Office Applications and Entertainment, Magic Cubes

3.8   Pandiagonal/Triagonal Magic Cubes (4 x 4 x 4)

3.8.1 Introduction

The cube shown below is a Pan Diagonal/Triagonal Magic Cube of the 4^th order, meaning a Magic Cube for which all (Pan) Diagonals and (Pan) Triagonals sum to the Magic Constant.

It is the first occurring cube while applying the method described in Section 3.12.5 based on the equations deducted in Section 3.8.2 below.

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

1. Compact : Every 2 x 2 x 2 block of cells (including wrap-around) sums to n*(n³ + 1) = 260 (Cubic Compact).
             The corners of all Sub-Cubes of orders 3 sum to n*(n³ + 1) = 260.
   
   
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 2.6).

A Pandiagonal/Triagonal Magic Cube can be transformed into another Pandiagonal/Triagonal Magic Cube by moving an orthogonal plane from one side of the cube to the other.

Consequently the cube belongs to a collection {A_ijk^m} 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 Pandiagonal/Triagonal Cube is shown in Attachment 3.8.1.

The Class of 64 elements which can be obtained by planar shifts of a Pandiagonal/Triagonal Cube is shown in Attachment 3.8.2. Each cube of Attachment 3.8.1 can be used as a Base for Attachment 3.8.2.

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).

3.8.2 Analytic Solution

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

The equations for a Pan Diagonal/Triagonal Magic Cube of the fourth order can be summarised as follows:

• All (Pan) Diagonals (96) sum to the Magic Sum (130).
  
  
• All (Pan) Triagonals (64) sum to the Magic Sum (130).

After deduction of the defining equations (160), the following set of linear equations - describing the Pan Diagonal/Triagonal Magic Cubes of the 4^th order - can be obtained:

a(54) =             a(56) + a(62) - a(64)
a(53) =             a(55) + a(61) - a(63)
a(52) =        s1 - a(55) - a(58) - a(61)
a(51) =        s1 - a(56) - a(57) - a(62)
a(50) =        s1 - a(55) - a(60) - a(61)
a(49) =        s1 - a(56) - a(59) - a(62)
a(46) =             a(48) - a(58) + a(60)
a(45) =             a(47) - a(57) + a(59)
a(44) =        s1 - a(47) - a(59) - a(64)
a(43) =        s1 - a(48) - a(60) - a(63)
a(42) =        s1 - a(47) - a(59) - a(62)
a(41) =        s1 - a(48) - a(60) - a(61)
a(40) =             a(48) - a(55) + a(63)
a(39) =             a(47) - a(56) + a(64)
a(38) =             a(48) - a(55) - a(58) + a(60) + a(63)
a(37) =             a(47) - a(56) - a(57) + a(59) + a(64)
a(36) =           - a(47) + a(56) + a(57) + a(62) - a(64)
a(35) =           - a(48) + a(55) + a(58) + a(61) - a(63)
a(34) =           - a(47) + a(56) + a(57)
a(33) =           - a(48) + a(55) + a(58)
a(32) =  0.5 * s1 - a(56) - a(62) + a(64)
a(31) =  0.5 * s1 - a(55) - a(61) + a(63)
a(30) =  0.5 * s1 - a(56)
a(29) =  0.5 * s1 - a(55)
a(28) = -0.5 * s1 + a(55) + a(60) + a(61)
a(27) = -0.5 * s1 + a(56) + a(59) + a(62)
a(26) = -0.5 * s1 + a(55) + a(58) + a(61)
a(25) = -0.5 * s1 + a(56) + a(57) + a(62)
a(24) =  0.5 * s1 - a(62)
a(23) =  0.5 * s1 - a(61)
a(22) =  0.5 * s1 - a(64)
a(21) =  0.5 * s1 - a(63)
a(20) =  0.5 * s1 - a(58)
a(19) =  0.5 * s1 - a(57)
a(18) =  0.5 * s1 - a(60)
a(17) =  0.5 * s1 - a(59)
a(16) =  0.5 * s1 - a(48) + a(55) + a(58) - a(60) - a(63)
a(15) =  0.5 * s1 - a(47) + a(56) + a(57) - a(59) - a(64)
a(14) =  0.5 * s1 - a(48) + a(55) - a(63)
a(13) =  0.5 * s1 - a(47) + a(56) - a(64)
a(12) =  0.5 * s1 + a(47) - a(56) - a(57)
a(11) =  0.5 * s1 + a(48) - a(55) - a(58)
a(10) =  0.5 * s1 + a(47) - a(56) - a(57) - a(62) + a(64)
a( 9) =  0.5 * s1 + a(48) - a(55) - a(58) - a(61) + a(63)
a( 8) =  0.5 * s1 - a(48) + a(58) - a(60)
a( 7) =  0.5 * s1 - a(47) + a(57) - a(59)
a( 6) =  0.5 * s1 - a(48)
a( 5) =  0.5 * s1 - a(47)
a( 4) = -0.5 * s1 + a(47) + a(59) + a(62)
a( 3) = -0.5 * s1 + a(48) + a(60) + a(61)
a( 2) = -0.5 * s1 + a(47) + a(59) + a(64)
a( 1) = -0.5 * s1 + a(48) + a(60) + a(63)

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

The solutions can be obtained by guessing:

   a(47), a(48), a(55) ... 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 ... 46, 49 ... 54
a(i) ≠ a(j)           for i ≠ j

which can be incorporated in a guessing routine, which might be used to find other 4^th order Pan Diagonal/Triagonal Magic Cubes.

However, the equations deducted above can be applied in a more efficient method to generate Pan Diagonal/Triagonal Magic Cubes, which will be discussed in Section 3.12.5.