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This page compares features of pandiagonal magic squares with pantriagonal
magic cubes.
Also, features of semi-pandiagonal magic squares with semi-pantriagonal magic
cubes.
Pandiagonal and pantriagonal |
Semi-pandiagonal and semi-pantriagonal |
The pandiagonals of a magic square are those diagonal
segments that are parallel to the main diagonals. Two segments that together
contain m cells are called a broken diagonal pair. For a magic square to
be considered pandiagonal, all 2m pairs must sum correctly.
A diagonal (or pandiagonal) is a line through space where two coordinates change
while the rest remain constant. This is obvious in a magic square. In a magic
cube, it means that the diagonal is confined to a particular plane.
The equivalent feature in three dimensions is the triagonal and pantriagonal. Because 3 coordinates change as the line moves through space, it is sometimes called a pan-3-agonal.
|
Equivalent similarities |
A simple magic square |
A simple magic cube |
|
Diagonal (2 main) |
Required |
Not required. If present, results in magic squares within the cube. If all diagonals are correct, there will be 3m planar simple magic squares and the cube is called diagonal magic. Order 5 is the smallest such cube possible. |
|
Pandiagonal |
Not required. If true, rows or columns may be moved from one edge of the square to the other without destroying magic. |
Not required. If present, results in pan-magic squares within the cube. If all pandiagonals are correct, there will be 3m planar pandiagonal magic squares and the cube is called pandiagonal magic. Order 7 is the smallest such cube possible. |
|
Triagonal (4 main) |
Not possible |
Required |
|
Pantriagonal |
Not possible |
Not required. If present, planes may be moved from one edge of the cube to the other without destroying magic. i.e. a pantriagonal magic cube is equivalent to a pandiagonal magic square! |
|
Number of segments |
Each pandiagonal has 1 or 2 segments |
A pantriagonal may have 1, 2 or 3 segments. |
|
Smallest order possible |
Order 4 |
Order 4 |
Examples:
|
Two Pantriagonal magic cubes
Triagonal 1 + 57 + 64 + 8 = 130 |
Triagonal 80 + 9 + 63 + 117 + 46 = 315 |
Semi-Pandiagonal magic square
Until I started writing this page, I hadn’t realized how
little work had been done with semi-pandiagonal magic squares. The term has been
around since at least 1910, when H. E. Dudeney published an article on order 4
classifications in the Queen. He later repeated these classifications and terms
in 1917 [1]. The term has almost always been mentioned in regards to orders 4
and 5, but not the higher orders.
I have seen no mention that all associated magic squares of double even order and odd orders >3
are semi-pandiagonal magic.
NOTE: All associated magic squares are semi-pandiagonal magic, but there are many semi-pandiagonal magic squares that are not associated. Presumably the same applies to magic cubes.
Semi-pandiagonal magic squares have the property that the sum of the cells in the opposite short diagonals are equal to the magic constant (in an even order hypercube). Opposite short diagonals are two diagonals parallel and on opposite sides of a main diagonal. Each short diagonal contains m/2 cells if the square or cube is even. If the hypercube is odd, the opposite short diagonals each contain (m-1)/2 or (m+1)/2.
In an even order square, the two opposite short diagonals will sum to the square's constant.
In an odd order square, these two opposite short diagonals, which together contain m-1 cells, will, when added to the center cell equal the square’s constant. The two opposite short diagonals, which together contain m+1 cells, will sum to the constant if the center cell is subtracted from their total.
The preceding three paragraphs may be adapted for magic cubes by substituting the word ‘triagonal’ for the word ‘diagonal’. Also, instead of 2 short diagonal pairs for each of the two main diagonals, there are 3 pairs for each of four main triagonals.
Of the 880 fundamental magic squares of order 4, 384 are semi-pan( 48 of these are also associative and 48 are bent diagonal).
I can find no references that mention that there are two
types of semi-pandiagonal squares. At least for double-even orders. One type has equal
short diagonals. The two halves of each main diagonal are also even so sum to S/2, thus producing bent-diagonal magic squares.
The other type of semi-pandiagonal squares have unequal short diagonals (which
together sun to S). These may be associated or simple magic.
In the case of order 4 squares, Group II is of the first type, Groups III to VI
are the second type (but Group VI also has simple magic squares).
[1] H. E. Dudeney, Amusements in Mathematics, Dover Publ. 1958, pp 119-121.
The following magic squares are all semi-pandiagonal. Red text shows an opposite short diagonal pair. In the case of the odd ordered 5 and 7 squares, I also show an opposite long diagonal pair (green). Actually, these pairs in odd order cubes are not pandiagonals because they do not have the correct number of cells per line. For over 100 years though, these squares have also been referred to as semi-pandiagonal!
Order 4 Order 4 03 16 09 22 15 04 11 05 14 01 14 08 11 20 08 21 14 02 06 13 03 12 15 04 10 05 07 25 13 01 19 09 02 16 07 12 07 13 02 24 12 05 18 06 15 08 10 01 06 09 03 16 11 04 17 10 23 bent diagonal associated Order 5, associated
58 12 51 01 47 29 38 24 04 29 12 37 20 45 28 06 56 15 61 19 33 26 44 35 11 36 19 44 27 03 36 18 41 27 53 07 64 14 10 42 18 43 26 02 34 32 46 21 39 09 59 04 50 41 17 49 25 01 33 09 11 57 02 52 30 48 23 37 16 48 24 07 32 08 40 55 05 62 16 34 20 43 25 47 23 06 31 14 39 15 17 35 28 42 08 54 13 63 22 05 30 13 38 21 46 45 31 40 22 60 10 49 03 Order 7, associated Order 8, not associated
Opposite short diagonals, Bent diagonals, Opposite long diagonals. Remember, for odd orders, the center cell is added or subtracted from the sum of the diagonal pair to obtain the magic constant.
Semi-Pantriagonal magic cube
I will now illustrate the magic cube equivalent of the
semi-pandiagonal magic square. Simply replace references to semi-pandiagonal in
the above definition with semi-pantriagonal . Also, for even order cubes, instead of
1 pair
parallel to each of the two main diagonals, there are 3 pairs parallel to each
of the four main triagonals.
For odd order cubes, instead of 1 short pair and 1 long pair
parallel to each of the two main diagonals, there are 3 pairs of each type parallel to each
of the four main triagonals. Note that for an odd-order cube, these short (and
long) diagonal pairs are not pandiagonals because they do not have the required
m cells. However, by tradition we will still refer to them as semi-pantriagonal
cubes
I will show just two examples of semi-pantriagonal magic cubes.
 One of the 4 main triagonals and the 2 parallel short triagonal pairs.
Example:
Main triagonal |
Short triagonal pair = 53 + 49 + 77 + 73 + 63 = 315 |
Addendum: On Jan. 16,2004, Guenter Stertinbrink advised me of several critical errors in this section. These have now been corrected.
This order 6 semi-pantriagonal, not associated magic cube was constructed using a method proposed by Adrian Smith [1]. I show one short triagonal pair.
Horizontal plane I - Top II III 10 161 195 199 80 6 23 192 205 185 3 43 144 67 74 63 148 155 159 190 125 105 55 17 196 14 102 142 149 48 65 54 139 11 216 166 197 42 19 8 177 208 147 25 86 12 214 167 76 137 153 157 56 72 172 107 33 37 134 168 104 138 124 158 57 70 90 121 128 117 94 101 78 136 98 132 1 206 7 176 129 61 95 183 146 81 58 200 135 31 35 15 181 170 204 46 174 106 5 93 133 140 130 191 99 103 2 126 IV V Horizontal plane VI - Bottom 91 26 114 118 215 87 212 30 16 50 165 178 171 175 47 36 40 182 24 109 71 186 82 179 34 122 210 88 41 156 173 162 4 119 108 85 116 96 100 89 123 127 66 187 113 201 52 32 49 164 180 184 29 45 145 188 60 64 53 141 131 84 97 77 111 151 9 13 209 198 202 20 213 163 152 51 28 44 169 68 21 115 203 75 38 27 193 92 189 112 62 69 154 143 150 73 39 160 194 120 79 59 211 110 18 22 83 207
Semi-pantriagonal cubes (and squares) exist for all orders greater then 3.
For even orders, they may be associated or not associated.
The order 6 cube shown above is semi-pantriagonal and is not associated.
The Weidemann order 6 cube shown on the Perfect-2 page is semi-pantriagonal and is associated.
Must all odd order semi-pantriagonal magic cubes (and squares) be associated?
[1] http://www.snaffles.demon.co.uk/mcubes.txt
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Harvey Heinz harveyheinz@shaw.ca
This page last updated
October 15, 2009
Copyright © 2003 by Harvey D. Heinz
