Pantriagonal 6x6x6 magic cube (Method of Luyendijk)
I have analyzed the pantriagonal 6x6x6 magic cube on the website of Jos Luyendijk (http://www.entertainmentmathematics.nl/index.html)
This 6x6x6 magic cube is based on a 3x3 magic square.
The first grid consists of the 3x3 magic square and the shifted and inverse (shifted) versions of the 3x3 magic square. See in the second level top left the 3x3 magic square and top right the inverse 3x3 magic square. In the first and third level the columns of the second level are shifted (or swapped). Level 4 up to 6 is the inverse of level 1 up to 3.
Take 1x number from first grid
| 30 | 30 | 30 | 30 | 30 | 30 | ||
| 1 | |||||||
| 30 | 1 | 8 | 6 | 9 | 2 | 4 | |
| 30 | 5 | 3 | 7 | 5 | 7 | 3 | |
| 30 | 9 | 4 | 2 | 1 | 6 | 8 | |
| 30 | 9 | 2 | 4 | 1 | 8 | 6 | |
| 30 | 5 | 7 | 3 | 5 | 3 | 7 | |
| 30 | 1 | 6 | 8 | 9 | 4 | 2 | |
| 30 | 30 | 30 | 30 | 30 | 30 | ||
| 2 | |||||||
| 30 | 6 | 1 | 8 | 4 | 9 | 2 | |
| 30 | 7 | 5 | 3 | 3 | 5 | 7 | |
| 30 | 2 | 9 | 4 | 8 | 1 | 6 | |
| 30 | 4 | 9 | 2 | 6 | 1 | 8 | |
| 30 | 3 | 5 | 7 | 7 | 5 | 3 | |
| 30 | 8 | 1 | 6 | 2 | 9 | 4 | |
| 30 | 30 | 30 | 30 | 30 | 30 | ||
| 3 | |||||||
| 30 | 8 | 6 | 1 | 2 | 4 | 9 | |
| 30 | 3 | 7 | 5 | 7 | 3 | 5 | |
| 30 | 4 | 2 | 9 | 6 | 8 | 1 | |
| 30 | 2 | 4 | 9 | 8 | 6 | 1 | |
| 30 | 7 | 3 | 5 | 3 | 7 | 5 | |
| 30 | 6 | 8 | 1 | 4 | 2 | 9 | |
| 30 | 30 | 30 | 30 | 30 | 30 | ||
| 4=1' | |||||||
| 30 | 9 | 2 | 4 | 1 | 8 | 6 | |
| 30 | 5 | 7 | 3 | 5 | 3 | 7 | |
| 30 | 1 | 6 | 8 | 9 | 4 | 2 | |
| 30 | 1 | 8 | 6 | 9 | 2 | 4 | |
| 30 | 5 | 3 | 7 | 5 | 7 | 3 | |
| 30 | 9 | 4 | 2 | 1 | 6 | 8 | |
| 30 | 30 | 30 | 30 | 30 | 30 | ||
| 5=2' | |||||||
| 30 | 4 | 9 | 2 | 6 | 1 | 8 | |
| 30 | 3 | 5 | 7 | 7 | 5 | 3 | |
| 30 | 8 | 1 | 6 | 2 | 9 | 4 | |
| 30 | 6 | 1 | 8 | 4 | 9 | 2 | |
| 30 | 7 | 5 | 3 | 3 | 5 | 7 | |
| 30 | 2 | 9 | 4 | 8 | 1 | 6 |
|
30 |
30 |
30 |
30 |
30 |
30 |
||
|
6=3' |
|||||||
|
30 |
2 |
4 |
9 |
8 |
6 |
1 |
|
|
30 |
7 |
3 |
5 |
3 |
7 |
5 |
|
|
30 |
6 |
8 |
1 |
4 |
2 |
9 |
|
|
30 |
8 |
6 |
1 |
2 |
4 |
9 |
|
|
30 |
3 |
7 |
5 |
7 |
3 |
5 |
|
|
30 |
4 |
2 |
9 |
6 |
8 |
1 |
|
In the second grid we need the numbers 1 up to 24 to construct (2x) 4 magic 3x3 squares with each time 3 different numbers in it. We must take care that each time the addition of 2 x 3 numbers give (6/2 x [1+24] = ) 75. We also must take care that the 5th up to the 8th magic 3x3 square consists of the inverse numbers of the 1st up to the 4th magic square and that there are no double numbers in it. So we use the table below to puzzle the right numbers.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | ||
| 24 | 23 | 22 | 21 | 20 | 19 | 18 | 17 | 16 | 15 | 14 | 13 | ||
| 21 | 1 | 5 | 15 | ||||||||||
| 75 | 54 | 23 | 19 | 12 | |||||||||
| 21 | 4 | 8 | 9 | ||||||||||
| 75 | 54 | 22 | 18 | 14 |
Notify that the first level of the second grid consists of the numbers from the table. In the second and third level the columns are shifted (or swapped). The 4th up to the 6th level is the same as the 1st up to the 3rd level, but the 4 magic 3x3 squares are swapped and the numbers are replaced by the inverse numbers.
+ 9x (digit -/- 1) from second grid
| 75 | 75 | 75 | 75 | 75 | 75 | ||
| 1 | |||||||
| 75 | 1 | 5 | 15 | 23 | 19 | 12 | |
| 75 | 5 | 15 | 1 | 19 | 12 | 23 | |
| 75 | 15 | 1 | 5 | 12 | 23 | 19 | |
| 75 | 22 | 18 | 14 | 4 | 8 | 9 | |
| 75 | 18 | 14 | 22 | 8 | 9 | 4 | |
| 75 | 14 | 22 | 18 | 9 | 4 | 8 | |
| 75 | 75 | 75 | 75 | 75 | 75 | ||
| 2 | |||||||
| 75 | 5 | 15 | 1 | 19 | 12 | 23 | |
| 75 | 15 | 1 | 5 | 12 | 23 | 19 | |
| 75 | 1 | 5 | 15 | 23 | 19 | 12 | |
| 75 | 18 | 14 | 22 | 8 | 9 | 4 | |
| 75 | 14 | 22 | 18 | 9 | 4 | 8 | |
| 75 | 22 | 18 | 14 | 4 | 8 | 9 | |
| 75 | 75 | 75 | 75 | 75 | 75 | ||
| 3 | |||||||
| 75 | 15 | 1 | 5 | 12 | 23 | 19 | |
| 75 | 1 | 5 | 15 | 23 | 19 | 12 | |
| 75 | 5 | 15 | 1 | 19 | 12 | 23 | |
| 75 | 14 | 22 | 18 | 9 | 4 | 8 | |
| 75 | 22 | 18 | 14 | 4 | 8 | 9 | |
| 75 | 18 | 14 | 22 | 8 | 9 | 4 | |
| 75 | 75 | 75 | 75 | 75 | 75 | ||
| 4=1' | |||||||
| 75 | 21 | 17 | 16 | 3 | 7 | 11 | |
| 75 | 17 | 16 | 21 | 7 | 11 | 3 | |
| 75 | 16 | 21 | 17 | 11 | 3 | 7 | |
| 75 | 2 | 6 | 13 | 24 | 20 | 10 | |
| 75 | 6 | 13 | 2 | 20 | 10 | 24 | |
| 75 | 13 | 2 | 6 | 10 | 24 | 20 | |
| 75 | 75 | 75 | 75 | 75 | 75 | ||
| 5=2' | |||||||
| 75 | 17 | 16 | 21 | 7 | 11 | 3 | |
| 75 | 16 | 21 | 17 | 11 | 3 | 7 | |
| 75 | 21 | 17 | 16 | 3 | 7 | 11 | |
| 75 | 6 | 13 | 2 | 20 | 10 | 24 | |
| 75 | 13 | 2 | 6 | 10 | 24 | 20 | |
| 75 | 2 | 6 | 13 | 24 | 20 | 10 | |
| 75 | 75 | 75 | 75 | 75 | 75 | ||
| 6=3' | |||||||
| 75 | 16 | 21 | 17 | 11 | 3 | 7 | |
| 75 | 21 | 17 | 16 | 3 | 7 | 11 | |
| 75 | 17 | 16 | 21 | 7 | 11 | 3 | |
| 75 | 13 | 2 | 6 | 10 | 24 | 20 | |
| 75 | 2 | 6 | 13 | 24 | 20 | 10 | |
| 75 | 6 | 13 | 2 | 20 | 10 | 24 |
= Pantriagonal 6x6x6 magic cube
| 1st level | |||||
| 1 | 44 | 132 | 207 | 164 | 103 |
| 41 | 129 | 7 | 167 | 106 | 201 |
| 135 | 4 | 38 | 100 | 204 | 170 |
| 198 | 155 | 121 | 28 | 71 | 78 |
| 158 | 124 | 192 | 68 | 75 | 34 |
| 118 | 195 | 161 | 81 | 31 | 65 |
| 2nd level | |||||
| 42 | 127 | 8 | 166 | 108 | 200 |
| 133 | 5 | 39 | 102 | 203 | 169 |
| 2 | 45 | 130 | 206 | 163 | 105 |
| 157 | 126 | 191 | 69 | 73 | 35 |
| 120 | 194 | 160 | 79 | 32 | 66 |
| 197 | 154 | 123 | 29 | 72 | 76 |
| 3th level | |||||
| 134 | 6 | 37 | 101 | 202 | 171 |
| 3 | 43 | 131 | 205 | 165 | 104 |
| 40 | 128 | 9 | 168 | 107 | 199 |
| 119 | 193 | 162 | 80 | 33 | 64 |
| 196 | 156 | 122 | 30 | 70 | 77 |
| 159 | 125 | 190 | 67 | 74 | 36 |
| 4th level | |||||
| 189 | 146 | 139 | 19 | 62 | 96 |
| 149 | 142 | 183 | 59 | 93 | 25 |
| 136 | 186 | 152 | 99 | 22 | 56 |
| 10 | 53 | 114 | 216 | 173 | 85 |
| 50 | 111 | 16 | 176 | 88 | 210 |
| 117 | 13 | 47 | 82 | 213 | 179 |
| 5th level | |||||
| 148 | 144 | 182 | 60 | 91 | 26 |
| 138 | 185 | 151 | 97 | 23 | 57 |
| 188 | 145 | 141 | 20 | 63 | 94 |
| 51 | 109 | 17 | 175 | 90 | 209 |
| 115 | 14 | 48 | 84 | 212 | 178 |
| 11 | 54 | 112 | 215 | 172 | 87 |
| 6th level | |||||
| 137 | 184 | 153 | 98 | 24 | 55 |
| 187 | 147 | 140 | 21 | 61 | 95 |
| 150 | 143 | 181 | 58 | 92 | 27 |
| 116 | 15 | 46 | 83 | 211 | 180 |
| 12 | 52 | 113 | 214 | 174 | 86 |
| 49 | 110 | 18 | 177 | 89 | 208 |
See in the download below that each row/column in each level and the pillars through the levels and all 144 pantriagonals (including the 4 main triagonals) give the same magic sum.
To prove that the analysis is correct, we construct another pantriagonal 6x6x6 magic cube.
We use another 3x3 magic square to build up the first grid.
Take 1x digit from 1st grid
| 30 | 30 | 30 | 30 | 30 | 30 | ||
| 30 | 1 | 6 | 8 | 9 | 4 | 2 | |
| 30 | 5 | 7 | 3 | 5 | 3 | 7 | |
| 30 | 9 | 2 | 4 | 1 | 8 | 6 | |
| 30 | 9 | 4 | 2 | 1 | 6 | 8 | |
| 30 | 5 | 3 | 7 | 5 | 7 | 3 | |
| 30 | 1 | 8 | 6 | 9 | 2 | 4 | |
| 30 | 30 | 30 | 30 | 30 | 30 | ||
| 30 | 8 | 1 | 6 | 2 | 9 | 4 | |
| 30 | 3 | 5 | 7 | 7 | 5 | 3 | |
| 30 | 4 | 9 | 2 | 6 | 1 | 8 | |
| 30 | 2 | 9 | 4 | 8 | 1 | 6 | |
| 30 | 7 | 5 | 3 | 3 | 5 | 7 | |
| 30 | 6 | 1 | 8 | 4 | 9 | 2 | |
| 30 | 30 | 30 | 30 | 30 | 30 | ||
| 30 | 6 | 8 | 1 | 4 | 2 | 9 | |
| 30 | 7 | 3 | 5 | 3 | 7 | 5 | |
| 30 | 2 | 4 | 9 | 8 | 6 | 1 | |
| 30 | 4 | 2 | 9 | 6 | 8 | 1 | |
| 30 | 3 | 7 | 5 | 7 | 3 | 5 | |
| 30 | 8 | 6 | 1 | 2 | 4 | 9 | |
| 30 | 30 | 30 | 30 | 30 | 30 | ||
| 30 | 9 | 4 | 2 | 1 | 6 | 8 | |
| 30 | 5 | 3 | 7 | 5 | 7 | 3 | |
| 30 | 1 | 8 | 6 | 9 | 2 | 4 | |
| 30 | 1 | 6 | 8 | 9 | 4 | 2 | |
| 30 | 5 | 7 | 3 | 5 | 3 | 7 | |
| 30 | 9 | 2 | 4 | 1 | 8 | 6 | |
| 30 | 30 | 30 | 30 | 30 | 30 | ||
| 30 | 2 | 9 | 4 | 8 | 1 | 6 | |
| 30 | 7 | 5 | 3 | 3 | 5 | 7 | |
| 30 | 6 | 1 | 8 | 4 | 9 | 2 | |
| 30 | 8 | 1 | 6 | 2 | 9 | 4 | |
| 30 | 3 | 5 | 7 | 7 | 5 | 3 | |
| 30 | 4 | 9 | 2 | 6 | 1 | 8 | |
| 30 | 30 | 30 | 30 | 30 | 30 | ||
| 30 | 4 | 2 | 9 | 6 | 8 | 1 | |
| 30 | 3 | 7 | 5 | 7 | 3 | 5 | |
| 30 | 8 | 6 | 1 | 2 | 4 | 9 | |
| 30 | 6 | 8 | 1 | 4 | 2 | 9 | |
| 30 | 7 | 3 | 5 | 3 | 7 | 5 | |
| 30 | 2 | 4 | 9 | 8 | 6 | 1 |
We create another adequate table with the numbers 1 up to 24 and build up the second grid.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | ||
| 24 | 23 | 22 | 21 | 20 | 19 | 18 | 17 | 16 | 15 | 14 | 13 | ||
| 25 | 1 | 11 | 13 | ||||||||||
| 75 | 50 | 22 | 20 | 8 | |||||||||
| 25 | 2 | 7 | 16 | ||||||||||
| 75 | 50 | 21 | 19 | 10 |
+ 9x number from second grid
| 75 | 75 | 75 | 75 | 75 | 75 | ||
| 75 | 1 | 11 | 13 | 22 | 20 | 8 | |
| 75 | 11 | 13 | 1 | 20 | 8 | 22 | |
| 75 | 13 | 1 | 11 | 8 | 22 | 20 | |
| 75 | 21 | 19 | 10 | 2 | 7 | 16 | |
| 75 | 19 | 10 | 21 | 7 | 16 | 2 | |
| 75 | 10 | 21 | 19 | 16 | 2 | 7 | |
| 75 | 75 | 75 | 75 | 75 | 75 | ||
| 75 | 11 | 13 | 1 | 20 | 8 | 22 | |
| 75 | 13 | 1 | 11 | 8 | 22 | 20 | |
| 75 | 1 | 11 | 13 | 22 | 20 | 8 | |
| 75 | 19 | 10 | 21 | 7 | 16 | 2 | |
| 75 | 10 | 21 | 19 | 16 | 2 | 7 | |
| 75 | 21 | 19 | 10 | 2 | 7 | 16 | |
| 75 | 75 | 75 | 75 | 75 | 75 | ||
| 75 | 13 | 1 | 11 | 8 | 22 | 20 | |
| 75 | 1 | 11 | 13 | 22 | 20 | 8 | |
| 75 | 11 | 13 | 1 | 20 | 8 | 22 | |
| 75 | 10 | 21 | 19 | 16 | 2 | 7 | |
| 75 | 21 | 19 | 10 | 2 | 7 | 16 | |
| 75 | 19 | 10 | 21 | 7 | 16 | 2 | |
| 75 | 75 | 75 | 75 | 75 | 75 | ||
| 75 | 23 | 18 | 9 | 4 | 6 | 15 | |
| 75 | 18 | 9 | 23 | 6 | 15 | 4 | |
| 75 | 9 | 23 | 18 | 15 | 4 | 6 | |
| 75 | 3 | 5 | 17 | 24 | 14 | 12 | |
| 75 | 5 | 17 | 3 | 14 | 12 | 24 | |
| 75 | 17 | 3 | 5 | 12 | 24 | 14 | |
| 75 | 75 | 75 | 75 | 75 | 75 | ||
| 75 | 18 | 9 | 23 | 6 | 15 | 4 | |
| 75 | 9 | 23 | 18 | 15 | 4 | 6 | |
| 75 | 23 | 18 | 9 | 4 | 6 | 15 | |
| 75 | 5 | 17 | 3 | 14 | 12 | 24 | |
| 75 | 17 | 3 | 5 | 12 | 24 | 14 | |
| 75 | 3 | 5 | 17 | 24 | 14 | 12 | |
| 75 | 75 | 75 | 75 | 75 | 75 | ||
| 75 | 9 | 23 | 18 | 15 | 4 | 6 | |
| 75 | 23 | 18 | 9 | 4 | 6 | 15 | |
| 75 | 18 | 9 | 23 | 6 | 15 | 4 | |
| 75 | 17 | 3 | 5 | 12 | 24 | 14 | |
| 75 | 3 | 5 | 17 | 24 | 14 | 12 | |
| 75 | 5 | 17 | 3 | 14 | 12 | 24 |
= pantriagonal 6x6x6 magic cube
| 1st level | |||||
| 1 | 96 | 116 | 198 | 175 | 65 |
| 95 | 115 | 3 | 176 | 66 | 196 |
| 117 | 2 | 94 | 64 | 197 | 177 |
| 189 | 166 | 83 | 10 | 60 | 143 |
| 167 | 84 | 187 | 59 | 142 | 12 |
| 82 | 188 | 168 | 144 | 11 | 58 |
| 2nd level | |||||
| 98 | 109 | 6 | 173 | 72 | 193 |
| 111 | 5 | 97 | 70 | 194 | 174 |
| 4 | 99 | 110 | 195 | 172 | 71 |
| 164 | 90 | 184 | 62 | 136 | 15 |
| 88 | 185 | 165 | 138 | 14 | 61 |
| 186 | 163 | 89 | 13 | 63 | 137 |
| 3th level | |||||
| 114 | 8 | 91 | 67 | 191 | 180 |
| 7 | 93 | 113 | 192 | 178 | 68 |
| 92 | 112 | 9 | 179 | 69 | 190 |
| 85 | 182 | 171 | 141 | 17 | 55 |
| 183 | 169 | 86 | 16 | 57 | 140 |
| 170 | 87 | 181 | 56 | 139 | 18 |
| 4th level | |||||
| 207 | 157 | 74 | 28 | 51 | 134 |
| 158 | 75 | 205 | 50 | 133 | 30 |
| 73 | 206 | 159 | 135 | 29 | 49 |
| 19 | 42 | 152 | 216 | 121 | 101 |
| 41 | 151 | 21 | 122 | 102 | 214 |
| 153 | 20 | 40 | 100 | 215 | 123 |
| 5th level | |||||
| 155 | 81 | 202 | 53 | 127 | 33 |
| 79 | 203 | 156 | 129 | 32 | 52 |
| 204 | 154 | 80 | 31 | 54 | 128 |
| 44 | 145 | 24 | 119 | 108 | 211 |
| 147 | 23 | 43 | 106 | 212 | 120 |
| 22 | 45 | 146 | 213 | 118 | 107 |
| 6th level | |||||
| 76 | 200 | 162 | 132 | 35 | 46 |
| 201 | 160 | 77 | 34 | 48 | 131 |
| 161 | 78 | 199 | 47 | 130 | 36 |
| 150 | 26 | 37 | 103 | 209 | 126 |
| 25 | 39 | 149 | 210 | 124 | 104 |
| 38 | 148 | 27 | 125 | 105 | 208 |
N.B.: You can use this method to contruct three dimensional magic cubes of double odd order.
With method of Luyendijk you can construct a pantriagonal magic cube of double odd order. See on this website the construction of:
6x6x6, 10x10x10, 14x14x14, 18x18x18, 22x22x22, 26x26x26 and 30x30x30