Office Applications and Entertainment, Magic Cubes

3.0    Magic Cubes (4 x 4 x 4)

       Introduction

In previous sections several Classes of order 4 Magic Cubes have been discussed with following main characteristics:

• Almost Perfect Magic Cubes, for which all rows, columns, pillars and the main diagonals of all orthogonal planes sum to the Magic Sum (ref. Section 3.2.1);
• Simple Magic Cubes, for which all rows, columns, pillars and space diagonals (main triagonals) sum to the Magic Sum (ref. Section 3.9);
• Pantriagonal Magic Cubes, being Simple Magic Cubes for which all main and broken triagonals sum to the Magic Sum (ref. Section 3.7 and 3.10).

Next sections will deal with some additional categories of Simple and Patriagonal Magic Cubes.

3.13   Simple Magic Cubes

3.13.1 Horizontal Associated Magic Planes

The defining properties result, after deduction of the corresponding equations, in following set of linear equations describing Simple Magic Cubes with Horizontal Associated Magic Planes.

a(61) = s1 - a(62) - a(63) - a(64)
a(59) = s1 - a(60) - a(63) - a(64)
a(58) = s1 - a(60) - a(62) - a(64)
a(57) = s1 - a(58) - a(59) - a(60)
a(45) = s1 - a(46) - a(47) - a(48)
a(43) = s1 - a(44) - a(47) - a(48)
a(42) = s1 - a(44) - a(46) - a(48)
a(41) = s1 - a(42) - a(43) - a(44)
a(31) = s1 - a(32) - a(46) - a(48) + a(61) - a(64)
a(29) = s1 - a(30) - a(31) - a(32)
a(28) =    - a(32) + a(44) - a(45) + a(62) + a(63)
a(27) =      a(32) - a(44) - a(47) + 2 * a(64)
a(26) = s1 - a(28) - a(30) - a(32)
a(25) = s1 - a(26) - a(27) - a(28)
a(16) = s1 - a(32) - a(48) - a(64)
a(15) = s1 - a(31) - a(47) - a(63)
a(14) = s1 - a(30) - a(46) - a(62)
a(13) =      a(30) + a(47) - a(64)
a(12) = s1 - a(28) - a(44) - a(60)
a(11) = s1 - a(27) - a(43) - a(59)
a(10) = s1 - a(26) - a(42) - a(58)
a(9)  = s1 - a(25) - a(41) - a(57)

with a(30), a(32), a(44), a(46) ... a(48), a(60), a(62) ... a(64) the independent variables.

The equations deducted above can be applied in an efficient method to generate subject cubes, which will be discussed in Section 3.16.1.

3.13.2 Horizontal Pan Magic Planes (3D-Compact)

The defining properties result, after deduction of the corresponding equations, in following set of linear equations describing Simple Magic Cubes with Horizontal Pan Magic Planes.

a(61) = s1 - a(62) - a(63) - a(64)
a(59) = s1 - a(60) - a(63) - a(64)
a(58) =      a(60) - a(62) + a(64)
a(57) =    - a(60) + a(62) + a(63)
a(46) = s1 - a(47) - a(62) - a(63)
a(45) =    - a(48) + a(62) + a(63)
a(44) = s1 - a(47) - a(60) - a(63)
a(43) =    - a(48) + a(60) + a(63)
a(42) =      a(48) - a(60) + a(62)
a(41) =      a(47) + a(60) - a(62)
a(29) = s1 - a(30) - a(31) - a(32)
a(28) = s1 - a(31) - 2 * a(32) + a(43) - a(48)
a(27) =      a(32) + 2 * a(48) - a(60) - a(63)
a(26) =      a(29) - 2 * a(48) + a(60) + a(63)
a(25) = s1 - a(26) - a(27) - a(28)
a(16) = s1 - a(32) - a(48) - a(64)
a(15) = s1 - a(31) - a(47) - a(63)
a(14) =    - a(30) + a(47) + a(63)
a(13) = s1 - a(14) - a(15) - a(16)
a(12) = s1 - a(28) - a(44) - a(60)
a(11) = s1 - a(27) - a(43) - a(59)
a(10) = s1 - a(26) - a(42) - a(58)
a(9)  = s1 - a(10) - a(11) - a(12)

with a(30) ... a(32), a(47), a(48), a(60) and a(62) ... a(64) the independent variables.

As the horizontal Pan Magic Planes are 2D-Compact, the resulting cubes are 3D-Compact as well.

Based on the above listed equations, a routine can be written to generate the defined Simple Magic Cubes of order 4 (ref. MgcCube42) .

Subject guessing routine counted 6668288 cubes within 32 hours, of which 126 are shown in Attachment 3.13.2.

An alternative method to generate Simple Magic Cubes, based on the equations deducted above, will be discussed in Section 3.16.2.

3.13.3 Associated with Horizontal Magic Planes

The defining properties result, after deduction of the corresponding equations, in following set of linear equations describing Associated Simple Magic Cubes with Horizontal Magic Planes.

a(61) = s1 - a(62) - a(63) - a(64)
a(57) = s1 - a(58) - a(59) - a(60)
a(55) =      a(56) - a(58) + a(60) - a(61) + a(64)
a(54) = s1 - a(55) - a(58) - a(59)
a(53) = s1 - a(54) - a(55) - a(56)
a(52) = s1 - a(56) - a(60) - a(64)
a(51) = s1 - a(55) - a(59) - a(63)
a(50) = s1 - a(54) - a(58) - a(62)
a(49) = s1 - a(50) - a(51) - a(52)
a(46) = s1 - a(47) - a(62) - a(63)
a(45) = s1 - a(46) - a(47) - a(48)
a(43) = s1 - a(48) - a(59) - a(64)
a(42) =      a(48) - a(58) + a(64)
a(41) = s1 - a(42) - a(43) - a(44)
a(40) = s1 - a(44) - a(56) - a(60)
a(39) =      a(48) - a(56) + a(58) - a(60) + a(61)
a(38) =    - a(39) + a(58) + a(59)
a(37) = s1 - a(38) - a(39) - a(40)
a(36) = s1 - a(39) - a(42) - a(45)
a(35) = s1 - a(39) - a(43) - a(47)
a(34) = s1 - a(38) - a(42) - a(46)
a(33) = s1 - a(34) - a(35) - a(36)

with a(44), a(47), a(48), a(56), a(58) ... a(60) and a(62) ... a(64) the independent variables.

The equations deducted above can be applied in an efficient method to generate subject cubes, which will be discussed in Section 3.16.3.

3.13.4 Associated and 3D-Compact

The defining properties result, after deduction of the corresponding equations, in following set of linear equations describing Associated 3D-Compact Simple Magic Cubes.

a(61) =   s1 - a(62) - a(63) - a(64)
a(58) =   s1 - a(59) - a(62) - a(63)
a(57) =      - a(60) + a(62) + a(63)
a(55) =   s1 - a(56) - a(59) - a(60)
a(53) =      - a(54) + a(59) + a(60)
a(52) =   s1 - a(56) - a(60) - a(64)
a(51) =      a(56) + a(60) - a(63)
a(50) =      - a(54) + a(59) + a(63)
a(49) =      a(54) - a(59) + a(64)
a(46) =   s1 - a(47) - a(62) - a(63)
a(45) =      - a(48) + a(62) + a(63)
a(43) = 2*s1 - a(44) - a(47) - a(48) - a(59) - a(60) - a(63) - a(64)
a(42) =      - a(43) + a(62) + a(63)
a(41) =   s1 - a(44) - a(62) - a(63)
a(40) =   s1 - a(44) - a(56) - a(60)
a(39) =      - a(43) + a(56) + a(60)
a(38) =      - a(39) - a(54) + a(56) + a(59) + a(60)
a(37) =        a(44) + a(54) - a(59)
a(36) =      - a(48) + a(56) + a(60)
a(35) =   s1 - a(47) - a(56) - a(60)
a(34) =        a(47) + a(54) - a(59)
a(33) =        a(48) - a(54) + a(59)

with a(44), a(47), a(48), a(54), a(56), a(59), a(60) and a(62) ... a(64) the independent variables.

The equations deducted above can be applied in an efficient method to generate subject cubes, which will be discussed in Section 3.16.4.

3.13.5 Plane Symmetrical

The defining properties result, after deduction of the corresponding equations, in following set of linear equations describing Plane Symmetrical Simple Magic Cubes.

a(61) = s1 - a(62) - a(63) - a(64)
a(57) = s1 - a(58) - a(59) - a(60)
a(53) = s1 - a(54) - a(55) - a(56)
a(52) = s1 - a(56) - a(60) - a(64)
a(51) = s1 - a(55) - a(59) - a(63)
a(50) = s1 - a(54) - a(58) - a(62)
a(49) = s1 - a(50) - a(51) - a(52)
a(45) = s1 - a(46) - a(47) - a(48)
a(41) = s1 - a(42) - a(43) - a(44)
a(39) =      a(42) + a(56) + a(60) - a(62) - a(63)
a(38) =      a(43) + a(53) + a(57) - a(62) - a(63)
a(37) = s1 - a(38) - a(39) - a(40)
a(36) = s1 - a(40) - a(44) - a(48)
a(35) = s1 - a(39) - a(43) - a(47)
a(34) = s1 - a(38) - a(42) - a(46)
a(33) = s1 - a(34) - a(35) - a(36)

with a(40), a(42) ... a(44), a(46) ... a(48), a(54) ... a(56), a(58) ... a(60) and a(62) ... a(64) the independent variables.

The equations deducted above can be applied in an efficient method to generate subject cubes, which will be discussed in Section 3.16.5.

3.14   Pantriagonal Magic Cubes

3.14.1 Complete

The defining properties result, after deduction of the corresponding equations, in following set of linear equations describing Complete Pantriagonal Magic Cubes.

a(61) =   s1 - a(62) - a(63) - a(64)
a(57) =   s1 - a(58) - a(59) - a(60)
a(53) =   s1 - a(54) - a(55) - a(56)
a(52) =   s1 - a(56) - a(60) - a(64)
a(51) =   s1 - a(55) - a(59) - a(63)
a(50) =   s1 - a(54) - a(58) - a(62)
a(49) =   s1 - a(53) - a(57) - a(61)
a(45) =   s1 - a(46) - a(47) - a(48)
a(42) = 2*s1 - a(44) - a(46) - a(48) - a(58) - a(60) - a(62) - a(64)
a(41) =   s1 - a(42) - a(43) - a(44)
a(40) =        a(46) - a(56) + a(62)
a(39) =        a(45) - a(55) + a(61)
a(38) =        a(48) - a(54) + a(64)
a(37) =        a(47) - a(53) + a(63)
a(36) =   s1 - a(44) - a(46) - a(48) + a(56) - a(62)
a(35) =      - a(43) + a(46) + a(48) + a(55) - a(61)
a(34) = - s1 + a(44) + a(54) + a(58) + a(60) + a(62)
a(33) =        a(43) + a(53) - a(58) - a(60) + a(61)

with a(43), a(44), a(46) ... a(48), a(54) ... a(56), a(58) ... a(60) and a(62) ... a(64) the independent variables.

The equations deducted above can be applied in an efficient method to generate subject cubes, which will be discussed in Section 3.16.6.

3.14.2 Complete with Horizontal Magic Planes

The defining properties result, after deduction of the corresponding equations, in following set of linear equations describing Complete Pantriagonal Magic Cubes with Horizontal Magic Planes.

a(61) = s1 - a(62) - a(63) - a(64)
a(57) = s1 - a(58) - a(59) - a(60)
a(55) =      a(56) - a(58) + a(60) - a(61) + a(64)
a(54) = s1 - a(55) - a(58) - a(59)
a(53) =    - a(56) + a(58) + a(59)
a(52) = s1 - a(56) - a(60) - a(64)
a(51) =      a(54) + a(58) - a(63)
a(50) = s1 - a(51) - a(62) - a(63)
a(49) =      a(56) + a(60) - a(61)
a(45) = s1 - a(46) - a(47) - a(48)
a(44) =      a(45) - a(46) - a(48) - a(60) + 2 * a(61) + a(63)
a(43) = s1 - a(48) - a(59) - a(64)
a(42) = s1 - a(45) - a(58) - a(61)
a(41) =      a(42) - a(43) - a(47) + 2 * a(58) + a(60) - a(63)
a(40) =      a(46) - a(56) + a(62)
a(39) =      a(45) - a(56) + a(58) - a(60) + 2 * a(61) - a(64)
a(38) =      a(48) - a(54) + a(64)
a(37) =      a(47) + a(56) - a(58) - a(59) + a(63)
a(36) = s1 - a(45) - a(54) - a(59)
a(35) =    - a(39) - a(47) + a(48) + a(59) + a(64)
a(34) = s1 - a(35) - a(46) - a(47)
a(33) =    - a(36) + a(46) + a(47)

with a(46) ... a(48), a(56), a(58) ... a(60) and a(62) ... a(64) the independent variables.

The equations deducted above can be applied in an efficient method to generate subject cubes, which will be discussed in Section 3.16.7.

3.14.3 2D-Compact

The defining properties result, after deduction of the corresponding equations, in following set of linear equations describing 2D-Compact Pantriagonal Magic Cubes.

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

with a(32), a(48), a(56), a(60) and a(62) ... a(64) the independent variables.

Based on the above listed equations, a routine can be written to generate 2D-Compact Pantriagonal Magic Cubes of order 4 (ref. MgcCube48) .

Subject guessing routine counted 322560 (= 105 * 3072) cubes, of which 120 are shown in Attachment 3.14.3.

3.14.4 2D-Compact and Plane Symmetrical

The defining properties result, after deduction of the corresponding equations, in following set of linear equations describing 2D-Compact Plane Symmetrical Pantriagonal Magic Cubes.

a(61) = s1 - a(62) - a(63) - a(64)
a(59) = s1 - 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) = s1 - a(56) - a(62) - a(63)
a(52) = s1 - a(56) - a(60) - a(64)
a(51) =      a(56) + a(60) - a(63)
a(50) = s1 - a(56) - a(60) - a(62)
a(49) =      a(56) + a(60) - a(61)
a(47) = s1 - a(48) - a(63) - a(64)
a(46) =      a(48) - a(62) + a(64)
a(45) =    - a(48) + a(62) + a(63)
a(44) = s1 - a(48) - a(60) - a(64)
a(43) =      a(48) - a(59) + a(64)
a(42) = s1 - a(48) - a(58) - a(64)
a(41) =      a(48) + a(58) - a(63)
a(40) =      a(48) - a(56) + a(64)
a(39) = s1 - a(48) - a(55) - a(64)
a(38) =      a(48) - a(54) + a(64)
a(37) =    - a(48) + a(54) + a(63)
a(36) =    - a(48) + a(56) + a(60)
a(35) =      a(48) - a(51) + a(64)
a(34) =    - a(48) + a(54) + a(60)
a(33) =      a(48) + a(53) - a(60)

with a(48), a(56), a(60) and a(62) ... a(64) the independent variables.

Based on the above listed equations, a routine can be written to generate 2D-Compact Plane Symmetrical Pantriagonal Magic Cubes of order 4 (ref. MgcCube49) .

Subject guessing routine produced 64 * 6 * 120 = 46080 cubes within 100 minutes, of which the first 120 are shown in Attachment 3.14.4.

3.15a  Spreadsheet Solutions (2)

The linear equations as deducted in previous sections have been applied in following Excel Spread Sheets:

• Simple Magic Cubes:
  
  CnstrSngl41, Horizontal Associated Magic Planes
  
  CnstrSngl42, Horizontal Pan Magic Planes (3D-Compact)
  
  CnstrSngl43, Associated with Horizontal Magic Planes
  
  CnstrSngl44, Associated and 3D-Compact
  
  CnstrSngl45, Plane Symmetrical
  
  
• Pantriagonal Magic Cubes:
  
  CnstrSngl46, Complete
  
  CnstrSngl47, Complete with Horizontal Magic Planes
  
  CnstrSngl48, 2D-Compact
  
  CnstrSngl49, 2D-Compact and Plane Symmetrical

Only the red figures have to be “guessed” to construct one of the applicable Magic Cubes of the 4^th order (wrong solutions are obvious).

3.15b  Summary

The obtained results regarding the miscellaneous types of order 4 Magic Cubes as deducted and discussed in previous sections are summarized in following table:

The Quaternary Solutions will be discussed in Section 3.16.