Magic Cubes - Pan and Semi-pan

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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

 Pandiagonal and 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

Triagonal 1 + 57 + 64 + 8 = 130
2 segment broken triagonal
          55 + 15 + 10 + 50 = 130
3 segment broken triagonal
          51 + 47 + 14 + 18 = 130

Triagonal
80 + 9 + 63 + 117 + 46 = 315

2 segment broken triagonal
 24 + 53 + 107 + 36 + 95 = 315

3 segment broken triagonal
87 + 16 + 75 + 104 + 33 = 315

 

 

 Semi-pandiagonal and semi-pantriagonal

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.

[1] H. E. Dudeney, Amusements in Mathematics, Dover Publ. 1958, pp 119-121.

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.

Example: A Main triagonal
1 + 23 + 42 + 64 = 130
The 3 short triagonal pairs
62 + 41 + 24 + 3 = 130
57 + 52 + 13 + 8 = 130
59 + 56 + 9 + 6 = 130

Short triagonal pair :
53 + 49 + 77 + 73 + 63 = 315

Long triagonal pair:
44 + 15 + 106 + 20 + 111
 + 82
- 63 = 315

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 (now no longer availabe)

This page was originally posted March 2003
It was last updated March 04, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz