Office Applications and Entertainment, Magic Cubes

7.0  Special Cubes, Prime Numbers

Prime Number Magic Cubes can be generated with comparable routines as develloped for Prime Number Magic Squares.

7.1  Magic Cubes (3 x 3 x 3)

The equations defining a Simple Magic Cube of the third order are:

a(25) =    s1   - a(26) - a(27) 
a(22) =    s1   - a(23) - a(24) 
a(21) =    s1   - a(24) - a(27) 
a(20) =    s1   - a(23) - a(26) 
a(19) = -  s1   + a(23) + a(24) + a(26) + a(27) 
a(18) = -2*s1/3 + a(23) + a(24) + a(26) 
a(17) =  4*s1/3 - a(23) - 2 * a(26) 
a(16) =  s1/3   - a(24) + a(26) 
a(15) =  4*s1/3 - a(23) - 2 * a(24) 
a(14) =  s1/3 
a(13) = -2*s1/3 + a(23) + 2 * a(24) 
a(12) =    s1/3 + a(24) - a(26) 
a(11) = -2*s1/3 + a(23) + 2 * a(26) 
a(10) =  4*s1/3 - a(23) - a(24) - a(26) 
a( 9) =  5*s1/3 - a(23) - a(24) - a(26) - a(27) 
a( 8) = -  s1/3 + a(23) + a(26) 
a( 7) = -  s1/3 + a(24) + a(27) 
a( 6) = -  s1/3 + a(23) + a(24) 
a( 5) =  2*s1/3 - a(23) 
a( 4) =  2*s1/3 - a(24) 
a( 3) = -  s1/3 + a(26) + a(27) 
a( 2) =  2*s1/3 - a(26) 
a( 1) =  2*s1/3 - a(27) 

with a(23), a(24), a(26) and a(27) the independent variables. The variable Magic Sum s1 should be divisible by 3.

The variable values {a_i} on which a 3^th order Prime Number Magic Cube might be based should contain 13 pairs and one corresponding center element.

Attachment 7.1 shows 78 Prime Number Magic Cubes of order 3, based on the first 1378 Prime Numbers (2 ... 11411), generated within 25 minutes (ref. PrimeCubes3).

The first occurring cubes, MC = 3309 and MC = 4659, were previously published by Akio Suzuki (1977).

7.2   Magic Cubes (4 x 4 x 4)

7.2.1 Simple Magic Cubes, Horizontal Pan Magic Planes

The equations defining a fourth order Simple Magic Cube with horizontal Pan Magic Planes are:

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.

The variable values {a_i} on which the defined fourth order Prime Number Magic Cube might be based should contain 32 pairs.

Based on the above listed equations, a routine can be written to generate the defined Prime Number Magic Cubes of order 4 (ref. PrimeCubes42).

Attachment 7.2.1 shows for miscellaneous Magic Sums the first occurring fourth order Prime Number Simple Magic Cube with Horizontal Pan Magic Planes.

7.2.2 Simple Magic Cubes, Horizontal Associated Magic Planes

The equations defining a fourth order Simple Magic Cube with horizontal Associated Magic Planes are:

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 variable values {a_i} on which the defined fourth order Prime Number Magic Cube might be based should contain 32 pairs.

Based on the above listed equations, a routine can be written to generate the defined Prime Number Magic Cubes of order 4 (ref. PrimeCubes41).

Attachment 7.2.2 shows for miscellaneous Magic Sums the first occurring fourth order Prime Number Simple Magic Cube with Horizontal Associated Magic Planes.

7.2.3 Simple Magic Cubes, Associated

The equations defining a fourth order Simple Associated Magic Cube are:

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

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

The variable values {a_i} on which the defined fourth order Prime Number Magic Cube might be based should contain 32 pairs.

Based on the above listed equations, a routine can be written to generate the defined Prime Number Magic Cubes of order 4 (ref. PrimeCubes4d).

Attachment 7.2.3 shows for miscellaneous Magic Sums the first occurring fourth order Prime Number Simple Associated Magic Cube.

7.2.4 Simple Magic Cubes, Associated and 3D-Compact

The equations defining a fourth order Simple Magic Cube as specified above are:

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), a(62) ... a(64) the independent variables.

The variable values {a_i} on which the defined fourth order Prime Number Magic Cube might be based should contain 32 pairs.

Based on the above listed equations, a routine can be written to generate the defined Prime Number Magic Cubes of order 4 (ref. PrimeCubes44).

Attachment 7.2.4 shows for miscellaneous Magic Sums the first occurring fourth order Prime Number Simple Associated and 3D-Compact Magic Cube.

7.2.5 Simple Magic Cubes, Associated with Horizontal Magic Planes

The equations defining a fourth order Simple Magic Cube as specified above are:

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 variable values {a_i} on which the defined fourth order Prime Number Magic Cube might be based should contain 32 pairs.

Based on the above listed equations, a routine can be written to generate the defined Prime Number Magic Cubes of order 4 (ref. PrimeCubes43).

Attachment 7.2.5 shows for miscellaneous Magic Sums the first occurring fourth order Prime Number Simple Associated Magic Cube with Horizontal Magic Squares.

7.2.6 Pantriagonal Magic Cubes, Complete

The equations defining a fourth order Pantriagonal and Complete Magic Cube are:

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 variable values {a_i} on which the defined fourth order Prime Number Magic Cube might be based should contain 32 pairs.

Based on the above listed equations, a routine can be written to generate the defined Prime Number Magic Cubes of order 4 (ref. PrimeCubes46).

Attachment 7.2.6 shows for miscellaneous Magic Sums the first occurring fourth order Prime Number Pantriagonal and Complete Magic Cube.

7.2.7 Pantriagonal Magic Cubes, Complete with Horizontal Magic Planes

The defining properties result, after deduction of the corresponding equations, in following set of linear equations describing Pantriagonal and Complete 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 variable values {a_i} on which the defined fourth order Prime Number Magic Cube might be based should contain 32 pairs.

Based on the above listed equations, a routine can be written to generate the defined Prime Number Magic Cubes of order 4 (ref. PrimeCubes47).

Attachment 7.2.7 shows for miscellaneous Magic Sums the first occurring fourth order Prime Number Pantriagonal and Complete Magic Cube with Horizontal Magic Planes.

7.2.8 Simple Magic Cubes, Plane Symmetrical with Horizontal Magic Planes

The defining properties result, after deduction of the corresponding equations, in following set of linear equations describing Plane Symmetrical Magic Cubes with Horizontal Magic Planes:

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

with a(43), a(44), a(46), a(47), a(48), a(54), a(58) ... a(61), a(63) and a(64) the independent variables.

The variable values {a_i} on which the defined fourth order Prime Number Magic Cube might be based should contain 32 pairs.

Based on the above listed equations, a routine can be written to generate the defined Prime Number Magic Cubes of order 4 (ref. PrimeCubes48).

Attachment 7.2.8 shows for miscellaneous Magic Sums the first occurring fourth order Prime Number Plane Symmetrical Magic Cube with Horizontal Magic Planes.

7.2.9 Almost Perfect Magic Cubes, Plane Symmetrical

Magic Cubes - as discussed in previous sections are (normally) referred to as Simple Magic Cubes, as not all diagonals sum to the Magic Constant.

A more strict defined Class of Magic Cubes is known as ‘Perfect Magic Cubes’, for which all (6n + 4) diagonals sum to the Magic Sum.

It can be proven that 4 x 4 x 4 (Prime Number) Perfect Magic Cubes can’t exist for distinct integers.

However Almost Perfect Magic Cubes for wich all rows, columns, pillars and diagonals with exception of the space diagonals sum to the Magic Sum are possible.

The defining properties for such a cube result, after deduction of the corresponding equations, in following set of linear equations describing Almost Perfect, Plane Symmetrical Magic Cubes:

a(61) =   s1   - a(62) - a(63) - a(64)
a(57) =   s1   - a(58) - a(59) - a(60)
a(56) =   s1/2 + a(58) - a(63) - a(64)
a(55) =  -s1/2 + a(60) + a(62) + a(64)
a(54) =   s1   - a(55) - a(58) - a(59)
a(53) =  -s1/2 + a(59) + a(63) + a(64)
a(52) =   s1/2 - a(58) - a(60) + a(63)
a(51) = 3*s1/2 - a(59) - a(60) - a(62) - a(63) - a(64)
a(50) =  -s1/2 + a(59) + a(60) + a(64)
a(49) =  -s1/2 + a(58) + a(60) + a(62)
a(46) =          a(47) - a(61) + a(64)
a(45) =   s1   - a(46) - a(47) - a(48)
a(44) =   s1   - a(47) - a(48) - 0.5 * a(58) + 0.5 * a(59) - 0.5 * a(62) + 0.5 * a(63) - a(64)
a(43) =   s1 - 0.5 * a(58) - 0.5 * a(59) - a(60) - 0.5 * a(62) - 0.5 * a(63)
a(42) =   s1 - a(43) - a(62) - a(63)
a(41) =      - a(44) + a(62) + a(63)
a(40) =   s1/2 + a(42) - a(47) - a(48)
a(39) =  -s1/2 - 0.5 * a(58) + 0.5 * a(59) + 0.5 * a(62) + 1.5 * a(63) + a(64)
a(38) =       - a(39) + a(62) + a(63)
a(37) =   s1  - a(40) - a(62) - a(63)
a(36) =   s1  - a(40) - a(44) - a(48)
a(35) =   s1  - a(39) - a(43) - a(47)
a(34) =   s1  - a(35) - 2 * a(47) + a(61) - a(64)
a(33) =   s1  - a(48) - a(51) - a(62)

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

The variable values {a_i} on which the defined fourth order Prime Number Magic Cube might be based should contain 32 pairs.

Based on the above listed equations, a routine can be written to generate the defined Prime Number Magic Cubes of order 4 (ref. PrimeCubes49).

Attachment 7.2.9 shows a few fourth order Prime Number Plane Symmetrical, Almost Perfect Magic Cubes for the magic constant MC = 12012.

7.2.10 Summary

The obtained results regarding the miscellaneous Prime Number Magic Cubes as deducted and discussed in previous sections are summarized in following table:

Comparable routines as listed above, can be used to generate Prime Number Magic Cubes of order 5, 6 and 7, which will be described in following sections.