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 3.0 Magic Cubes (4 x 4 x 4) 3.1 Historical Background Magic Cubes as discussed in section 2.1 for which not all diagonals sum to the Magic Constant are normally referred to as Simple Magic Cubes. 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. For 4 x 4 x 4 Magic Cubes it has been proven that Perfect Magic Cubes are impossible (refer also section 3.3). However in 2004 Walter Trump found a cube that is almost a Perfect Magic Cube, as all rows, columns, pillars and diagonals with exception of the space diagonals sum to the Magic Sum. The cube and the 12 orthogonal planes are shown below:

Plane 11 (Top)

 59 30 40 1 21 4 58 47 34 55 13 28 16 41 19 54

Plane 21 (Left)

 16 34 21 59 36 57 14 23 29 8 51 42 49 31 44 6

Plane 31 (Back)

 59 30 40 1 23 2 60 45 42 63 5 20 6 35 25 64

Plane 12

 23 2 60 45 14 43 17 56 57 32 38 3 36 53 15 26

Plane 22

 41 55 4 30 53 32 43 2 12 33 22 63 24 10 61 35

Plane 32

 21 4 58 47 14 43 17 56 51 22 48 9 44 61 7 18

Plane 13

 42 63 5 20 51 22 48 9 8 33 27 62 29 12 50 39

Plane 23

 19 13 58 40 15 38 17 60 50 27 48 5 46 52 7 25

Plane 33

 34 55 13 28 57 32 38 3 8 33 27 62 31 10 52 37

Plane 14

 6 35 25 64 44 61 7 18 31 10 52 37 49 24 46 11

Plane 24

 54 28 47 1 26 3 56 45 39 62 9 20 11 37 18 64

Plane 34

 16 41 19 54 36 53 15 26 29 12 50 39 49 24 46 11
 Each of the 12 orthogonal planes contain a Non Normal Magic Square, meaning the Magic Squares contain distinct but not consecutive integers. In addition to this the cube is plane symmetrical (horizontal planes). The cube belongs to a Class of 48 elements which can be found by means of rotation and/or reflection which is visualised in Attachment 3.1.3 and Attachment 3.1.4. 3.2 Analytic Solution In general Magic Cubes of order 4 can be represented as follows:

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 the numbers a(i), i = 1 ... 64 for each of the rows, columns and (long) plane diagonals of the 12 orthogonal planes sum to the same constant (130) this results in following linear equations:
 Plane 11 (Top) a(49)+a(50)+a(51)+a(52) = 130 a(53)+a(54)+a(55)+a(56) = 130 a(57)+a(58)+a(59)+a(60) = 130 a(61)+a(62)+a(63)+a(64) = 130 a(49)+a(53)+a(57)+a(61) = 130 a(50)+a(54)+a(58)+a(62) = 130 a(51)+a(55)+a(59)+a(63) = 130 a(52)+a(56)+a(60)+a(64) = 130 a(49)+a(54)+a(59)+a(64) = 130 a(52)+a(55)+a(58)+a(61) = 130 Plane 21 (Left) a(61)+a(57)+a(53)+a(49) = 130 a(45)+a(41)+a(37)+a(33) = 130 a(29)+a(25)+a(21)+a(17) = 130 a(13)+a(9 )+a(5 )+a( 1) = 130 a(61)+a(45)+a(29)+a(13) = 130 a(57)+a(41)+a(25)+a( 9) = 130 a(53)+a(37)+a(21)+a( 5) = 130 a(49)+a(33)+a(17)+a( 1) = 130 a(61)+a(41)+a(21)+a( 1) = 130 a(49)+a(37)+a(25)+a(13) = 130 Plane 31 (Back) a(49)+a(50)+a(51)+a(52) = 130 a(33)+a(34)+a(35)+a(36) = 130 a(17)+a(18)+a(19)+a(20) = 130 a( 1)+a( 2)+a( 3)+a( 4) = 130 a(49)+a(33)+a(17)+a( 1) = 130 a(50)+a(34)+a(18)+a( 2) = 130 a(51)+a(35)+a(19)+a( 3) = 130 a(52)+a(36)+a(20)+a( 4) = 130 a(49)+a(34)+a(19)+a( 4) = 130 a(52)+a(35)+a(18)+a( 1) = 130 Plane 12 a(33)+a(34)+a(35)+a(36) = 130 a(37)+a(38)+a(39)+a(40) = 130 a(41)+a(42)+a(43)+a(44) = 130 a(45)+a(46)+a(47)+a(48) = 130 a(33)+a(37)+a(41)+a(45) = 130 a(34)+a(38)+a(42)+a(46) = 130 a(35)+a(39)+a(43)+a(47) = 130 a(36)+a(40)+a(44)+a(48) = 130 a(33)+a(38)+a(43)+a(48) = 130 a(36)+a(39)+a(42)+a(45) = 130 Plane 22 a(62)+a(58)+a(54)+a(50) = 130 a(46)+a(42)+a(38)+a(34) = 130 a(30)+a(26)+a(22)+a(18) = 130 a(14)+a(10)+a( 6)+a( 2) = 130 a(62)+a(46)+a(30)+a(14) = 130 a(58)+a(42)+a(26)+a(10) = 130 a(54)+a(38)+a(22)+a( 6) = 130 a(50)+a(34)+a(18)+a( 2) = 130 a(62)+a(42)+a(22)+a( 2) = 130 a(50)+a(38)+a(26)+a(14) = 130 Plane 32 a(53)+a(54)+a(55)+a(56) = 130 a(37)+a(38)+a(39)+a(40) = 130 a(21)+a(22)+a(23)+a(24) = 130 a( 5)+a( 6)+a( 7)+a( 8) = 130 a(53)+a(37)+a(21)+a( 5) = 130 a(54)+a(38)+a(22)+a( 6) = 130 a(55)+a(39)+a(23)+a( 7) = 130 a(56)+a(40)+a(24)+a( 8) = 130 a(53)+a(38)+a(23)+a( 8) = 130 a(56)+a(39)+a(22)+a( 5) = 130 Plane 13 a(17)+a(18)+a(19)+a(20) = 130 a(21)+a(22)+a(23)+a(24) = 130 a(25)+a(26)+a(27)+a(28) = 130 a(29)+a(30)+a(31)+a(32) = 130 a(17)+a(21)+a(25)+a(29) = 130 a(18)+a(22)+a(26)+a(30) = 130 a(19)+a(23)+a(27)+a(31) = 130 a(20)+a(24)+a(28)+a(32) = 130 a(17)+a(22)+a(27)+a(32) = 130 a(20)+a(23)+a(26)+a(29) = 130 Plane 23 a(63)+a(59)+a(55)+a(51) = 130 a(47)+a(43)+a(39)+a(35) = 130 a(31)+a(27)+a(23)+a(19) = 130 a(15)+a(11)+a( 7)+a( 3) = 130 a(63)+a(47)+a(31)+a(15) = 130 a(59)+a(43)+a(27)+a(11) = 130 a(55)+a(39)+a(23)+a( 7) = 130 a(51)+a(35)+a(19)+a( 3) = 130 a(63)+a(43)+a(23)+a( 3) = 130 a(51)+a(39)+a(27)+a(15) = 130 Plane 33 a(57)+a(58)+a(59)+a(60) = 130 a(41)+a(42)+a(43)+a(44) = 130 a(25)+a(26)+a(27)+a(28) = 130 a( 9)+a(10)+a(11)+a(12) = 130 a(57)+a(41)+a(25)+a( 9) = 130 a(58)+a(42)+a(26)+a(10) = 130 a(59)+a(43)+a(27)+a(11) = 130 a(60)+a(44)+a(28)+a(12) = 130 a(57)+a(42)+a(27)+a(12) = 130 a(60)+a(43)+a(26)+a( 9) = 130 Plane 14 a( 1)+a( 2)+a( 3)+a( 4) = 130 a( 5)+a( 6)+a( 7)+a( 8) = 130 a( 9)+a(10)+a(11)+a(12) = 130 a(13)+a(14)+a(15)+a(16) = 130 a( 1)+a( 5)+a( 9)+a(13) = 130 a( 2)+a( 6)+a(10)+a(14) = 130 a( 3)+a( 7)+a(11)+a(15) = 130 a( 4)+a( 8)+a(12)+a(16) = 130 a( 1)+a( 6)+a(11)+a(16) = 130 a( 4)+a( 7)+a(10)+a(13) = 130 Plane 24 a(64)+a(60)+a(56)+a(52) = 130 a(48)+a(44)+a(40)+a(36) = 130 a(32)+a(28)+a(24)+a(20) = 130 a(16)+a(12)+a( 8)+a( 4) = 130 a(64)+a(48)+a(32)+a(16) = 130 a(60)+a(44)+a(28)+a(12) = 130 a(56)+a(40)+a(24)+a( 8) = 130 a(52)+a(36)+a(20)+a( 4) = 130 a(64)+a(44)+a(24)+a( 4) = 130 a(52)+a(40)+a(28)+a(16) = 130 Plane 34 a(61)+a(62)+a(63)+a(64) = 130 a(45)+a(46)+a(47)+a(48) = 130 a(29)+a(30)+a(31)+a(32) = 130 a(13)+a(14)+a(15)+a(16) = 130 a(61)+a(45)+a(29)+a(13) = 130 a(62)+a(46)+a(30)+a(14) = 130 a(63)+a(47)+a(31)+a(15) = 130 a(64)+a(48)+a(32)+a(16) = 130 a(61)+a(46)+a(31)+a(16) = 130 a(64)+a(47)+a(30)+a(13) = 130

3.2.1 Almost Perfect Cubes, Plane Symmetrical

For Plane Symmetrical Cubes also following equations are applicable:

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

which results, after deduction, in following set of linear equations describing the 'Almost Perfect Magic Cubes' as found by Walter Trump:

a(61) = 130 - a(62) - a(63) - a(64)
a(57) = 130 - a(58) - a(59) - a(60)
a(56) =  65 + a(58) - a(63) - a(64)
a(55) = -65 + a(60) + a(62) + a(64)
a(54) = 130 - a(55) - a(58) - a(59)
a(53) = -65 + a(59) + a(63) + a(64)
a(52) =  65 - a(58) - a(60) + a(63)
a(51) = 130 - a(53) - a(60) - a(62)
a(50) = -65 + a(59) + a(60) + a(64)
a(49) = -65 + a(58) + a(60) + a(62)
a(46) =       a(47) - a(61) + a(64)
a(45) = 130 - a(46) - a(47) - a(48)
a(44) = 130 - a(47) - a(48) - 0.5 * a(58) + 0.5 * a(59) - 0.5 * a(62) + 0.5 * a(63) - a(64)
a(43) = 260 - a(44) - a(47) - a(48) - a(58) - a(60) - a(62) - a(64)
a(42) = 130 - a(43) - a(62) - a(63)
a(41) =     - a(44) + a(62) + a(63)
a(40) =  65 + a(42) - a(47) - a(48)
a(39) =  65 - a(41) + a(47) + a(48) - 2 * a(61)
a(38) =     - a(39) + a(62) + a(63)
a(37) = 130 - a(40) - a(62) - a(63)
a(36) = 130 - a(40) - a(44) - a(48)
a(35) = 130 - a(39) - a(43) - a(47)
a(34) = 130 - a(35) - 2 * a(47) + a(61) - a(64)
a(33) = 130 - a(48) - a(51) - a(62)

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

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(58) ... a(60) and 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, ... 46, 49 ... 57 and 61
Int(a(i)) = a(i)      for i = 44
a(i) ≠ a(j)           for i ≠ j

which have been incorporated in an optimized guessing routine (MgcCube4c).

Subject guessing routine produced 184320 (= 64 * 15 * 48 * 4) Magic Cubes within 2.7 hours, of which the first 192 are shown in Attachment 3.2.1 and Attachment 3.2.2.

3.2.2 Almost Perfect Cubes, Generalised

If only the numbers a(i), i = 1 ... 64 for each of the rows, columns and (long) plane diagonals of the 12 orthogonal planes sum to the same constant (130) the linear equations shown in section 3.2 above can be deducted to:

a(61) =  130 - a(62) - a(63) - a(64)
a(57) =  130 - a(58) - a(59) - a(60)
a(55) = -130 + a(56) - a(58) + a(60) + a(62) + a(63) + 2 * a(64)
a(54) =  130 - a(55) - a(58) - a(59)
a(53) =      - a(56) + a(58) + a(59)
a(52) =  130 - a(56) - a(60) - a(64)
a(51) =  130 - a(55) - a(59) - a(63)
a(50) =        a(55) + a(59) - a(62)
a(49) =        a(55) + a(58) - a(64)
a(45) =  130 - a(46) - a(47) - a(48)
a(43) =  130 - a(44) - a(46) - a(48) - a(58) - a(60) + a(63) + a(64)
a(42) =  130 - a(43) - a(62) - a(63)
a(41) =      - a(44) + a(62) + a(63)
a(40) =  130 + a(42) - a(47) - a(48) - a(56) + a(58) - a(63) - a(64)
a(39) =  130 - a(43) - a(56) - a(60)
a(38) =      - a(42) + a(56) + a(60)
a(37) =  130 - a(40) - a(62) - a(63)
a(36) =  130 - a(40) - a(44) - a(48)
a(35) =      - a(47) + a(56) + a(60)
a(34) =  130 - a(46) - a(56) - a(60)
a(33) =      - a(36) + a(46) + a(47)
a(32) =  195 + a(44) + 1.5 * a(46) - 0.5 * a(47) - a(56) + 1.5 * a(58) - 0.5*a(59) - a(62) - 3*a(63) - 3*a(64)
a(31) = -130 + a(32) - a(46) + a(48) + a(62) + a(63) + 2 * a(64)
a(30) =  130 - a(31) - a(46) - a(47)
a(29) =      - a(32) + a(46) + a(47)
a(28) =      - a(30) - a(44) - a(46) + 2 * a(56) - 2 * a(58) + 2 * a(63) + 2 * a(64)
a(27) =      - a(32) + a(44) + a(46) + a(58) + a(60) - a(63) - a(64)
a(26) =      - a(27) + a(62) + a(63)
a(25) =      - a(27) - a(46) - a(48) + a(59) + a(60) + a(63) + a(64)
a(24) =  260 - a(32) - a(44) - a(48) - a(56) - a(60) - 2 * a(64)
a(23) =      - a(27) + a(56) + a(60)
a(22) =  130 - a(23) - a(62) - a(63)
a(21) =      - a(24) + a(62) + a(63)
a(20) =  130 - a(25) + a(44) - a(46) - a(47) - a(48) - a(56) + a(58) + a(59) + a(60) - a(62) - a(63)
a(19) =      - a(21) + a(44) + a(46)
a(18) =        a(21) - a(44) + a(47)
a(17) =  130 - a(20) - a(46) - a(47)
a(16) = -130 - a(18) + a(47) + a(56) + a(60) + a(62) + a(63) + a(64)
a(15) =        a(19) - a(47) + a(56) + a(60) - a(63)
a(14) =  130 - a(15) - a(62) - a(63)
a(13) =      - a(16) + a(62) + a(63)
a(12) =  260 + a(20) - 2 * a(44) - a(48) - a(56) - 2 * a(60) - 2 * a(64)
a(11) =  130 + a(13) - a(59) - a(62) - a(63) - a(64)
a(10) =  130 - a(11) - a(58) - a(59)
a( 9) =      - a(17) + a(48) + a(56)
a( 8) =  130 - a(12) - a(56) - a(60)
a( 7) =      - a( 8) - a(46) + a(47) - a(58) - a(60) + 2 * a(63) + 2 * a(64)
a( 6) = -260 + a(11) + a(56) + 2 * a(59) + a(60) + a(62) + a(63) + 2 * a(64)
a( 5) =  390 - a(12) - 2 * a(44) - a(46) - a(47) - 2 * a(48) - a(56) - a(60) - 2 * a(64)
a( 4) =  130 + a(6) - a(59) - a(62) - a(63) - a(64)
a( 3) =      - a(5) + a(60) + a(62)
a( 2) =      - a(3) + a(62) + a(63)
a( 1) =  130 - a(4) - a(62) - 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(44), a(46) ... a(48), a(56), a(58) ... a(60) and 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, ... 43, 45, 49 ... 55, 57 and 61
Int(a(i)) = a(i)      for i = 32
a(i) ≠ a(j)           for i ≠ j

which have been incorporated in an optimized guessing routine (MgcCube4a). Due to the absence of the Plane Symmetrical restrictions this guessing routine appeared to be disappointing slow.

However, the equations deducted above can be applied in a much more efficient method to generate Almost Perfect Magic Cubes, which will be discussed in Section 3.12.

3.3 Perfect Magic Cubes of order 4 don't exist (Alternative Proof)

If the equations for the 4 Space Diagonals:

a(13) + a(26) + a(39) + a(52) = 130
a(16) + a(27) + a(38) + a(49) = 130
a( 1) + a(22) + a(43) + a(64) = 130
a( 4) + a(23) + a(42) + a(61) = 130

are added to the equations of the 'Almost Perfect Magic Cubes', the following set of linear equations will result:

a(62) =   65 - a(63)
a(61) =   65 - a(64)
a(57) =  130 - a(58) - a(59) - a(60)
a(56) =   65 - a(60)
a(55) =      - a(58) + 2 * a(64)
a(54) =  130 - a(59) - 2 * a(64)
a(53) =  -65 + a(58) + a(59) + a(60)
a(52) =   65 - a(64)
a(51) =  130 + a(58) - a(59) - a(63) - 2 * a(64)
a(50) =  -65 - a(58) + a(59) + a(63) + 2 * a(64)
a(49) =        a(64)
a(45) =  130 - a(46) - a(47) - a(48)
a(44) = -1.5 * a(46) + 0.5 * a(47) - a(48) - 1.5 * a(58) + 0.5 * a(59) - a(60) + 2 * a(63) + 3 * a(64)
a(43) =  130 + 0.5 * a(46) - 0.5 * a(47) + 0.5 * a(58) - 0.5 * a(59) - a(63) - 2 * a(64)
a(42) =  -65 - 0.5 * a(46) + 0.5 * a(47) - 0.5 * a(58) + 0.5 * a(59) + a(63) + 2 * a(64)
a(41) =   65 + 1.5 * a(46) - 0.5 * a(47) + a(48) + 1.5 * a(58) - 0.5 * a(59) + a(60) - 2 * a(63) - 3 * a(64)
a(40) =      - 0.5 * a(46) - 0.5 * a(47) - a(48) + 0.5 * a(58) + 0.5 * a(59) + a(60) + a(64)
a(39) =  -65 - 0.5 * a(46) + 0.5 * a(47) - 0.5 * a(58) + 0.5 * a(59) + a(63) + 2 * a(64)
a(38) =  130 + 0.5 * a(46) - 0.5 * a(47) + 0.5 * a(58) - 0.5 * a(59) - a(63) - 2 * a(64)
a(37) =   65 + 0.5 * a(46) + 0.5 * a(47) + a(48) - 0.5 * a(58) - 0.5 * a(59) - a(60) - a(64)
a(36) =  130 + 2 * a(46) + a(48) + a(58) - a(59) - 2 * a(63) - 4 * a(64)
a(35) =   65 - a(47)
a(34) =   65 - a(46)
a(33) = -130 - a(46) + a(47) - a(48) - a(58) + a(59) + 2 * a(63) + 4 * a(64)
a(32) =   65 - a(48)
a(31) =      - a(46) + 2 * a(64)
a(30) =  130 - a(47) - 2 * a(64)
a(29) =  -65 + a(46) + a(47) + a(48)
a(28) =  0.5 * a(46) + 0.5 * a(47) + a(48) - 0.5 * a(58) - 0.5 * a(59) - a(60) + a(64)
a(27) =  -65 - 0.5 * a(46) + 0.5 * a(47) - 0.5 * a(58) + 0.5 * a(59) + a(63) + 2 * a(64)
a(26) =  130 + 0.5 * a(46) - 0.5 * a(47) + 0.5 * a(58) - 0.5 * a(59) - a(63) - 2 * a(64)
a(25) =   65 - 0.5 * a(46) - 0.5 * a(47) - a(48) + 0.5 * a(58) + 0.5 * a(59) + a(60) - a(64)
a(24) =  130 + 1.5 * a(46) - 0.5 * a(47) + a(48) + 1.5 * a(58) - 0.5 * a(59) + a(60) - 2 * a(63) - 5 * a(64)
a(23) =  130 + 0.5 * a(46) - 0.5 * a(47) + 0.5 * a(58) - 0.5 * a(59) - a(63) - 2 * a(64)
a(22) =  -65 - 0.5 * a(46) + 0.5 * a(47) - 0.5 * a(58) + 0.5 * a(59) + a(63) + 2 * a(64)
a(21) =  -65 - 1.5 * a(46) + 0.5 * a(47) - a(48) - 1.5 * a(58) + 0.5 * a(59) - a(60) + 2 * a(63) + 5 * a(64)
a(20) =  -65 - 2 * a(46) - a(48) - a(58) + a(59) + 2 * a(63) + 4 * a(64)
a(19) =   65 + a(46) - 2 * a(64)
a(18) =  -65 + a(47) + 2 * a(64)
a(17) =  195 + a(46) - a(47) + a(48) + a(58) - a(59) - 2 * a(63) - 4 * a(64)
a(16) =   65 - a(64)
a(15) =  130 + a(46) - a(47) - a(63) - 2 * a(64)
a(14) =  -65 - a(46) + a(47) + a(63) + 2 * a(64)
a(13) =        a(64)
a(12) =  130 + a(46) - a(47) + 2 * a(58) + a(60) - 2 * a(63) - 4 * a(64)
a(11) =   65 - a(59)
a(10) =   65 - a(58)
a( 9) = -130 - a(46) + a(47) - a(58) + a(59) - a(60) + 2 * a(63) + 4 * a(64)
a( 8) =  -65 - a(46) + a(47) - 2 * a(58) - a(60) + 2 * a(63) + 4 * a(64)
a( 7) =   65 + a(58) - 2 * a(64)
a( 6) =  -65 + a(59) + 2 * a(64)
a( 5) =  195 + a(46) - a(47) + a(58) - a(59) + a(60) - 2 * a(63) - 4 * a(64)
a( 4) =        a(64)
a( 3) = -130 - a(46) + a(47) - a(58) + a(59) + a(63) + 4 * a(64)
a( 2) =  195 + a(46) - a(47) + a(58) - a(59) - a(63) - 4 * a(64)
a( 1) =   65 - a(64)

The equations above will definitely not return sets of distinct integers as:

a( 1) = a(16) = a(52) = a(61)
a( 4) = a(13) = a(49) = a(64)
a(22) = a(27) = a(39) = a(42)
a(23) = a(26) = a(38) = a(43)

Consequently Perfect Magic Cubes of the 4th order don’t exist.

The same method can be used to proof that 4th order Magic Cubes with Pan Magic Orthogonal Squares don’t exist either.