16x16 magic square

Khajuraho methode

 

Use the famous Khajuraho 4x4 panmagic square to construct larger magic squares which are a multiple of 4 (= 8x8, 12x12, 16x16, 20x20, … magic square).

 

Rewrite the Khajuraho magic square as follows:

 

 

Khajuraho magic square                Basic magic square 

7

12

1

14

   

7

h-4

1

h-2

2

13

8

11

   

2

h-3

8

h-5

16

3

10

5

   

h

3

h-6

5

9

6

15

4

   

h-7

6

h-1

4

  

 

To construct an 16x16 panmagic square, you need the basic square and 15 extending magic squares:

 

 

7
h-4 1 h-2 8 -8 8 -8 16 -16 16 -16 24 -24 24 -24
2 h-3 8 h-5 8 -8 8 -8 16 -16 16 -16 24 -24 24 -24
h 3 h-6 5 -8 8 -8 8 -16 16 -16 16 -24 24 -24 24
h-7 6 h-1 4 -8 8 -8 8 -16 16 -16 16 -24 24 -24 24
32 -32 32 -32 40 -40 40 -40 48 -48 48 -48 56 -56 56 -56
32 -32 32 -32 40 -40 40 -40 48 -48 48 -48 56 -56 56 -56
-32 32 -32 32 -40 40 -40 40 -48 48 -48 48 -56 56 -56 56
-32 32 -32 32 -40 40 -40 40 -48 48 -48 48 -56 56 -56 56
64 -64 64 -64 72 -72 72 -72 80 -80 80 -80 88 -88 88 -88
64 -64 64 -64 72 -72 72 -72 80 -80 80 -80 88 -88 88 -88
-64 64 -64 64 -72 72 -72 72 -80 80 -80 80 -88 88 -88 88
-64 64 -64 64 -72 72 -72 72 -80 80 -80 80 -88 88 -88 88
96 -96 96 -96 104 -104 104 -104 112 -112 112 -112 120 -120 120 -120
96 -96 96 -96 104 -104 104 -104 112 -112 112 -112 120 -120 120 -120
-96 96 -96 96 -104 104 -104 104 -112 112 -112 112 -120 120 -120 120
-96 96 -96 96 -104 104 -104 104 -112 112 -112 112 -120 120 -120 120

 

 

The highest number in the 16x16 square is 256. Fill in 256 for h and calculate all the numbers. You get the following 16x16 panmagic square. 

 

 

Panmagic 16x16 square

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

 

 

This magic square is almost Franklin panmagic. Only not all 2x2 sub-squares give 1/4 of the magic sum (1/4 x 2056 = 514). If you swap the colours you get the following most perfect (Franklin pan)magic 16x16 square:

 

 

Most perfect (Franklin pan)magic 16x16 square 

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

 

 

Use the Khajuraho method to construct magic squares of order is multiple of 4 from 8x8 to infinity. See 8x8, 12x12, 16x16, 20x20, 24x24, 28x28 and 32x32

 

 

It is possible to use each 4x4 panmagic square to construct a 16x16 Franklin panmagic square.

 

See above how to construct the almost perfect 16x16 Franklin panmagic square (replace the numbers 9 up to 16 of the 4x4 panmagic square by 249 up to 256 to create the first 4x4 sub-square and add each time 8 to the eight low numbers and -/- 8 to the eight high numbers to create the 15 other 4x4 sub-squares).

 

You must swap half of the numbers to get a perfect 16x16 Franklin panmagic square. Which numbers you must swap and how to swap the numbers, depends on the place of the 1 and the 8 in the 4x4 panmagic square.

 

 

1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16

 

 

Holger Danielsson showed me how to swap numbers in the 16x16, 24x24, 32x32, ... square (see also on his website https://www.magicsquares.info/construction/pandiagonal

5.html).

 

If the 1 and the 8 are in the same column, than you must swap half of the numbers of sub-square 1/5/9/13 with 4/8/12/16 and 2/6/10/14 with 3/7/11/15 (= horizontally).

 

If the 1 and the 8 are in the same row, than you must swap half of the numbers of sub-square 1/2/3/4 with 13/14/15/16 and 5/6/7/8 with 9/10/11/12 (= vertically).

 

 

Correction sheet 1

                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               

 

 

Correction sheet 2

                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               

 

 

If the 1 and the 8 are in position 1 & 2 or 3 & 4 of the row/column, than you must use correction sheet 1.

 

If the 1 and the 8 are in position 2 & 3 or 1 & 4 of the row/column, than you must use correction sheet 2.

 

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16x16 magic square