Office Applications and Entertainment, Magic Squares of Subtraction

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Composed Squares of Subtraction

In his paper "Magic Squares of Subtraction of Adam Adamandy Kochanski" Henryk Fuks describes how an order 8 Square of Subtraction can be constructed based on an order 4 Square of Subtraction.

Following sections will describe for miscellaneous orders possible construction methods for higher order Composed Magic Squares of Subtraction.

12.5 Squares of Subtraction (12 x 12)

Order 12 Magic Squares of Subtraction can be composed out of 9 order 4 Magic Squares of Subtraction, each with 16 consecutive integers and residuum Res4 = 8.

The construction method can be summarised as follows:

  1. Construct a square C composed out of 9 identical order 4 Magic Squares of Subtraction Cj = A (j = 1 ... 9);
  2. Construct an order 3 Simple Magic Square B with elements bj (j = 1 ... 9);
  3. Replace each element cij of Sub Square Cj by cij = cij + (bj - 1) * 42 (i = 1 ... 9);
  4. The result will be an order 12 Magic Square of Subtraction with 144 consecutive integers and residuum
    Res12 = 3 * 8 = 24.

An example obtained by subject method is shown below:

A, Res4 = 8
1 6 13 2
10 5 14 9
7 12 11 16
8 3 4 15
C, Res12 = 24
49 54 61 50
58 53 62 57
55 60 59 64
56 51 52 63
129 134 141 130
138 133 142 137
135 140 139 144
136 131 132 143
17 22 29 18
26 21 30 25
23 28 27 32
24 19 20 31
33 38 45 34
42 37 46 41
39 44 43 48
40 35 36 47
65 70 77 66
74 69 78 73
71 76 75 80
72 67 68 79
97 102 109 98
106 101 110 105
103 108 107 112
104 99 100 111
113 118 125 114
122 117 126 121
119 124 123 128
120 115 116 127
1 6 13 2
10 5 14 9
7 12 11 16
8 3 4 15
81 86 93 82
90 85 94 89
87 92 91 96
88 83 84 95
B, s3 = 15
4 9 2
3 5 7
8 1 6

Notes

  1. The method applied above for Magic Squares of Subtraction has been adopted from the method for (Additive) Magic Squares as described in Section 9.9.1.
  2. With 8 possible squares for B and n4 for Cj (j = 1 ... 9), the resulting number of 12th order Magic Squares of Subtraction with Res12 = 24 will be 8 * n49 with n4 = 8 * 4160 (ref. Exhibit 4).
  3. However for order 12 Squares of Subtraction it is sufficient that B contains the numbers (1 ... 9) in any sequence. Consequently the possible number of resulting Squares of Subtraction will be 9! * n49.
  4. With square B Simple Magic and square A Associated, the resulting square C will be an Associated Magic Square of Subtraction, as illustrated in Attachment 12.1.

16.0 Squares of Subtraction (16 x 16)

Comparable with Section 12.0 above, order 16 Magic Squares of Subtraction with Res16 = 32 can be constructed based on 16 order 4 Magic Squares of Subtraction each with Res4 = 8.

A, Res4 = 8
1 6 13 2
10 5 14 9
7 12 11 16
8 3 4 15
C, Res16 = 32
1 6 13 2
10 5 14 9
7 12 11 16
8 3 4 15
17 22 29 18
26 21 30 25
23 28 27 32
24 19 20 31
33 38 45 34
42 37 46 41
39 44 43 48
40 35 36 47
49 54 61 50
58 53 62 57
55 60 59 64
56 51 52 63
65 70 77 66
74 69 78 73
71 76 75 80
72 67 68 79
81 86 93 82
90 85 94 89
87 92 91 96
88 83 84 95
97 102 109 98
106 101 110 105
103 108 107 112
104 99 100 111
113 118 125 114
122 117 126 121
119 124 123 128
120 115 116 127
129 134 141 130
138 133 142 137
135 140 139 144
136 131 132 143
145 150 157 146
154 149 158 153
151 156 155 160
152 147 148 159
161 166 173 162
170 165 174 169
167 172 171 176
168 163 164 175
177 182 189 178
186 181 190 185
183 188 187 192
184 179 180 191
193 198 205 194
202 197 206 201
199 204 203 208
200 195 196 207
209 214 221 210
218 213 222 217
215 220 219 224
216 211 212 223
225 230 237 226
234 229 238 233
231 236 235 240
232 227 228 239
241 246 253 242
250 245 254 249
247 252 251 256
248 243 244 255
B
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16

Notes

  1. Square B might contain the numbers (1 ... 16) in any sequence. Consequently the possible number of resulting Squares of Subtraction will be 16! * n416.
  2. With square B Associated Magic and square A Associated, the resulting square C will be an Associated Magic Square of Subtraction, as illustrated in Attachment 16.1.
  3. With square B Pan Magic and square A with Pan Magic Diagonals, the resulting square C will be a Magic Square of Subtraction with Pan Magic Diagonals, as illustrated in Attachment 16.2.

18.0 Squares of Subtraction (18 x 18)

Comparable with Section 12.0 above, order 18 Magic Squares of Subtraction with Res18 = 45 can be constructed based on 9 order 6 Magic Squares of Subtraction each with Res6 = 15.

With A an Additive Magic Square of Subtraction and B a Simple (Additive) Magic Square, the resulting Square of Subtraction C will be Additve Magic as well.

A, s6 = 111, Res6 = 15
1 19 9 25 28 29
4 26 15 31 2 33
24 8 34 11 18 16
30 12 20 21 5 23
35 32 6 13 22 3
17 14 27 10 36 7
C, s18 = 2925, Res18 = 45
109 127 117 133 136 137
112 134 123 139 110 141
132 116 142 119 126 124
138 120 128 129 113 131
143 140 114 121 130 111
125 122 135 118 144 115
289 307 297 313 316 317
292 314 303 319 290 321
312 296 322 299 306 304
318 300 308 309 293 311
323 320 294 301 310 291
305 302 315 298 324 295
37 55 45 61 64 65
40 62 51 67 38 69
60 44 70 47 54 52
66 48 56 57 41 59
71 68 42 49 58 39
53 50 63 46 72 43
73 91 81 97 100 101
76 98 87 103 74 105
96 80 106 83 90 88
102 84 92 93 77 95
107 104 78 85 94 75
89 86 99 82 108 79
145 163 153 169 172 173
148 170 159 175 146 177
168 152 178 155 162 160
174 156 164 165 149 167
179 176 150 157 166 147
161 158 171 154 180 151
217 235 225 241 244 245
220 242 231 247 218 249
240 224 250 227 234 232
246 228 236 237 221 239
251 248 222 229 238 219
233 230 243 226 252 223
253 271 261 277 280 281
256 278 267 283 254 285
276 260 286 263 270 268
282 264 272 273 257 275
287 284 258 265 274 255
269 266 279 262 288 259
1 19 9 25 28 29
4 26 15 31 2 33
24 8 34 11 18 16
30 12 20 21 5 23
35 32 6 13 22 3
17 14 27 10 36 7
181 199 189 205 208 209
184 206 195 211 182 213
204 188 214 191 198 196
210 192 200 201 185 203
215 212 186 193 202 183
197 194 207 190 216 187

B, s3 = 15
4 9 2
3 5 7
8 1 6

Note
With 8 possible squares for B and n6 for Cj (j = 1 ... 9), the resulting number of 18th order Magic Squares of Subtraction (s18 = 2925, Res18 = 45) will be 8 * n69 with n6 = 8 * 24 * 1933 (Walter Trump).

64.0 Bimagic Squares of Subtraction (64 x 64)

Comparable with Section 18.0 above, order 64 Bimagic Squares of Subtraction with Res64 = 256 can be constructed based on 64 order 8 Bimagic Squares of Subtraction each with Res8 = 32.

With A an Additive Bimagic Square of Subtraction and B an Additive Bimagic Square, the resulting Square of Subtraction C will be Additve Bimagic as well (s1 = 131104, s2 = 358045024, Res64 = 256).

With nB the possible squares for B and nA for Cj (j = 1 ... 64), the resulting number of 64th order (Additive) Bimagic Squares of Subtraction will be nB * nA64, in which nB = 8 * 192 * 136244 and nA = 8 * 192 * 608.

Associated Bimagic Squares of Subtraction

With A an Additive Associated Bimagic Square of Subtraction and B an Additive Associated Bimagic Square, the resulting Square of Subtraction C will be Additve, Associated and Bimagic as well.

A, Res8 = 32, s1 = 260, s2 = 11180
2 21 46 57 52 39 32 11
25 14 53 34 43 64 7 20
60 47 24 3 10 29 38 49
35 56 15 28 17 6 61 42
23 4 59 48 37 50 9 30
16 27 36 55 62 41 18 5
45 58 1 22 31 12 51 40
54 33 26 13 8 19 44 63
B, s1 = 260, s2 = 11180
1 20 16 61 53 40 37 32
26 18 43 62 51 41 6 13
42 50 21 31 48 7 56 5
35 57 10 27 29 2 46 54
11 19 63 36 38 55 8 30
60 9 58 17 34 44 15 23
52 59 24 14 3 22 47 39
33 28 25 12 4 49 45 64

The resulting Additve Associated Bimagic Square of Subtraction C is shown in Attachment 64.1.

Square C corresponds with nB * nA64 Additve Associated Bimagic Squares of Subtraction, in which nB = 8 * 192 * 841 and nA = 8 * 192 * 32.

Pan Magic Bimagic Squares of Subtraction

With A an Additive, Pan Magic and Bimagic Square of Subtraction and B an Additive, Pan Magic and Bimagic Square, the resulting Square of Subtraction C will be Additve, Pan Magic and Bimagic as well.

A, Res8 = 32, s1 = 260, s2 = 11180
2 13 19 32 44 39 57 54
27 24 10 5 49 62 36 47
56 59 37 42 30 17 15 4
45 34 64 51 7 12 22 25
21 26 8 11 63 52 46 33
16 3 29 18 38 41 55 60
35 48 50 61 9 6 28 23
58 53 43 40 20 31 1 14
B, s1 = 260, s2 = 11180
2 11 57 52 30 23 37 48
60 49 3 10 40 45 31 22
46 39 21 32 50 59 9 4
24 29 47 38 12 1 51 58
35 42 28 17 63 54 8 13
25 20 34 43 5 16 62 55
15 6 56 61 19 26 44 33
53 64 14 7 41 36 18 27

The resulting Additve, Pan Magic and Bimagic Square of Subtraction C is shown in Attachment 64.2.

Square C corresponds with nB * nA64 Additve, Pan Magic and Bimagic Squares of Subtraction, in which nB = 8 * (29376 + 7288) and nA = 8 * 256.

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