Below two new perfect 8×8 panmagic squares are shown, with both
franklin
and normal
panmagic properties. For example,
all rows*,
columns*,
diagonals*,
subdiagonals*,
and franklin-diagonals*
show the sum 260.
All half-rows*,
half-columns*,
half-diagonals*,
2×2 squares*
show the sum 130.
The quadrants are panmagic as well.
(A subdiagonal is formed by
e.g.
the numbers 47-40-18-27-61-54-4-9; a franklin-diagonal is
e.g.
47-40-18-27-6-15-57-50)
62 | 4 | 13 | 51 | 46 | 20 | 29 | 35 |
5 | 59 | 54 | 12 | 21 | 43 | 38 | 28 |
52 | 14 | 3 | 61 | 36 | 30 | 19 | 45 |
11 | 53 | 60 | 6 | 27 | 37 | 44 | 22 |
64 | 2 | 15 | 49 | 48 | 18 | 31 | 33 |
7 | 57 | 56 | 10 | 23 | 41 | 40 | 26 |
50 | 16 | 1 | 63 | 34 | 32 | 17 | 47 |
9 | 55 | 58 | 8 | 25 | 39 | 42 | 24 |
60 | 6 | 11 | 53 | 44 | 22 | 27 | 37 |
13 | 51 | 62 | 4 | 29 | 35 | 46 | 20 |
54 | 12 | 5 | 59 | 38 | 28 | 21 | 43 |
3 | 61 | 52 | 14 | 19 | 45 | 36 | 30 |
58 | 8 | 9 | 55 | 42 | 24 | 25 | 39 |
15 | 49 | 64 | 2 | 31 | 33 | 48 | 18 |
56 | 10 | 7 | 57 | 40 | 26 | 23 | 41 |
1 | 63 | 50 | 16 | 17 | 47 | 34 | 32 |
Both squares have been constructed in April 2007 by Willem Barink, author of the puzzle-game Medjig (strongly magic square related). Inspiration was due to three young master-class students in Nijmegen, the Netherlands, who in March 2007 reached world news because they had constructed a spectacular 12×12 panmagic square (HSA square).
Below you see the first square broken down in its grids, after p = a × 8 + r.
6 | 4 | 5 | 3 | 6 | 4 | 5 | 3 | 7 | 0 | 1 | 6 | 5 | 2 | 3 | 4 | |
5 | 3 | 6 | 4 | 5 | 3 | 6 | 4 | 0 | 7 | 6 | 1 | 2 | 5 | 4 | 3 | |
4 | 6 | 3 | 5 | 4 | 6 | 3 | 5 | 6 | 1 | 0 | 7 | 4 | 3 | 2 | 5 | |
3 | 5 | 4 | 6 | 3 | 5 | 4 | 6 | 1 | 6 | 7 | 0 | 3 | 4 | 5 | 2 | |
8 | 2 | 7 | 1 | 8 | 2 | 7 | 1 | 7 | 0 | 1 | 6 | 5 | 2 | 3 | 4 | |
7 | 1 | 8 | 2 | 7 | 1 | 8 | 2 | 0 | 7 | 6 | 1 | 2 | 5 | 4 | 3 | |
2 | 8 | 1 | 7 | 2 | 8 | 1 | 7 | 6 | 1 | 0 | 7 | 4 | 3 | 2 | 5 | |
1 | 7 | 2 | 8 | 1 | 7 | 2 | 8 | 1 | 6 | 7 | 0 | 3 | 4 | 5 | 2 | |
grid for the r-numbers | grid for the a-numbers |
---|
In both grids the rows and columns are filled with 8, and two times 4 different numbers (8/4 system). This is different from all more or less panmagic 8×8 squares I (Willem Barink) found on internet, and which, when broken down, showed grids with 8, and an alternation of 2 different numbers in either the rows, or the columns. The panmagic properties of the existing 8×8 squares could still be improved!
In the above 8/4 system I applied my knowledge about latin squares. There are 12 (and only 12) different structures in latin squares of order 4. I examined the patterns of all of them for their usefulness in constructing the grids, in order to avoid the problem of duplo-numbering when composing the final square. I found three (and only three) latins which were suitable:
Structure A | Structure B | Structure C | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A | B | C | D | A | B | C | D | A | B | C | D | ||
B | A | D | C | C | D | A | B | C | D | A | B | ||
D | C | B | A | B | A | D | C | D | C | B | A | ||
C | D | A | B | D | C | B | A | B | A | D | C |
Regarding the grids of the first 8×8 square, you find out that they have been constructed by using 2×2 structure A for the a-numbers, and 2×2 structure B for the r-numbers. Decompose the second 8×8 square yourself, and you will find out that its grids have been constructed by using 2×2 structure C and its reflection. Consequently the first square may be called type AB, the second type CC.
I found one difference in magic properties between the two squares AB and CC. In type AB in the columns the sum of pairs of two consecutive numbers is 67, 63, 71, and 59, while in the rows it is an alternation of only 66 and 64. In the type CC square both in the rows and columns there is an alternation of just two sums of pairs (73 / 57 and 66 / 64). I call this feature isotropic (a term borrowed from mineralogy and crystallography). Due to these just two sums of pairs the second square contains more magic than the first: 5 semi-magic 4×4 units versus 2 semi-magic units extra magic (except for the diagonals, e.g. the 4×4 square between the numbers 5, 28, 33 and 64 in the second).
The above method can also be applied for constructing a 12×12 magic square (totals of rows, columns, etc. 870). The one shown below has been constructed by using 3×3 structure C for both the a-numbers and the r-numbers (its reflection).
14 | 11 | 133 | 132 | 38 | 35 | 109 | 108 | 62 | 59 | 85 | 84 |
121 | 144 | 2 | 23 | 97 | 120 | 26 | 47 | 73 | 96 | 50 | 71 |
12 | 13 | 131 | 134 | 36 | 37 | 107 | 110 | 60 | 61 | 83 | 86 |
143 | 122 | 24 | 1 | 119 | 98 | 48 | 25 | 95 | 74 | 72 | 49 |
16 | 9 | 135 | 130 | 40 | 33 | 111 | 106 | 64 | 57 | 87 | 82 |
123 | 142 | 4 | 21 | 99 | 118 | 28 | 45 | 75 | 94 | 52 | 69 |
10 | 15 | 129 | 136 | 34 | 39 | 105 | 112 | 58 | 63 | 81 | 88 |
141 | 124 | 22 | 3 | 117 | 100 | 46 | 27 | 93 | 76 | 70 | 51 |
18 | 7 | 137 | 128 | 42 | 31 | 113 | 104 | 66 | 55 | 89 | 80 |
125 | 140 | 6 | 19 | 101 | 116 | 30 | 43 | 77 | 92 | 54 | 67 |
8 | 17 | 127 | 138 | 32 | 41 | 103 | 114 | 56 | 65 | 79 | 90 |
139 | 126 | 20 | 5 | 115 | 102 | 44 | 29 | 91 | 78 | 68 | 53 |
The nine 4×4 subsquares, and also the four 8×8 subsquares are all perfectly panmagic. So, the above 12×12 square contains in fact 14 panmagic subsquares!
Note that the square is isotropic in the sense that both in rows and columns the sum of pairs of consecutive numbers is an alternation of six sums (135, 155, 139, 151, 143 and 147 for the columns, 25, 265, 73, 217, 121 and 169 for the rows). Also here this isotropic feature is an easy way to recognize a type CC construction. It is also possible to construct a type CC 12×12 square with horizontally and vertically an alternation of just two sums, see the next square.
138 | 8 | 17 | 127 | 114 | 32 | 41 | 103 | 90 | 56 | 65 | 79 |
19 | 125 | 140 | 6 | 43 | 101 | 116 | 30 | 67 | 77 | 92 | 54 |
128 | 18 | 7 | 137 | 104 | 42 | 31 | 113 | 80 | 66 | 55 | 89 |
5 | 139 | 126 | 20 | 29 | 115 | 102 | 44 | 53 | 91 | 78 | 68 |
136 | 10 | 15 | 129 | 112 | 34 | 39 | 105 | 88 | 58 | 63 | 81 |
21 | 123 | 142 | 4 | 45 | 99 | 118 | 28 | 69 | 75 | 94 | 52 |
130 | 16 | 9 | 135 | 106 | 40 | 33 | 111 | 82 | 64 | 57 | 87 |
3 | 141 | 124 | 22 | 27 | 117 | 100 | 46 | 51 | 93 | 76 | 70 |
134 | 12 | 13 | 131 | 110 | 36 | 37 | 107 | 86 | 60 | 61 | 83 |
23 | 121 | 144 | 2 | 47 | 97 | 120 | 26 | 71 | 73 | 96 | 50 |
132 | 14 | 11 | 133 | 108 | 38 | 35 | 109 | 84 | 62 | 59 | 85 |
1 | 143 | 122 | 24 | 25 | 119 | 98 | 48 | 49 | 95 | 74 | 72 |
With this square we have a lot more magic than in foregoing 12×12 square: 16 semi-magic 4×4 units more!
Relative to the well-known Morris 12×12 and the HSA 12×12, this abundance of panmagic and semi-magic subsquares is quite an extra panmagic quality; on the other hand, the square lacks the constant sum of franklin-diagonals*. This lack is due to the position of the diameters, dividing panmagic 4×4 units into non-magic 2×2 parts.
This flaw would not occur when constructing a 16×16 with magic properties as shown above. Consequently, the perfect 16×16 panmagic squares of Benjamin Franklin (1706-1790), recently discovered, and the more recent one of Morris can probably be improved! You see the result in my square below.
For comparison, see the franklin-square, and the panmagic Morris 16×16, and its description on the websites:
In the construction of my 16×16 panmagic square I kept the sums of pairs restricted to two in both rows and columns, as in the second 12×12 square above. The result is a square containing 30 panmagic (sub)squares, and 33 semi-magic 4×4 units! Enjoy the beautiful pattern in the positioning of the numbers, going from 1 to 256.
248 | 10 | 23 | 233 | 216 | 42 | 55 | 201 | 184 | 74 | 87 | 169 | 152 | 106 | 119 | 137 |
25 | 231 | 250 | 8 | 57 | 199 | 218 | 40 | 89 | 167 | 186 | 72 | 121 | 135 | 154 | 104 |
234 | 24 | 9 | 247 | 202 | 56 | 41 | 215 | 170 | 88 | 73 | 183 | 138 | 120 | 105 | 151 |
7 | 249 | 232 | 26 | 39 | 217 | 200 | 58 | 71 | 185 | 168 | 90 | 103 | 153 | 136 | 122 |
246 | 12 | 21 | 235 | 214 | 44 | 53 | 203 | 182 | 76 | 85 | 171 | 150 | 108 | 117 | 139 |
27 | 229 | 252 | 6 | 59 | 197 | 220 | 38 | 91 | 165 | 188 | 70 | 123 | 133 | 156 | 102 |
236 | 22 | 11 | 245 | 204 | 54 | 43 | 213 | 172 | 86 | 75 | 181 | 140 | 118 | 107 | 149 |
5 | 251 | 230 | 28 | 37 | 219 | 198 | 60 | 69 | 187 | 166 | 92 | 101 | 155 | 134 | 124 |
244 | 14 | 19 | 237 | 212 | 46 | 51 | 205 | 180 | 78 | 83 | 173 | 148 | 110 | 115 | 141 |
29 | 227 | 254 | 4 | 61 | 195 | 222 | 36 | 93 | 163 | 190 | 68 | 125 | 131 | 158 | 100 |
238 | 20 | 13 | 243 | 206 | 52 | 45 | 211 | 174 | 84 | 77 | 179 | 142 | 116 | 109 | 147 |
3 | 253 | 228 | 30 | 35 | 221 | 196 | 62 | 67 | 189 | 164 | 94 | 99 | 157 | 132 | 126 |
242 | 16 | 17 | 239 | 210 | 48 | 49 | 207 | 178 | 80 | 81 | 175 | 146 | 112 | 113 | 143 |
31 | 225 | 256 | 2 | 63 | 193 | 224 | 34 | 95 | 161 | 192 | 66 | 127 | 129 | 160 | 98 |
240 | 18 | 15 | 241 | 208 | 50 | 47 | 209 | 176 | 82 | 79 | 177 | 144 | 114 | 111 | 145 |
1 | 255 | 226 | 32 | 33 | 223 | 194 | 64 | 65 | 191 | 162 | 96 | 97 | 159 | 130 | 128 |
The 8/4, 12/4, 16/4 technique, as described above, for the construction of panmagic squares of order 4k (k≥2) is new, and gives very good results. The method is based on three useful structures in latin squares of order 4, which I called structure A, B and C. In the construction two different combinations of structures are possible: AB and CC. A type CC construction leads to the most impressive magic results.
Willem Barink,
Amsterdam, April 2007.
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