Office Applications and Entertainment, Magic Cubes

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3.7   Pantriagonal Magic Cubes (4 x 4 x 4)

3.7.1 Historical Background


Magic Cubes as discussed in Section 2.1 for which basically only the rows, columns, pillars and 4 space diagonals (main triagonals) sum to the Magic Constant are normally referred to as Simple Magic Cubes.

John Hendricks introduced the concept of Pantriagonal Magic Cubes, being Simple Magic Cubes for which all main and broken triagonals sum to the Magic Constant.

In 1972 he published a 4th order Pantriagonal Magic Cube, which is shown below together with the 12 orthogonal planes:

Magic Cube, John Hendricks (1972)

Plane 11 (Top)

1 32 49 48
56 41 8 25
13 20 61 36
60 37 12 21

Plane 21 (Left)

60 13 56 1
7 50 11 62
57 16 53 4
6 51 10 63

Plane 31 (Back)

1 32 49 48
62 35 14 19
4 29 52 45
63 34 15 18


Plane 12

62 35 14 19
11 22 59 38
50 47 2 31
7 26 55 42


Plane 22

37 20 41 32
26 47 22 35
40 17 44 29
27 46 23 34


Plane 32

56 41 8 25
11 22 59 38
53 44 5 28
10 23 58 39


Plane 13

4 29 52 45
53 44 5 28
16 17 64 33
57 40 9 24


Plane 23

12 61 8 49
55 2 59 14
9 64 5 52
54 3 58 15


Plane 33

13 20 61 36
50 47 2 31
16 17 64 33
51 46 3 30


Plane 14

63 34 15 18
10 23 58 39
51 46 3 30
6 27 54 43


Plane 24

21 36 25 48
42 31 38 19
24 33 28 45
43 30 39 18


Plane 34

60 37 12 21
7 26 55 42
57 40 9 24
6 27 54 43


The cube shown above has following additional properties:

  1. Compact : Every 2 x 2 sub square in each of the orthogonal planes sums to the Magic Constant.

  2. Complete: Every (pan)triagonal contains n/2 complementary pairs spaced n/2 apart along the (pan)triagonal.
              Comparable with Most Perfect Magic Squares as discussed in 'Magic Squares', Section 2.4).

As a consequence of the defining properties mentioned above, the cube has also following properties:

  1. Every 2 x 2 x 2 block of cells (including wrap-around) sums to n*(n3 + 1) = 260 (Cubic Compact).

  2. The corners of all Sub-Cubes of orders 3 sum to n*(n3 + 1) = 260.

A Pantriagonal Magic Cube can be transformed into another Pantriagonal Magic Cube by moving an orthogonal plane from one side of the cube to the other. (Comparable with the row and column movements for Pandiagonal Magic Squares as discussed in 'Magic Squares' Section 3.4).

Consequently a Pantriagonal Magic Cube belongs to a collection {Aijkm} of 43 * 48 = 3072 elements which can be found by means of rotation, reflection or plane movements.

The Class of 48 elements which can be obtained by rotation / reflection of a Pantriagonal Cube is shown in Attachment 3.7.1.

The Class of 64 elements which can be obtained by planar shifts of a Pantriagonal Cube is shown in Attachment 3.7.2. Each cube of Attachment 3.7.1 can be used as a Base for Attachment 3.7.2.

It should be noted that the planar shifts are from left to right (L1, L2, L3), from back to front (B1, B2, B3) and from bottom to top (T1, T2, T3).

3.7.2 Analytic Solution

In general Magic Cubes of order 4 can be represented as follows:

Magic Cube (4 x 4 x 4)

Plane 11 (Top)

a49 a50 a51 a52
a53 a54 a55 a56
a57 a58 a59 a60
a61 a62 a63 a64

Plane 21 (Left)

a61 a57 a53 a49
a45 a41 a37 a33
a29 a25 a21 a17
a13 a9 a5 a1

Plane 31 (Back)

a49 a50 a51 a52
a33 a34 a35 a36
a17 a18 a19 a20
a1 a2 a3 a4


Plane 12

a33 a34 a35 a36
a37 a38 a39 a40
a41 a42 a43 a44
a45 a46 a47 a48


Plane 22

a62 a58 a54 a50
a46 a42 a38 a34
a30 a26 a22 a18
a14 a10 a6 a2


Plane 32

a53 a54 a55 a56
a37 a38 a39 a40
a21 a22 a23 a24
a5 a6 a7 a8


Plane 13

a17 a18 a19 a20
a21 a22 a23 a24
a25 a26 a27 a28
a29 a30 a31 a32


Plane 23

a63 a59 a55 a51
a47 a43 a39 a35
a31 a27 a23 a19
a15 a11 a7 a3


Plane 33

a57 a58 a59 a60
a41 a42 a43 a44
a25 a26 a27 a28
a9 a10 a11 a12


Plane 14

a1 a2 a3 a4
a5 a6 a7 a8
a9 a10 a11 a12
a13 a14 a15 a16


Plane 24

a64 a60 a56 a52
a48 a44 a40 a36
a32 a28 a24 a20
a16 a12 a8 a4


Plane 34

a61 a62 a63 a64
a45 a46 a47 a48
a29 a30 a31 a32
a13 a14 a15 a16


As all 2 x 2 squares in each of the orthogonal planes sum to 130 and all complementary pairs of all (pan)triagonals sum to 65, this results in linear equations like:

Plane 11

a49 + a50 + a53 + a54 = 130
a53 + a54 + a57 + a58 = 130
a57 + a58 + a61 + a62 = 130
a61 + a62 + a49 + a50 = 130

a50 + a51 + a54 + a55 = 130
a54 + a55 + a58 + a59 = 130
a58 + a59 + a62 + a63 = 130
a62 + a63 + a50 + a51 = 130

a51 + a52 + a55 + a56 = 130
a55 + a56 + a59 + a60 = 130
a59 + a60 + a63 + a64 = 130
a63 + a64 + a51 + a52 = 130

a52 + a49 + a56 + a53 = 130
a56 + a53 + a60 + a57 = 130
a60 + a57 + a64 + a61 = 130
a64 + a61 + a52 + a49 = 130


Pantriagonal pairs from bottom/left/front to top/right/back

a1  + a43 = 65
a5  + a47 = 65
a9  + a35 = 65
a13 + a39 = 65

a30 + a56 = 65
a18 + a60 = 65
a22 + a64 = 65
a26 + a52 = 65

a2  + a44 = 65
a6  + a48 = 65
a10 + a36 = 65
a14 + a40 = 65

a31 + a53 = 65
a19 + a57 = 65
a23 + a61 = 65
a27 + a49 = 65

a3  + a41 = 65
a7  + a45 = 65
a11 + a33 = 65
a15 + a37 = 65

a32 + a54 = 65
a20 + a58 = 65
a24 + a62 = 65
a28 + a50 = 65

a4  + a42 = 65
a8  + a46 = 65
a12 + a34 = 65
a16 + a38 = 65

a29 + a55 = 65
a17 + a59 = 65
a21 + a63 = 65
a25 + a51 = 65

The equations for the other orthogonal planes and pantriagonals are comparable.

The complete set of equations results, after deduction, in following set of linear equations describing the "Pantriagonal Magic Cubes" as published by John Hendricks:

a(61) =  130 - a(62) - a(63) - a(64)
a(59) =  130 - a(60) - a(63) - a(64)
a(58) =        a(60) - a(62) + a(64)
a(57) =      - a(60) + a(62) + a(63)
a(55) =      - a(56) + a(63) + a(64)
a(54) =        a(56) + a(62) - a(64)
a(53) =  130 - a(56) - a(62) - a(63)
a(52) =  130 - a(56) - a(60) - a(64)
a(51) =        a(56) + a(60) - a(63)
a(50) =  130 - a(56) - a(60) - a(62)
a(49) = -130 + a(56) + a(60) + a(62) + a(63) + a(64)
a(47) =  130 - a(48) - a(63) - a(64)
a(46) =        a(48) - a(62) + a(64)
a(45) =      - a(48) + a(62) + a(63)
a(44) =  130 - a(48) - a(60) - a(64)
a(43) = -130 + a(48) + a(60) + a(63) + 2 * a(64)
a(42) =  130 - a(48) - a(60) + a(62) - 2 * a(64)
a(41) =        a(48) + a(60) - a(62) - a(63) + a(64)
a(40) =        a(48) - a(56) + a(64)
a(39) =  130 - a(48) + a(56) - a(63) - 2 * a(64)
a(38) =        a(48) - a(56) - a(62) + 2 * a(64)
a(37) =      - a(48) + a(56) + a(62) + a(63) - a(64)
a(36) =      - a(48) + a(56) + a(60)
a(35) =        a(48) - a(56) - a(60) + a(63) + a(64)
a(34) =      - a(48) + a(56) + a(60) + a(62) - a(64)
a(33) =  130 + a(48) - a(56) - a(60) - a(62) - a(63)
a(32) =   65 - a(56) - a(62) + a(64)
a(31) = - 65 + a(56) + a(62) + a(63)
a(30) =   65 - a(56)
a(29) =   65 + a(56) - a(63) - a(64)
a(28) = - 65 + a(56) + a(60) + a(62)
a(27) =  195 - a(56) - a(60) - a(62) - a(63) - a(64)
a(26) = - 65 + a(56) + a(60) + a(64)
a(25) =   65 - a(56) - a(60) + a(63)
a(24) =   65 - a(62)
a(23) = - 65 + a(62) + a(63) + a(64)
a(22) =   65 - a(64)
a(21) =   65 - a(63)
a(20) =   65 - a(60) + a(62) - a(64)
a(19) =   65 + a(60) - a(62) - a(63)
a(18) =   65 - a(60)
a(17) = - 65 + a(60) + a(63) + a(64)
a(16) =   65 - a(48) + a(56) + a(62) - 2 * a(64)
a(15) =   65 + a(48) - a(56) - a(62) - a(63) + a(64)
a(14) =   65 - a(48) + a(56) - a(64)
a(13) = - 65 + a(48) - a(56) + a(63) + 2 * a(64)
a(12) =   65 + a(48) - a(56) - a(60) - a(62) + a(64)
a(11) = - 65 - a(48) + a(56) + a(60) + a(62) + a(63)
a(10) =   65 + a(48) - a(56) - a(60)
a( 9) =   65 - a(48) + a(56) + a(60) - a(63) - a(64)
a( 8) =   65 - a(48) + a(62) - a(64)
a( 7) =   65 + a(48) - a(62) - a(63)
a( 6) =   65 - a(48)
a( 5) = - 65 + a(48) + a(63) + a(64)
a( 4) = - 65 + a(48) + a(60) - a(62) + 2 * a(64)
a( 3) =   65 - a(48) - a(60) + a(62) + a(63) - a(64)
a( 2) = - 65 + a(48) + a(60) + a(64)
a( 1) =  195 - a(48) - a(60) - a(63) - 2 * a(64)

The linear equations shown above, are ready to be solved, for the Magic Constant 130.

The solutions can be obtained by guessing:

   a(48), a(56), a(60), a(62) ... a(64)

and filling out these guesses in the abovementioned equations.

For distinct integers also following relations are applicable:

0 < a(i) =< 64        for i = 1, 2 ... 47, 49 ... 55, 57 ... 59, 61
a(i) ≠ a(j)           for i ≠ j

which have been incorporated in an optimized guessing routine (MgcCube4d).

Subject guessing routine produced 64 * 6 * 120 = 46080 Pantriagonal Magic Cubes within 45 minutes, of which the first 120 are shown in Attachment 3.7.3.

An alternative method to generate Pantriagonal Magic Cubes, based on the equations deducted above, will be discussed in Section 3.12.4.


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