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Paulus Gerdes introduced the Liki magic square (see http://plus.maths.org/content/new-designs-africa). He showed that it is possible to transform a square with consecutive digits into a magic square by swapping half of the numbers symmetrically. You can use this method to construct magic squares which are a multiple of 4 (= 4x4, 8x8, 12x12, 16x16, ... magic square).
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Paulus Gerdes constructed the following symmetric 8x8 magic square:
8x8 square with consecutive numbers
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 |  |
232 |
240 |
248 |
256 |
264 |
272 |
280 |
288 |
 |
 |
260 |
 |  |  |  |  |  |  |  |
260 |
36 |
 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
 |
100 |
 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
 |
164 |
 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
24 |
 |
228 |
 |
25 |
26 |
27 |
28 |
29 |
30 |
31 |
32 |
 |
292 |
 |
33 |
34 |
35 |
36 |
37 |
38 |
39 |
40 |
 |
356 |
 |
41 |
42 |
43 |
44 |
45 |
46 |
47 |
48 |
 |
420 |
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49 |
50 |
51 |
52 |
53 |
54 |
55 |
56 |
 |
484 |
 |
57 |
58 |
59 |
60 |
61 |
62 |
63 |
64 |
 |
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Symmetric 8x8 magic square
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 |  |
260 |
260 |
260 |
260 |
260 |
260 |
260 |
260 |
 |
 |
260 |
 |  |  |  |  |  |  |  |
260 |
260 |
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1 |
63 |
3 |
61 |
60 |
6 |
58 |
8 |
 |
260 |
 |
56 |
55 |
11 |
12 |
13 |
14 |
50 |
49 |
 |
260 |
 |
17 |
18 |
46 |
45 |
44 |
43 |
23 |
24 |
 |
260 |
 |
40 |
26 |
38 |
28 |
29 |
35 |
31 |
33 |
 |
260 |
 |
32 |
34 |
30 |
36 |
37 |
27 |
39 |
25 |
 |
260 |
 |
41 |
42 |
22 |
21 |
20 |
19 |
47 |
48 |
 |
260 |
 |
16 |
15 |
51 |
52 |
53 |
54 |
10 |
9 |
 |
260 |
 |
57 |
7 |
59 |
5 |
4 |
62 |
2 |
64 |
 |
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I used Paulus' method to construct a symmetric 16x16 magic square:
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 |  | 1936 | 1952 | 1968 | 1984 | 2000 | 2016 | 2032 | 2048 | 2064 | 2080 | 2096 | 2112 | 2128 | 2144 | 2160 | 2176 |  |
 | 2056 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 2056 |
136 | Â | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | Â |
392 | Â | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | Â |
648 | Â | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | Â |
904 | Â | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | Â |
1160 | Â | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | Â |
1416 | Â | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | Â |
1672 | Â | 97 | 98 | 99 | 100 | 101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 | 111 | 112 | Â |
1928 | Â | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 | 121 | 122 | 123 | 124 | 125 | 126 | 127 | 128 | Â |
2184 | Â | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 | Â |
2440 | Â | 145 | 146 | 147 | 148 | 149 | 150 | 151 | 152 | 153 | 154 | 155 | 156 | 157 | 158 | 159 | 160 | Â |
2696 | Â | 161 | 162 | 163 | 164 | 165 | 166 | 167 | 168 | 169 | 170 | 171 | 172 | 173 | 174 | 175 | 176 | Â |
2952 | Â | 177 | 178 | 179 | 180 | 181 | 182 | 183 | 184 | 185 | 186 | 187 | 188 | 189 | 190 | 191 | 192 | Â |
3208 | Â | 193 | 194 | 195 | 196 | 197 | 198 | 199 | 200 | 201 | 202 | 203 | 204 | 205 | 206 | 207 | 208 | Â |
3464 | Â | 209 | 210 | 211 | 212 | 213 | 214 | 215 | 216 | 217 | 218 | 219 | 220 | 221 | 222 | 223 | 224 | Â |
3720 | Â | 225 | 226 | 227 | 228 | 229 | 230 | 231 | 232 | 233 | 234 | 235 | 236 | 237 | 238 | 239 | 240 | Â |
3976 | Â | 241 | 242 | 243 | 244 | 245 | 246 | 247 | 248 | 249 | 250 | 251 | 252 | 253 | 254 | 255 | 256 | Â |
 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
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 |  | 2056 | 2056 | 2056 | 2056 | 2056 | 2056 | 2056 | 2056 | 2056 | 2056 | 2056 | 2056 | 2056 | 2056 | 2056 | 2056 |  |
 | 2056 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 2056 |
2056 | Â | 256 | 255 | 3 | 253 | 5 | 6 | 7 | 249 | 248 | 10 | 11 | 12 | 244 | 14 | 242 | 241 | Â |
2056 | Â | 17 | 18 | 19 | 20 | 236 | 235 | 234 | 233 | 232 | 231 | 230 | 229 | 29 | 30 | 31 | 32 | Â |
2056 | Â | 224 | 223 | 222 | 221 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 212 | 211 | 210 | 209 | Â |
2056 | Â | 208 | 207 | 206 | 205 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 196 | 195 | 194 | 193 | Â |
2056 | Â | 65 | 66 | 67 | 68 | 188 | 187 | 186 | 185 | 184 | 183 | 182 | 181 | 77 | 78 | 79 | 80 | Â |
2056 | Â | 176 | 175 | 174 | 84 | 172 | 86 | 87 | 88 | 89 | 90 | 91 | 165 | 93 | 163 | 162 | 161 | Â |
2056 | Â | 97 | 98 | 99 | 100 | 156 | 155 | 154 | 153 | 152 | 151 | 150 | 149 | 109 | 110 | 111 | 112 | Â |
2056 | Â | 113 | 114 | 142 | 141 | 117 | 139 | 138 | 120 | 121 | 135 | 134 | 124 | 132 | 131 | 127 | 128 | Â |
2056 | Â | 129 | 130 | 126 | 125 | 133 | 123 | 122 | 136 | 137 | 119 | 118 | 140 | 116 | 115 | 143 | 144 | Â |
2056 | Â | 145 | 146 | 147 | 148 | 108 | 107 | 106 | 105 | 104 | 103 | 102 | 101 | 157 | 158 | 159 | 160 | Â |
2056 | Â | 96 | 95 | 94 | 164 | 92 | 166 | 167 | 168 | 169 | 170 | 171 | 85 | 173 | 83 | 82 | 81 | Â |
2056 | Â | 177 | 178 | 179 | 180 | 76 | 75 | 74 | 73 | 72 | 71 | 70 | 69 | 189 | 190 | 191 | 192 | Â |
2056 | Â | 64 | 63 | 62 | 61 | 197 | 198 | 199 | 200 | 201 | 202 | 203 | 204 | 52 | 51 | 50 | 49 | Â |
2056 | Â | 48 | 47 | 46 | 45 | 213 | 214 | 215 | 216 | 217 | 218 | 219 | 220 | 36 | 35 | 34 | 33 | Â |
2056 | Â | 225 | 226 | 227 | 228 | 28 | 27 | 26 | 25 | 24 | 23 | 22 | 21 | 237 | 238 | 239 | 240 | Â |
2056 | Â | 16 | 15 | 243 | 13 | 245 | 246 | 247 | 9 | 8 | 250 | 251 | 252 | 4 | 254 | 2 | 1 | Â |
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Though it is a beautiful 16x16 magic square, it is only a simple, symmetric 16x16 magic square. If you swap the numbers in the same cells of each 4x4 sub-square, you get a magic 16x16 square with more magic features:
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16x16Â square with consecutive numbers
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163 |
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232 |
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237 |
238 |
239 |
240 |
241 |
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243 |
244 |
245 |
246 |
247 |
248 |
249 |
250 |
251 |
252 |
253 |
254 |
255 |
256 |
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Symmetric 16x16 magic square
1 |
255 |
254 |
4 |
5 |
251 |
250 |
8 |
9 |
247 |
246 |
12 |
13 |
243 |
242 |
16 |
240 |
18 |
19 |
237 |
236 |
22 |
23 |
233 |
232 |
26 |
27 |
229 |
228 |
30 |
31 |
225 |
224 |
34 |
35 |
221 |
220 |
38 |
39 |
217 |
216 |
42 |
43 |
213 |
212 |
46 |
47 |
209 |
49 |
207 |
206 |
52 |
53 |
203 |
202 |
56 |
57 |
199 |
198 |
60 |
61 |
195 |
194 |
64 |
65 |
191 |
190 |
68 |
69 |
187 |
186 |
72 |
73 |
183 |
182 |
76 |
77 |
179 |
178 |
80 |
176 |
82 |
83 |
173 |
172 |
86 |
87 |
169 |
168 |
90 |
91 |
165 |
164 |
94 |
95 |
161 |
160 |
98 |
99 |
157 |
156 |
102 |
103 |
153 |
152 |
106 |
107 |
149 |
148 |
110 |
111 |
145 |
113 |
143 |
142 |
116 |
117 |
139 |
138 |
120 |
121 |
135 |
134 |
124 |
125 |
131 |
130 |
128 |
129 |
127 |
126 |
132 |
133 |
123 |
122 |
136 |
137 |
119 |
118 |
140 |
141 |
115 |
114 |
144 |
112 |
146 |
147 |
109 |
108 |
150 |
151 |
105 |
104 |
154 |
155 |
101 |
100 |
158 |
159 |
97 |
96 |
162 |
163 |
93 |
92 |
166 |
167 |
89 |
88 |
170 |
171 |
85 |
84 |
174 |
175 |
81 |
177 |
79 |
78 |
180 |
181 |
75 |
74 |
184 |
185 |
71 |
70 |
188 |
189 |
67 |
66 |
192 |
193 |
63 |
62 |
196 |
197 |
59 |
58 |
200 |
201 |
55 |
54 |
204 |
205 |
51 |
50 |
208 |
48 |
210 |
211 |
45 |
44 |
214 |
215 |
41 |
40 |
218 |
219 |
37 |
36 |
222 |
223 |
33 |
32 |
226 |
227 |
29 |
28 |
230 |
231 |
25 |
24 |
234 |
235 |
21 |
20 |
238 |
239 |
17 |
241 |
15 |
14 |
244 |
245 |
11 |
10 |
248 |
249 |
7 |
6 |
252 |
253 |
3 |
2 |
256 |
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This 16x16 magic square is not only symmetric, but each 1/4 row/column gives 1/4 of the magic sum.
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But we can do better. If you refine the starting position of the consecutive numbers, you can get an ultra magic 16x16 square:
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Starting position 16x16 magic square
1 | 2 | 6 | 5 | 10 | 9 | 13 | 14 | 17 | 18 | 22 | 21 | 26 | 25 | 29 | 30 |
3 | 4 | 8 | 7 | 12 | 11 | 15 | 16 | 19 | 20 | 24 | 23 | 28 | 27 | 31 | 32 |
35 | 36 | 40 | 39 | 44 | 43 | 47 | 48 | 51 | 52 | 56 | 55 | 60 | 59 | 63 | 64 |
33 | 34 | 38 | 37 | 42 | 41 | 45 | 46 | 49 | 50 | 54 | 53 | 58 | 57 | 61 | 62 |
67 | 68 | 72 | 71 | 76 | 75 | 79 | 80 | 83 | 84 | 88 | 87 | 92 | 91 | 95 | 96 |
65 | 66 | 70 | 69 | 74 | 73 | 77 | 78 | 81 | 82 | 86 | 85 | 90 | 89 | 93 | 94 |
97 | 98 | 102 | 101 | 106 | 105 | 109 | 110 | 113 | 114 | 118 | 117 | 122 | 121 | 125 | 126 |
99 | 100 | 104 | 103 | 108 | 107 | 111 | 112 | 115 | 116 | 120 | 119 | 124 | 123 | 127 | 128 |
129 | 130 | 134 | 133 | 138 | 137 | 141 | 142 | 145 | 146 | 150 | 149 | 154 | 153 | 157 | 158 |
131 | 132 | 136 | 135 | 140 | 139 | 143 | 144 | 147 | 148 | 152 | 151 | 156 | 155 | 159 | 160 |
163 | 164 | 168 | 167 | 172 | 171 | 175 | 176 | 179 | 180 | 184 | 183 | 188 | 187 | 191 | 192 |
161 | 162 | 166 | 165 | 170 | 169 | 173 | 174 | 177 | 178 | 182 | 181 | 186 | 185 | 189 | 190 |
195 | 196 | 200 | 199 | 204 | 203 | 207 | 208 | 211 | 212 | 216 | 215 | 220 | 219 | 223 | 224 |
193 | 194 | 198 | 197 | 202 | 201 | 205 | 206 | 209 | 210 | 214 | 213 | 218 | 217 | 221 | 222 |
225 | 226 | 230 | 229 | 234 | 233 | 237 | 238 | 241 | 242 | 246 | 245 | 250 | 249 | 253 | 254 |
227 | 228 | 232 | 231 |