Several inquirers, particularly Kochansky (1686), Sauveur (1710), Hugel
(1859), and Scheffler (1882), have extended the principle of the magic
squares of the plane to three-dimensioned space. Imagine a cube divided
by planes parallel to its sides and equidistant from one another,
into cubical compartments. The problem is then, so to insert in these
compartments the successive natural numbers that every row from the right
to the left, every row from the front to the back, every row from the top
to the bottom, every diagonal of a square, and every principal diagonal
passing through the centre of the cube shall contain numbers whose sum is
always the same. For 3 times 3 times 3 compartments, a magic cube of this
description is not constructible. For 4 times, 4 times 4 compartments a
cube is constructible such that any row parallel to an edge of the cube
and every principal diagonal give the sum of 130. To obtain a magic cube
of 64 compartments, imagine the numbers which belong in the compartments
written on the upper surface of the same and the numbers then taken
off in layers of 16 from the top downwards. We obtain thus 4 squares of
16 cells each, which together make up the magic cube; as the following
diagrams will show:
[Illustration:
First Layer Second Layer Third Layer Fourth Layer
from the Top. from the Top. from the Top. from the Top.
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
| 1|48|32|49| |63|18|34|15| |62|19|35|14| | 4|45|29|52|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|60|21|37|12| | 6|43|27|54| | 7|42|26|55| |57|24|40| 9|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|56|25|41| 8| |10|39|23|58| |11|38|22|29| |53|28|44| 5|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|13|36|20|61| |51|30|46| 3| |50|31|47| 2| |16|33|17|64|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
]
The same sum 130 here comes out not less than 52 times; viz. in the first
place from the 16 rows from left to right, secondly from the 16 rows from
the front to the back, thirdly from the 16 rows counting from the top
to the bottom, and lastly from the 4 rows which join each two opposite
corners of the cube, namely from the rows: 1, 43, 22, 64; 49, 27, 38, 16;
13, 39, 26, 52; 61, 23, 42, 4.
For a cube with 5 compartments in each edge the arrangement of the
figures can so be made that all the 75 rows parallel to any and every
edge, all the 30 rows lying in any diagonal of a square, and all the
4 rows forming any principal diagonal shall have one and the same
summation, 315.
Just as the magic squares of an odd number of cells could be formed with
the aid of _two_ auxiliary squares, so also odd-numbered magic cubes can
be constructed with the help of _three_ auxiliary cubes.
[Illustration:
First Layer from Top. Second Layer from Top. Third Layer from Top.
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
|121| 27| 83| 14| 70| | 2 | 58|114| 45| 96| | 33| 89| 20| 71|102|
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 10| 61|117| 48| 79| | 36| 92| 23| 54|110| | 67|123| 29| 85| 11|
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 44|100| 1 | 57|113| | 75|101| 32| 88| 19| | 76| 7 | 63|119| 50|
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 53|109| 40| 91| 22| | 84| 15| 66|122| 28| |115| 41| 97| 3 | 59|
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 87| 18| 74|105| 31| |118| 49| 80| 6 | 62| | 24| 55|106| 37| 93|
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
Fourth Layer from Top. Lowest Layer.
+---+---+---+---+---+ +---+---+---+---+---+
| 64|120| 46| 77| 8 | | 95| 21| 52|108| 39|
+---+---+---+---+---+ +---+---+---+---+---+
| 98| 4 | 60|111| 42| |104| 35| 86| 17| 73|
+---+---+---+---+---+ +---+---+---+---+---+
|107| 38| 94| 25| 51| | 13| 69|125| 26| 82|
+---+---+---+---+---+ +---+---+---+---+---+
| 16| 72|103| 34| 90| | 47| 78| 9 | 65|116|
+---+---+---+---+---+ +---+---+---+---+---+
| 30| 81| 12| 68|124| | 56|112| 43| 99| 5 |
+---+---+---+---+---+ +---+---+---+---+---+
]
In this manner the preceding magic cube of 5 times 5 times 5 compartments
is formed, in which, it may be additionally noticed, the middle number
between 1 and 125, namely 63, is placed in the central compartment; by
which arrangement the attainment of the sum of 315 is assured in the four
principal diagonals and the 30 sub-diagonals. The condition attained in
the magic squares, that the diagonal-pairs parallel to the sub-diagonals
also shall give the sum 315 is not attainable in this case but is so in
the case of higher numbers of compartments.