PM p112, p153 3 PM p209, p213, p216 4 PM p118, p209, p215, p241 5 PM p001, p085, p118, p178 , p209 6 PM p070, p153, p172, p178, p215, p248 1 Regular p241 3 Regular p001, p082, p085 4 Regular p153, p172, p178, p248

Note: The squares listed above contain also other patterns, however not in all four quadrants.

Subject squares were filtered from the order 17 (Pan) Magic Squares, which can be constructed based on Latin Diagonal Squares, as discussed in Section 17.2.1.

Following sections will describe and illustrate how comparable squares, not necessarily listed in the table shown above, can be constructed or generated.

The quadrant properties defined in Section 12.8.1 above can be described by the linear equations as listed in Attachment 12.8.3.

Subject equations can be combined with the equations describing miscellaneous types of order 17 Magic Squares.

Subject equations might be incorporated in procedures to determine all occurring patterns of (previously) generated Quadrant Magic Squares.

12.9.4 Pan Magic, Example 6

The occurring patterns for the above mentioned Pan Magic Square 'Example 6' can be summarised as follows (ref. ChkPtrn17):

• The number of patterns which occur in all 4 Quadrants is nQ4 = 61,
• All 253 patterns occur in the first Quadrant (left/top).

Attachment 12.8.41 shows for 'Example 6'(separately) the 61 Quadrant Magic Patterns.

Attachment 12.8.42 shows for 'Example 6'(separately) the 253 Patterns occurring in the first Quadrant (left/top).

12.9.5 Ultra Magic

In Section 17.2.2 is described how order 17 Ultra Magic Squares can be constructed based on pairs of Orthogonal Latin Diagonal Squares A and T(A).

Note: Square A is of type R2, L2, L3, R3, L4, R4, L5, R5, L6, R6, L7, R7, L8 and R8. T(A) is the transposed of A.

With the type R2 based routine UltraLat17 10.321.920 Quadrant Ultra Magic Squares could be generated as summarised in Attachment 12.8.52 and following graph:

An example of a Quadrant Ultra Magic Square with nQ4 = 2 (P73/P229) is shown below:

P73
 289 35 84 100 150 200 215 250 8 26 61 108 160 124 174 224 257 3 19 68 103 169 134 184 234 266 284 42 77 95 142 194 209 242 260 276 37 70 102 137 203 219 252 13 28 63 110 162 129 176 228 244 7 22 55 105 155 136 171 237 270 286 47 79 97 144 196 214 230 265 278 41 73 89 139 189 221 239 16 32 65 115 164 131 178 216 246 9 27 57 109 158 123 173 223 272 273 50 83 99 149 198 183 232 267 280 43 78 91 143 192 208 241 2 34 52 118 168 133 202 218 251 11 29 59 111 163 125 177 226 259 275 36 85 86 152 120 186 236 269 285 45 80 93 145 197 210 245 5 21 54 104 170 138 204 205 254 15 31 64 113 165 127 179 231 261 279 39 72 88 157 122 172 238 256 288 49 82 98 147 199 212 247 10 23 58 107 92 141 191 207 240 17 18 67 117 167 132 181 233 263 281 44 74 112 159 126 175 225 258 274 51 69 101 151 201 217 249 12 25 60 76 94 146 193 211 243 4 20 53 119 154 135 185 235 268 283 46 62 114 161 128 180 227 262 277 38 71 87 153 188 220 253 14 30 48 81 96 148 195 213 248 6 24 56 106 156 121 187 222 271 287 33 66 116 166 130 182 229 264 282 40 75 90 140 190 206 255 1
P229
 289 35 84 100 150 200 215 250 8 26 61 108 160 124 174 224 257 3 19 68 103 169 134 184 234 266 284 42 77 95 142 194 209 242 260 276 37 70 102 137 203 219 252 13 28 63 110 162 129 176 228 244 7 22 55 105 155 136 171 237 270 286 47 79 97 144 196 214 230 265 278 41 73 89 139 189 221 239 16 32 65 115 164 131 178 216 246 9 27 57 109 158 123 173 223 272 273 50 83 99 149 198 183 232 267 280 43 78 91 143 192 208 241 2 34 52 118 168 133 202 218 251 11 29 59 111 163 125 177 226 259 275 36 85 86 152 120 186 236 269 285 45 80 93 145 197 210 245 5 21 54 104 170 138 204 205 254 15 31 64 113 165 127 179 231 261 279 39 72 88 157 122 172 238 256 288 49 82 98 147 199 212 247 10 23 58 107 92 141 191 207 240 17 18 67 117 167 132 181 233 263 281 44 74 112 159 126 175 225 258 274 51 69 101 151 201 217 249 12 25 60 76 94 146 193 211 243 4 20 53 119 154 135 185 235 268 283 46 62 114 161 128 180 227 262 277 38 71 87 153 188 220 253 14 30 48 81 96 148 195 213 248 6 24 56 106 156 121 187 222 271 287 33 66 116 166 130 182 229 264 282 40 75 90 140 190 206 255 1

Attachment 12.8.51 shows (separately) all Quadrant Magic Patterns for the first occurring Quadrant Ultra Magic Square with nQ4 = 98 (maximum).

Notes:

1. The squares described above contain also numerous other patterns, however not in all four quadrants.
2. It can be noticed that all Ultra Magic Squares constructed as described in Section 17.2.2 are (Multi) Quadrant Ultra Magic.

12.9.7 Bordered, Centre Square Order 13

Order 17 Bordered Quadrant Magic Squares with order 13 Ultra Magic Centre Squares can be constructed based on:

• Latin Square based Ultra Magic Centre Squares as discussed in Section 13.2.2
• Semi Latin Square based borders as discussed in Section 17.2.4

The construction method is dependent from the extend in which the Quadrant Magic Property is effected by variations in the border(s).

Case 1 P61, P140, P143, P147, P148, P154

The Quadrant Magic Property based on the patterns listed above is invariant for variations of the borders as illustrated by following example:

P154
 9 2 3 4 277 278 279 280 282 283 284 285 14 15 16 17 137 18 26 20 21 260 261 262 25 27 266 267 268 31 32 271 138 272 35 36 253 88 116 148 180 198 39 60 132 211 76 159 225 254 255 52 135 40 55 134 207 82 165 231 249 90 111 149 177 195 155 238 69 67 228 246 91 106 151 173 201 46 61 130 209 77 166 223 221 102 169 196 47 58 127 210 72 168 224 252 97 112 147 175 121 188 119 70 164 226 247 98 109 144 176 191 49 54 133 216 78 220 171 136 118 182 197 45 56 128 217 75 161 227 242 100 105 150 172 154 170 186 71 167 233 248 96 107 145 183 194 42 57 123 219 104 120 187 87 140 185 190 48 63 129 215 73 162 234 245 93 108 203 103 204 189 212 74 157 236 241 99 114 146 181 192 43 64 126 101 86 205 240 115 143 178 193 38 66 122 218 80 163 232 243 94 50 85 222 206 124 213 81 160 229 244 89 117 139 184 199 44 62 84 68 239 257 95 113 141 179 200 41 59 125 208 83 156 235 250 33 51 256 237 65 131 214 79 158 230 251 92 110 142 174 202 37 53 34 289 152 270 269 30 29 28 265 263 24 23 22 259 258 19 264 1 153 288 287 286 13 12 11 10 8 7 6 5 276 275 274 273 281

Suitable Centre Squares can be selected from the 5760 unique Ultra Magic Squares with routine ChkPtrn17a.

Attachment 12.8.71 shows the first occurring Bordered Quadrant Magic Square for each of the patterns listed above.

Case 2 P38, P39, P45, P51, P134, P135, P138, P139

The Quadrant Magic Property based on the patterns listed above is invariant for variations of the outer border.

Exhibit P38 describes how P38 Quadrant Bordered Magic Squares, with Ultra Magic Centre Squares, can be constructed based on Semi Latin Borders as discussed in Section 17.2.4.

Centre Squares as described for Case 1 above (ref. Attachment 12.8.71) will result in two way Quadrant Bordered Magic Squares e.g. (P38/P61) as illustrated below:

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

Attachment 12.8.73 shows for each of the patterns listed above the first occurring Bordered Quadrant Magic Square for miscellaneous centre squares.

Case 3 P112, P115, P116, P119, P120, P126, P244

Also the patterns listed above will result in Quadrant Magic Properties which are invariant for variations of the outer border.

Quadrant Bordered Magic Squares, with Ultra Magic Centre Squares, can be constructed based on Semi Latin Borders as discussed in Section 17.2.4 and a method comparable with the one described in Exhibit P38.

Centre Squares as described for Case 1 above (ref. Attachment 12.8.71) will result in two way Quadrant Bordered Magic Squares e.g. (P112/P61).

Attachment 12.8.75 shows for each of the patterns listed above the first occurring Bordered Quadrant Magic Square for miscellaneous centre squares.

12.9.8 Summary

The obtained results regarding the miscellaneous types of order 17 Quadrant Magic Squares as deducted and discussed in previous sections are summarised in following table:

 Type Characteristics Subroutine Results Patterns P01 - P070, P71 - P238, P239 - P253 nQ4 = 61 Pan Magic (Latin Square Based) nQ4 = 98 Ultra Magic (Latin Square Based) Bordered Ultra Magic Centre Square (1) Ultra Magic Centre Square (2) Ultra Magic Centre Square (3) - - - -
 Comparable routines as listed above, can be used to generate alternative types of order 17 Magic Squares.