### Pantriagonal 6x6x6 magic cube

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

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.

6x6x6, pantriagonal, analysis.xlsx

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.

6x6x6, pantriagonal.xlsx