Office Applications and Entertainment, Magic Cubes

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7.0   Special Cubes, Prime Numbers

7.3   Bordered Magic Cubes (5 x 5 x 5)

7.3.1 Introduction

A Bordered Magic Cube of order 5 consists of an Embedded Magic Cube of order 3 with a border around it.

The equations defining Bordered Magic Cubes of the 5th order with 6 Magic Surface Planes (s-Magic) have been deducted in Section 4.3 for natural numbers.

Subject linear equations can be incorporated in a guessing routine, which might be used to generate Prime Number Bordered Magic Cubes of order 5 for miscellaneous Magic Sums.

However the total number of independent variables (22) is quite high to find results within a reasonable time.

7.3.2 Semi Magic Surface Planes

In spite of the above, Natalia Makarova's study 'Concentric Magic Cubes of Prime Numbers' contains a variety of Bordered Magic Cubes with Semi Magic Surface Planes of order 5, 6, 7 and 8.

The tabled equations for order 5, with 28 independent variables, result in solutions for miscellaneous Magic Sums, of which a few examples are shown in Attachment 7.3.1

7.3.3 Magic Top/Bottom Planes

Rather than applying comparable equations as deducted in Section 4.3, Bordered Magic Cubes can be constructed based on Complementary Anti Symmetric Magic Squares of order 5, as discussed in Section 14.3.10.

The construction method, based on this principle, can be summarized as follows:

  • Generate, for the applicable Magic Sums, pair collections reduced with the pairs required for one of the possible inner cubes of order 3 (ref. Attachment 7.1);
  • Construct, based on these reduced collections, Anti Symmetric Magic Squares of order 5, which can be considered as possible top squares for the border (ref. Attachment 7.3.2);
  • Determine, based on the selected top - and resulting bottom square, the back - and front square;
  • Determine, based on the top -, bottom -, back - and front squares, the left - and resulting right square.

The relation between opposite surface squares (symmetry) can be represented as follows:

c1 c2 c3 c4 c5
c6 c7 c8 c9 c10
c11 c12 c13 c14 c15
c16 c17 c18 c19 c20
c21 c22 c23 c24 c25
Pr5 - c25 Pr5 - c22 Pr5 - c23 Pr5 - c24 Pr5 - c21
Pr5 - c10 Pr5 - c7 Pr5 - c8 Pr5 - c9 Pr5 - c6
Pr5 - c15 Pr5 - c12 Pr5 - c13 Pr5 - c14 Pr5 - c11
Pr5 - c20 Pr5 - c17 Pr5 - c18 Pr5 - c19 Pr5 - c16
Pr5 - c5 Pr5 - c2 Pr5 - c3 Pr5 - c4 Pr5 - c1

with Pr5 = 2 * s5 / 5 the pair sum for the corresponding Magic Sum s5.

With c(i) the cube variables and the substitution:

a(1) a(2) a(3) a(4) a(5)
a(6) a(7) a(8) a(9) a(10)
a(11) a(12) a(13) a(14) a(15)
a(16) a(17) a(18) a(19) a(20)
a(21) a(22) a(23) a(24) a(25)
=
c(1) c(2) c(3) c(4) c(5)
c(26) c(27) c(28) c(29) c(30)
c(51) c(52) c(53) c(54) c(55)
c(76) c(77) c(78) c(79) c(80)
c(101) c(102) c(103) c(104) c(105)

the defining equations of the Back Square (Semi Magic) can be written as: 

a( 6) = s5 - a(11) - a(16) - a( 1) - a(21)
a( 7) = s5 - a(12) - a(17) - a( 2) - a(22)
a( 8) = s5 - a(13) - a(18) - a( 3) - a(23)
a( 9) = s5 - a(14) - a(19) - a( 4) - a(24)
a(10) = s5 - a(15) - a(20) - a( 5) - a(25)
a(11) = s5 - a(12) - a(13) - a(14) - a(15)
a(16) = s5 - a(17) - a(18) - a(19) - a(20)

with a(i) independent for i = 12 ... 15 and i = 17 ... 20;
and  a(i) defined     for i =  1 ...  5 and i = 21 ... 25.

Based on a comparable substitution:

a(1) a(2) a(3) a(4) a(5)
a(6) a(7) a(8) a(9) a(10)
a(11) a(12) a(13) a(14) a(15)
a(16) a(17) a(18) a(19) a(20)
a(21) a(22) a(23) a(24) a(25)
=
c(1) c(6) c(11) c(16) c(21)
c(26) c(31) c(36) c(41) c(46)
c(51) c(56) c(61) c(66) c(71)
c(76) c(81) c(86) c(91) c(96)
c(101) c(106) c(111) c(116) c(121)

the defining equations of the Left Square (Semi Magic) can be written as: 

a( 7) = s5 - a(12) - a(17) - a( 2) - a(22)
a( 8) = s5 - a(13) - a(18) - a( 3) - a(23)
a( 9) = s5 - a(14) - a(19) - a( 4) - a(24)
a(12) = s5 - a(13) - a(14) - a(11) - a(15)
a(17) = s5 - a(18) - a(19) - a(16) - a(20)

with a(i) independent for i = 13, 14, 18 and 19;
and  a(i) defined     for i = 1 ... 6, 10, 11, 15, 16 and 20 ... 25.

Based on the equations listed above, a guessing routine can be written to generate Prime Number Bordered Magic Cubes of order 5 within a reasonable time (PrimeCubes5a).

Attachment 7.3.3, page 1 shows - for miscellaneous Magic Sums and Center Cubes - the first occurring 5th order Prime Number Bordered Magic Cube with Magic Top and Bottom Planes;
Attachment 7.3.3, page 2 shows - for some higher Magic Sums - the first occurring 5th order Prime Number Bordered Magic Cube, which can be used as Center Cube for Concentric Magic Cubes of order 7 (ref. Section 7.5.2).

7.3.4 Magic Top/Bottom and Back/Front Planes

For Magic Back and Front Squares, and based on the same substitution:

a(1) a(2) a(3) a(4) a(5)
a(6) a(7) a(8) a(9) a(10)
a(11) a(12) a(13) a(14) a(15)
a(16) a(17) a(18) a(19) a(20)
a(21) a(22) a(23) a(24) a(25)
=
c(1) c(2) c(3) c(4) c(5)
c(26) c(27) c(28) c(29) c(30)
c(51) c(52) c(53) c(54) c(55)
c(76) c(77) c(78) c(79) c(80)
c(101) c(102) c(103) c(104) c(105)

the defining equations of the Back Square can be written as: 

a( 6) = s5 - a( 7) - a( 8) - a( 9) - a(10)
a( 7) = s5 - a(13) - a(19) - a( 1) - a(25)
a( 8) = s5 - a(13) - a(18) - a( 3) - a(23)
a( 9) = s5 - a(14) - a(19) - a( 4) - a(24)
a(10) = s5 - a(15) - a(20) - a( 5) - a(25)
a(11) = s5 - a( 6) - a(16) - a( 1) - a(21)
a(12) = s5 - a( 7) - a(17) - a( 2) - a(22)
a(13) =      a(14) - a(17) + a(19) + a( 4) - a(5) - a(21) + a(24)
a(16) = s5 - a(17) - a(18) - a(19) - a(20)

with a(i) independent for i = 14, 15  and i = 17 ... 20;
and  a(i) defined     for i = 1 ... 5 and i = 21 ... 25.

Based on the equations listed above, a comparable routine can be written to generate also these more strict defined Prime Number Bordered Magic Cubes within a reasonable time (PrimeCubes5b).

Attachment 7.3.4 shows, for miscellaneous Magic Sums, the first occurring 5th order Prime Number Bordered Magic Cubes with four Magic Surface Planes.

7.3.5 Magic Surface Planes (s-Magic)

The Anti Symmetric Magic Squares applied in Section 7.3.3 and 7.3.4 above can be considered as possible top squares for s-Magic Borders.

The (pre) selection requires however a faster routine as applied in subject sections.

With c(i) the cube variables and the substitution:

a(1) a(2) a(3) a(4) a(5)
a(6) a(7) a(8) a(9) a(10)
a(11) a(12) a(13) a(14) a(15)
a(16) a(17) a(18) a(19) a(20)
a(21) a(22) a(23) a(24) a(25)
=
c(1) c(2) c(3) c(4) c(5)
c(26) c(27) c(28) c(29) c(30)
c(51) c(52) c(53) c(54) c(55)
c(76) c(77) c(78) c(79) c(80)
c(101) c(102) c(103) c(104) c(105)

the defining equations of the Magic Back Square can be written as: 

a( 7) = s5 - a(13) - a(19) - a( 1) - a(25)
a( 9) = s5 - a(17) - a(13) - a( 5) - a(21)
a(14) =      a(17) + a(13) - a(19) - a( 4) + a(5) + a(21) - a(24)
a(12) = s5 - a(17) - a( 7) - a( 2) - a(22)
a(16) = s5 - a(18) - a(20) - a(17) - a(19)
a( 8) = s5 - a(18) - a(13) - a( 3) - a(23)
a(11) = s5 - a(12) - a(14) - a(15) - a(13)
a( 6) = s5 - a(11) - a(16) - a( 1) - a(21)
a(10) = s5 - a(15) - a(20) - a( 5) - a(25)

with a(i) independent for i = 13, 15  and i = 17 ... 20;
and  a(i) defined     for i = 1 ... 5 and i = 21 ... 25.

Based on a comparable substitution:

a(1) a(2) a(3) a(4) a(5)
a(6) a(7) a(8) a(9) a(10)
a(11) a(12) a(13) a(14) a(15)
a(16) a(17) a(18) a(19) a(20)
a(21) a(22) a(23) a(24) a(25)
=
c(1) c(6) c(11) c(16) c(21)
c(26) c(31) c(36) c(41) c(46)
c(51) c(56) c(61) c(66) c(71)
c(76) c(81) c(86) c(91) c(96)
c(101) c(106) c(111) c(116) c(121)

the defining equations of the Left Magic Square can be written as: 

a( 7) = s5 - a(13) -  a(19) - a( 1) - a(25)
a( 8) = s5 - a(13) -  a(18) - a( 3) - a(23)
a( 9) = s5 - a(14) -  a(19) - a( 4) - a(24)
a(12) = s5 - a(13) -  a(14) - a(11) - a(15)
a(13) =      a(14) -  a(17) + a(19) + a(4) -   a( 5) -   a(21) +  a(24)
a(14) =  (-3*a(18) -6*a(19) - a(11) - a(15)- 3*a(16) - 3*a(20) +  a( 1) + 
                                           + 3*a( 2) + 2*a( 3) +4*a( 5) + 4*a(21) + 3*a(22) + 2*a(23) + a(25))/3
a(17) = s5 - a(18) - a(19) - a(16) - a(20)

with a(i) independent for i = 18 and 19;
and  a(i) defined     for i = 1 ... 6, 10, 11, 15, 16 and 20 ... 25.

Based on the equations listed above, a guessing routine can be written to generate Prime Number Bordered Magic Cubes of order 5 - with magic surface planes - within a reasonable time (PrimeCubes5c).

Attachment 7.3.5 shows, for a few Magic Sums, the first occurring 5th order Prime Number Bordered Magic Cubes with six Magic Surface Planes (s-Magic).

7.4   Bordered Magic Cubes (6 x 6 x 6)

7.4.1 Introduction

Bordered Magic Cubes of order 6 - with Semi Magic Surface Planes - can be constructed based on the application of 3th order Semi Magic Sub Squares (6 magic lines).

7.4.2 Construction Method, Semi Magic Surface Planes

The construction method, based on this principle, can be summarized as follows:

  • Generate, for the applicable Magic Sums, pair collections reduced with the pairs required for one of the possible inner cubes of order 4 (ref. Section 7.2.9);
  • Generate, based on these reduced collections, sets of Anti Symmetric Semi Magic Squares of order 3, and construct the order 6 top -, bottom -, back - and front squares (ref. Attachment 7.4.10);
  • Determine, based on the top -, bottom -, back - and front squares, the left - and resulting right square.

The relation between opposite surface squares (symmetry) can be represented as follows:

c1 c2 c3 c4 c5 c6
c7 c8 c9 c10 c11 c12
c13 c14 c15 c16 c17 c18
c19 c20 c21 c22 c23 c24
c25 c26 c27 c28 c29 c30
c31 c32 c33 c34 c35 c36
Pr3 - c36 Pr3 - c32 Pr3 - c33 Pr3 - c34 Pr3 - c35 Pr3 - c31
Pr3 - c12 Pr3 - c8 Pr3 - c9 Pr3 - c10 Pr3 - c11 Pr3 - c7
Pr3 - c18 Pr3 - c14 Pr3 - c15 Pr3 - c16 Pr3 - c17 Pr3 - c13
Pr3 - c24 Pr3 - c20 Pr3 - c21 Pr3 - c22 Pr3 - c23 Pr3 - c19
Pr3 - c30 Pr3 - c26 Pr3 - c27 Pr3 - c28 Pr3 - c29 Pr3 - c25
Pr3 - c6 Pr3 - c2 Pr3 - c3 Pr3 - c4 Pr3 - c5 Pr3 - c1

with Pr3 = s6 / 3 the pair sum for the corresponding Magic Sum s6.

Based on the procedure described above, Partly Completed Borders (top -, bottom -, back - and front square), can be generated quite fast (PrimeCubes6a).

With c(i) the cube variables and the substitution:

a(1) a(2) a(3) a(4) a(5) a(6)
a(7) a(8) a(9) a(10) a(11) a(12)
a(13) a(14) a(15) a(16) a(17) a(18)
a(19) a(20) a(21) a(22) a(23) a(24)
a(25) a(26) a(27) a(28) a(29) a(30)
a(31) a(32) a(33) a(34) a(35) a(36)
=
c(1) c(7) c(13) c(19) c(25) c(31)
c(37) c(43) c(49) c(55) c(61) c(67)
c(73) c(79) c(85) c(91) c(97) c(103)
c(109) c(115) c(121) c(127) c(133) c(139)
c(145) c(151) c(157) c(163) c(169) c(175)
c(181) c(187) c(193) c(199) c(205) c(211)

the defining equations of the Left Surface Square (Semi Magic) can be written as: 

a(26) = s6 - a(27) - a(28) - a(29) - a(25) - a(30)
a(20) = s6 - a(21) - a(22) - a(23) - a(19) - a(24)
a(14) = s6 - a(15) - a(16) - a(17) - a(13) - a(18)
a(11) = s6 - a(17) - a(23) - a(29) - a(35) - a( 5)
a(10) = s6 - a(16) - a(22) - a(28) - a(34) - a( 4)
a( 9) = s6 - a(15) - a(21) - a(27) - a(33) - a( 3)
a( 8) = s6 - a(14) - a(20) - a(26) - a(32) - a( 2)

with a(i) independent for i = 15, 16, 17; 21, 22, 23; 27, 28, 29;
and  a(i) defined     for i = 1 ... 7, 12, 13, 18, 19, 24, 25 and i = 30 ... 36.

which can be incorporated in a guessing routine to complete the borders within a reasonable time (PrimeCubes6b).

Based on available 4th order Magic Cubes (ref. Section 7.2.9) and the procedures explained above, 6th order Bordered Magic Cubes could be generated for:

  • Simple Center Cubes:
    - Horizontal Pan Magic Planes                 (ref. Attachment 7.4.1)
    - Horizontal Associated Planes                (ref. Attachment 7.4.2)

  • Simple Center Cubes:
    - Associated                                  (ref. Attachment 7.4.3)
    - Associated and 3D-Compact                   (ref. Attachment 7.4.4)
    - Associated with Horizontal Magic Planes     (ref. Attachment 7.4.5)

  • Pantriagonal Center Cubes:
    - Complete                                    (ref. Attachment 7.4.6)
    - Complete with Horizontal Magic Planes       (ref. Attachment 7.4.7)

Each cube shown corresponds with numerous cubes for the same Magic Sum (n4 * 48 * 48).

7.4.3 Construction Method, Magic Top/Bottom Planes

Alternatively, for higher Magic Sums, the top squares might be based on sets of 3th order Anti Symmetric Semi Magic Squares with 7 magic lines (ref. Attachment 7.4.11).

Attachment 7.4.8 contains a few Bordered Magic Cubes with magic top - and bottom planes, which could be constructed based on the procedure explained in Section 7.4.2 above (PrimeCubes6c).

7.4.4 Construction Method, Magic Surface Planes (s-Magic)

Based on the Anti Symmetric (Semi-) Magic Surface Planes as constructed in Section 7.4.3 above, numerous Anti Symmetric Magic Squares can be obtained by means of row and column permutations.

Subject Anti Symmetric Magic Squares can be considered as possible top squares for s-Magic Borders. A few examples are shown in Attachment 7.4.9a.

With c(i) the cube variables and the substitution:

a(1) a(2) a(3) a(4) a(5) a(6)
a(7) a(8) a(9) a(10) a(11) a(12)
a(13) a(14) a(15) a(16) a(17) a(18)
a(19) a(20) a(21) a(22) a(23) a(24)
a(25) a(26) a(27) a(28) a(29) a(30)
a(31) a(32) a(33) a(34) a(35) a(36)
=
c(1) c(2) c(3) c(4) c(5) c(6)
c(37) c(38) c(39) c(40) c(41) c(42)
c(73) c(74) c(75) c(76) c(77) c(78)
c(109) c(110) c(111) c(112) c(113) c(114)
c(145) c(146) c(147) c(148) c(149) c(150)
c(181) c(182) c(183) c(184) c(185) c(186)

the defining equations of the Back Square (Magic) can be written as: 

a(8)  = s6 - a(15) - a(22) - a(29) - a(1)  - a(36)
a(11) = s6 - a(17) - a(23) - a(29) - a(5)  - a(35)
a(16) = s6 - a(21) - a(26) - a(11) - a(6)  - a(31)
a(14) = s6 - a(20) - a(26) - a(8)  - a(2)  - a(32)
a(10) = s6 - a(28) - a(16) - a(22) - a(4)  - a(34)
a(9)  = s6 - a(27) - a(21) - a(15) - a(3)  - a(33)
a(25) = s6 - a(30) - a(27) - a(28) - a(26) - a(29)
a(19) = s6 - a(24) - a(20) - a(21) - a(23) - a(22)
a(13) = s6 - a(18) - a(14) - a(16) - a(17) - a(15)
a(12) = s6 - a(30) - a(18) - a(24) - a(6)  - a(36)
a(7)  = s6 - a(12) - a(9)  - a(10) - a(11) - a(8)

with a(i) independent for i = 26 ... 30,  20 ... 24 and i = 15, 17, 18
and  a(i) defined     for i = 1 ... 6 and i = 31 ... 36

Based on a comparable substitution:

a(1) a(2) a(3) a(4) a(5) a(6)
a(7) a(8) a(9) a(10) a(11) a(12)
a(13) a(14) a(15) a(16) a(17) a(18)
a(19) a(20) a(21) a(22) a(23) a(24)
a(25) a(26) a(27) a(28) a(29) a(30)
a(31) a(32) a(33) a(34) a(35) a(36)
=
c(1) c(7) c(13) c(19) c(25) c(31)
c(37) c(43) c(49) c(55) c(61) c(67)
c(73) c(79) c(85) c(91) c(97) c(103)
c(109) c(115) c(121) c(127) c(133) c(139)
c(145) c(151) c(157) c(163) c(169) c(175)
c(181) c(187) c(193) c(199) c(205) c(211)

the defining equations of the Left Square (Magic) can be written as: 

a(8)  = s6 -  a(1)  - a(15) - a(22) - a(29) - a(36)
a(16) = s6 -  a(21) - a(15) - a(22) +
           - (a(1)  + a(3)  + a(4)  + a(6)  - a(7) - a(12) - a(25) - a(30) + a(31) + a(33) + a(34) + a(36))/2
a(11) = s6 -  a(6)  - a(16) - a(21) - a(26) - a(31)
a(27) = s6 -  a(25) - a(26) - a(28) - a(29) - a(30)
a(10) = s6 -  a(16) - a(22) - a(28) - a(4)  - a(34)
a(9)  = s6 -  a(15) - a(21) - a(27) - a(3)  - a(33)
a(20) = s6 -  a(21) - a(22) - a(23) - a(19) - a(24)
a(17) = s6 -  a(11) - a(23) - a(29) - a(5)  - a(35)
a(14) = s6 -  a(15) - a(16) - a(17) - a(13) - a(18)

with a(i) independent for i = 26, 28, 29,  21 ... 23 and i = 15
and  a(i) defined     for i = 1 ... 6, i = 31 ... 36 and i = 7, 12, 13, 18, 19, 24, 25, 30

Based on the equations listed above, guessing routines can be written to generate Prime Number Bordered Magic Cubes of order 6 - with magic surface planes - within a reasonable time (PrimeCubes61).

Attachment 7.4.9b shows, for miscellaneous Magic Sums, the first occurring 6th order Prime Number Bordered Magic Cubes with six Magic Surface Planes (s-Magic).

7.5   Bordered Magic Cubes (7 x 7 x 7)

7.5.1 Introduction

Bordered Magic Cubes of order 7 - with (Semi-) Magic Surface Planes - can be constructed based on Complementary Anti Symmetric (Semi-) Magic Squares.

7.5.2 Construction Method, Semi Magic Surface Planes

The construction method, based on this principle, can be summarized as follows:

  • Generate, for the applicable Magic Sums, pair collections reduced with the pairs required for one of the possible inner cubes of order 5 (ref. Attachment 7.3.3);
  • Construct, based on these reduced collections, Anti Symmetric Semi Magic Squares of order 7, which can be considered as possible top squares for the border (ref. Attachment 7.5.1);
  • Determine, based on the selected top - and resulting bottom square, the back - and front square;
  • Determine, based on the top -, bottom -, back - and front squares, the left - and resulting right square.

The relation between opposite surface squares (symmetry) can be represented as follows:

c1 c2 c3 c4 c5 c6 c7
c8 c9 c10 c11 c12 c13 c14
c15 c16 c17 c18 c19 c20 c21
c22 c23 c24 c25 c26 c27 c28
c29 c30 c31 c32 c33 c34 c35
c36 c37 c38 c39 c40 c41 c42
c43 c44 c45 c46 c47 c48 c49
Pr7 - c49 Pr7 - c44 Pr7 - c45 Pr7 - c46 Pr7 - c47 Pr7 - c48 Pr7 - c43
Pr7 - c14 Pr7 - c9 Pr7 - c10 Pr7 - c11 Pr7 - c12 Pr7 - c13 Pr7 - c8
Pr7 - c21 Pr7 - c16 Pr7 - c17 Pr7 - c18 Pr7 - c19 Pr7 - c20 Pr7 - c15
Pr7 - c28 Pr7 - c23 Pr7 - c24 Pr7 - c25 Pr7 - c26 Pr7 - c27 Pr7 - c22
Pr7 - c35 Pr7 - c30 Pr7 - c31 Pr7 - c32 Pr7 - c33 Pr7 - c34 Pr7 - c29
Pr7 - c42 Pr7 - c37 Pr7 - c38 Pr7 - c39 Pr7 - c40 Pr7 - c41 Pr7 - c36
Pr7 - c7 Pr7 - c2 Pr7 - c3 Pr7 - c4 Pr7 - c5 Pr7 - c6 Pr7 - c1

with Pr7 = 2 * s7 / 7 the pair sum for the corresponding Magic Sum s7.

With c(i) the cube variables and the substitution a(i) = c(i), i = 1 ... 49,
the defining equations of the Top Square (Semi Magic, Composed, ref. Attachment 7.5.1) can be written as:

a(22) = s4 - a(23) - a(24) - a(25)
a(15) = s4 - a(16) - a(17) - a(18)
a( 8) = s4 - a( 9) - a(10) - a(11)
a( 4) = s4 - a(11) - a(18) - a(25)
a( 3) = s4 - a(10) - a(17) - a(24)
a( 2) = s4 - a( 9) - a(16) - a(23)
a( 1) = s4 - a( 8) - a(15) - a(22)

a(47) = s3 - a(48) - a(49)
a(40) = s3 - a(41) - a(42)
a(35) = s3 - a(42) - a(49)
a(34) = s3 - a(41) - a(48)
a(33) = s3 - a(40) - a(47)

a(43) = s4 - a(44) - a(45) - a(46)
a(36) = s4 - a(37) - a(38) - a(39)
a(32) = s3 - a(39) - a(46)
a(31) = s3 - a(38) - a(45)
a(30) = s3 - a(37) - a(44)
a(29) = s3 - a(36) - a(43)

a( 7) = s4 - a(14) - a(21) - a(28)
a( 6) = s4 - a(13) - a(20) - a(27)
a(26) = s3 - a(27) - a(28)
a(19) = s3 - a(20) - a(21)
a(12) = s3 - a(13) - a(14)
a( 5) = s3 - a( 6) - a( 7)

with a(i) independent for i =  9, 10, 11, 16, 17, 18, 23, 24, 25 and 41, 42, 48, 49 (sub squares)
and                       i = 13, 14, 20, 21, 27, 28 and 37, 38, 39, 44, 45, 46     (magic rectangles)

With c(i) the cube variables and the substitution:

a1 a2 a3 a4 a5 a6 a7
a8 a9 a10 a11 a12 a13 a14
a15 a16 a17 a18 a19 a20 a21
a22 a23 a24 a25 a26 a27 a28
a29 a30 a31 a32 a33 a34 a35
a36 a37 a38 a39 a40 a41 a42
a43 a44 a45 a46 a47 a48 a49
=
c1 c2 c3 c4 c5 c6 c7
c50 c51 c52 c53 c54 c55 c56
c99 c100 c101 c102 c103 c104 c105
c148 c149 c150 c151 c152 c153 c154
c197 c198 c199 c200 c201 c202 c203
c246 c247 c248 c249 c250 c251 c252
c295 c296 c297 c298 c299 c300 c301

the defining equations of the Back Square (Semi Magic, Broken Rows) can be written as:

a(36) = s4 - a(37) - a(38) - a(39)
a(29) = s4 - a(30) - a(31) - a(32)
a(22) = s4 - a(23) - a(24) - a(25)
a(15) = s4 - a(16) - a(17) - a(18)

a(40) = s3 - a(41) - a(42)
a(33) = s3 - a(34) - a(35)
a(26) = s3 - a(27) - a(28)
a(19) = s3 - a(20) - a(21) 

a(14) = s7 - a(7) - a(21) - a(28) - a(35) - a(42) - a(49)
a(13) = s7 - a(6) - a(20) - a(27) - a(34) - a(41) - a(48)
a(12) = s7 - a(5) - a(19) - a(26) - a(33) - a(40) - a(47)
a(11) = s7 - a(4) - a(18) - a(25) - a(32) - a(39) - a(46)
a(10) = s7 - a(3) - a(17) - a(24) - a(31) - a(38) - a(45)
a( 9) = s7 - a(2) - a(16) - a(23) - a(30) - a(37) - a(44)
a( 8) = s7 - a(9) - a(10) - a(11) - a(12) - a(13) - a(14)

with a(i) independent for i = 16 ... 18, 23 ... 25, 30 ... 32, 37 ... 39, 20, 21, 27, 28, 34, 35, 41, 42
and  a(i) defined     for i = 1 ... 7 and 43 ... 49

Based on a comparable substitution:

a1 a2 a3 a4 a5 a6 a7
a8 a9 a10 a11 a12 a13 a14
a15 a16 a17 a18 a19 a20 a21
a22 a23 a24 a25 a26 a27 a28
a29 a30 a31 a32 a33 a34 a35
a36 a37 a38 a39 a40 a41 a42
a43 a44 a45 a46 a47 a48 a49
=
c1 c8 c15 c22 c29 c36 c43
c50 c57 c64 c71 c78 c85 c92
c99 c106 c113 c120 c127 c134 c141
c148 c155 c162 c169 c176 c183 c190
c197 c204 c211 c218 c225 c232 c239
c246 c253 c260 c267 c274 c281 c288
c295 c302 c309 c316 c323 c330 c337

the defining equations of the Left Square (Semi Magic) can be written as: 

a(13) = s7 - a( 6) - a(20) - a(27) - a(34) - a(41) - a(48)
a(12) = s7 - a( 5) - a(19) - a(26) - a(33) - a(40) - a(47)
a(11) = s7 - a( 4) - a(18) - a(25) - a(32) - a(39) - a(46)
a(37) = s7 - a(36) - a(38) - a(39) - a(40) - a(41) - a(42)
a(30) = s7 - a(29) - a(31) - a(32) - a(33) - a(34) - a(35)
a(23) = s7 - a(22) - a(24) - a(25) - a(26) - a(27) - a(28)
a(10) = s7 - a( 3) - a(17) - a(24) - a(31) - a(38) - a(45)
a(16) = s7 - a(15) - a(17) - a(18) - a(19) - a(20) - a(21)
a( 9) = s7 - a( 8) - a(10) - a(11) - a(12) - a(13) - a(14)

with a(i) independent for i = 17 ... 20, 24 ... 27, 31 ... 34 and 38 ... 41;
and  a(i) defined     for i = 1 ... 8, 14, 15, 21, 22, 28, 29, 35, 36 and 42 ... 49

Based on the equations listed above, guessing routines can be written to generate Prime Number Bordered Magic Cubes of order 7 within a reasonable time (PrimeCubes7).

Attachment 7.5.2 shows, for miscellaneous Magic Sums, the first occurring 7th order Prime Number Bordered Magic Cubes with Semi Magic Surface Planes.

7.5.3 Construction Method, Magic Surface Planes (s-Magic)

Based on the Anti Symmetric Semi Magic Surface Planes as constructed in Section 7.5.2 above, numerous Anti Symmetric Magic Squares can be obtained by means of row and column permutations.

Subject Anti Symmetric Magic Squares can be considered as possible top squares for s-Magic Borders. A few examples are shown in Attachment 7.5.3.

With c(i) the cube variables and the substitution:

a1 a2 a3 a4 a5 a6 a7
a8 a9 a10 a11 a12 a13 a14
a15 a16 a17 a18 a19 a20 a21
a22 a23 a24 a25 a26 a27 a28
a29 a30 a31 a32 a33 a34 a35
a36 a37 a38 a39 a40 a41 a42
a43 a44 a45 a46 a47 a48 a49
=
c1 c2 c3 c4 c5 c6 c7
c50 c51 c52 c53 c54 c55 c56
c99 c100 c101 c102 c103 c104 c105
c148 c149 c150 c151 c152 c153 c154
c197 c198 c199 c200 c201 c202 c203
c246 c247 c248 c249 c250 c251 c252
c295 c296 c297 c298 c299 c300 c301

the defining equations of the Back Square (Magic, Broken Rows) can be written as:

a(36) = s4 - a(37) - a(38) - a(39)
a(29) = s4 - a(30) - a(31) - a(32)
a(22) = s4 - a(23) - a(24) - a(25)
a(15) = s4 - a(16) - a(17) - a(18)

a(40) = s3 - a(41) - a(42)
a(33) = s3 - a(34) - a(35)
a(26) = s3 - a(27) - a(28)
a(19) = s3 - a(20) - a(21) 

a(14) = s7 - a( 7) - a(21) - a(28) - a(35) - a(42) - a(49)
a(13) = s7 - a( 6) - a(20) - a(27) - a(34) - a(41) - a(48)
a(12) = s7 - a( 5) - a(19) - a(26) - a(33) - a(40) - a(47)
a( 9) = s7 - a(17) - a(25) - a(33) - a(41) - a( 1) - a(49)
a(31) = s7 - a( 7) - a(13) - a(19) - a(25) - a(37) - a(43)
a(11) = s7 - a( 4) - a(18) - a(25) - a(32) - a(39) - a(46)
a(10) = s7 - a( 3) - a(17) - a(24) - a(31) - a(38) - a(45)
a(16) = s7 - a( 2) - a( 9) - a(23) - a(30) - a(37) - a(44)
a( 8) = s7 - a( 9) - a(10) - a(11) - a(12) - a(13) - a(14)

with a(i) independent for i = 17, 18, 23 ... 25, 30, 32, 37 ... 39, 20, 21, 27, 28, 34, 35, 41, 42
and  a(i) defined     for i = 1 ... 7 and 43 ... 49

Based on a comparable substitution:

a1 a2 a3 a4 a5 a6 a7
a8 a9 a10 a11 a12 a13 a14
a15 a16 a17 a18 a19 a20 a21
a22 a23 a24 a25 a26 a27 a28
a29 a30 a31 a32 a33 a34 a35
a36 a37 a38 a39 a40 a41 a42
a43 a44 a45 a46 a47 a48 a49
=
c1 c8 c15 c22 c29 c36 c43
c50 c57 c64 c71 c78 c85 c92
c99 c106 c113 c120 c127 c134 c141
c148 c155 c162 c169 c176 c183 c190
c197 c204 c211 c218 c225 c232 c239
c246 c253 c260 c267 c274 c281 c288
c295 c302 c309 c316 c323 c330 c337

the defining equations of the Left Square (Magic) can be written as: 

a( 9) = s7 - a(17) - a(25) - a(33) - a(41) - a( 1) - a(49)
a(13) = s7 - a(19) - a(25) - a(31) - a(37) - a( 7) - a(43)
a(16) = s7 - a(17) - a(18) - a(19) - a(20) - a(15) - a(21)
a(30) = s7 - a( 9) - a(16) - a(23) - a(37) - a( 2) - a(44)
a(34) = s7 - a(41) - a(27) - a(20) - a(13) - a( 6) - a(48)
a(32) = s7 - a(30) - a(31) - a(33) - a(34) - a(29) - a(35)
a(24) = s7 - a(23) - a(25) - a(26) - a(27) - a(22) - a(28)
a(39) = s7 - a(11) - a(18) - a(25) - a(32) - a( 4) - a(46)
a(38) = s7 - a(10) - a(17) - a(24) - a(31) - a( 3) - a(45)
a(12) = s7 - a( 9) - a(10) - a(11) - a(13) - a( 8) - a(14)
a(40) = s7 - a(37) - a(38) - a(39) - a(41) - a(36) - a(42)

with a(i) independent for i = 10, 11, 17 ... 20, 23, 25 ... 27, 31, 33, 37 and 41;
and  a(i) defined     for i = 1 ... 8, 14, 15, 21, 22, 28, 29, 35, 36 and 42 ... 49

Based on the equations listed above, guessing routines can be written to generate Prime Number Bordered Magic Cubes of order 7 - with magic surface planes - within a reasonable time (PrimeCubes71).

Attachment 7.5.4 shows, for miscellaneous Magic Sums, the first occurring 7th order Prime Number Bordered Magic Cubes with six Magic Surface Planes (s-Magic).

7.6   Bordered Magic Cubes (8 x 8 x 8)

7.6.1 Introduction

Bordered Magic Cubes of order 8 - with Semi Magic Surface Planes - can be constructed based on Complementary Anti Symmetric Semi Magic Squares.

7.6.2 Construction Method, Semi Magic Surface Planes

The construction method, based on this principle, can be summarized as follows:

  • Generate, for the applicable Magic Sums, pair collections reduced with the pairs required for one of the possible inner cubes of order 6 (ref. Section 7.4.2);
  • Construct, based on these reduced collections, Anti Symmetric Composed Semi Magic Squares of order 8, which can be considered as possible top squares for the border;
  • Determine, based on the selected top - and resulting bottom square, the back - and front square;
  • Determine, based on the top -, bottom -, back - and front squares, the left - and resulting right square.

The relation between opposite surface squares (symmetry) can be represented as follows:

c1 c2 c3 c4 c5 c6 c7 c8
c9 c10 c11 c12 c13 c14 c15 c16
c17 c18 c19 c20 c21 c22 c23 c24
c25 c26 c27 c28 c29 c30 c31 c32
c33 c34 c35 c36 c37 c38 c39 c40
c41 c42 c43 c44 c45 c46 c47 c48
c49 c50 c51 c52 c53 c54 c55 c56
c57 c58 c59 c60 c61 c62 c63 c64
p - c64 p - c58 p - c59 p - c60 p - c61 p - c62 p - c63 p - c57
p - c16 p - c10 p - c11 p - c12 p - c13 p - c14 p - c15 p - c9
p - c24 p - c18 p - c19 p - c20 p - c21 p - c22 p - c23 p - c17
p - c32 p - c26 p - c27 p - c28 p - c29 p - c30 p - c31 p - c25
p - c40 p - c34 p - c35 p - c36 p - c37 p - c38 p - c39 p - c33
p - c48 p - c42 p - c43 p - c44 p - c45 p - c46 p - c47 p - c41
p - c56 p - c50 p - c51 p - c52 p - c53 p - c54 p - c55 p - c49
p - c8 p - c2 p - c3 p - c4 p - c5 p - c6 p - c7 p - c1

with p = s8 / 4 the pair sum for the corresponding Magic Sum s8.

With c(i) the cube variables and the substitution:

a1(1) a1(2) a1(3) a1(4) a2(1) a2(2) a2(3) a2(4)
a1(5) a1(6) a1(7) a1(8) a2(5) a2(6) a2(7) a2(8)
a1(9) a1(10) a1(11) a1(12) a2(9) a2(10) a2(11) a2(12)
a1(13) a1(14) a1(15) a1(16) a2(13) a2(14) a2(15) a2(16)
a3(1) a3(2) a3(3) a3(4) a4(1) a4(2) a4(3) a4(4)
a3(5) a3(6) a3(7) a3(8) a4(5) a4(6) a4(7) a4(8)
a3(9) a3(10) a3(11) a3(12) a4(9) a4(10) a4(11) a4(12)
a3(13) a3(14) a3(15) a3(16) a4(13) a4(14) a4(15) a4(16)
=
c1 c2 c3 c4 c5 c6 c7 c8
c9 c10 c11 c12 c13 c14 c15 c16
c17 c18 c19 c20 c21 c22 c23 c24
c25 c26 c27 c28 c29 c30 c31 c32
c33 c34 c35 c36 c37 c38 c39 c40
c41 c42 c43 c44 c45 c46 c47 c48
c49 c50 c51 c52 c53 c54 c55 c56
c57 c58 c59 c60 c61 c62 c63 c64

the defining equations of the Top Square (Composed, Semi Magic) can be written as: 

a'(1) =       a'( 6) + a'( 7) + a'( 8) - a'( 9) - a'(13)
a'(2) =  s4 - a'( 6) - a'(10) - a'(14)
a'(3) =  s4 - a'( 7) - a'(11) - a'(15)
a'(4) = -s4 - a'( 8) + a'( 9) + a'(10) + a'(11) + a'(13) + a'(14) + a'(15)
a'(5) =  s4 - a'( 6) - a'( 7) - a'( 8)
a'(9) =  s4 - a'(10) - a'(11) - a'(12)
a'(13) = s4 - a'(14) - a'(15) - a'(16)

with a'(i) = aj(i) independent for i = 6, 7, 8, 10, 11, 12, 14, 15, 16 and j = 1 ... 4

The same equations can be applied for the Back Square (Composed, Semi Magic), based on following comparable substitution:

a1(13) a1(14) a1(15) a1(16) a2(13) a2(14) a2(15) a2(16)
a1(9) a1(10) a1(11) a1(12) a2(9) a2(10) a2(11) a2(12)
a1(5) a1(6) a1(7) a1(8) a2(5) a2(6) a2(7) a2(8)
a1(1) a1(2) a1(3) a1(4) a2(1) a2(2) a2(3) a2(4)
a3(1) a4(2) a4(3) a4(4) a3(2) a3(3) a3(4) a4(1)
a3(5) a4(6) a4(7) a4(8) a3(6) a3(7) a3(8) a4(5)
a3(9) a4(10) a4(11) a4(12) a3(10) a3(11) a3(12) a4(9)
a3(13) a4(14) a4(15) a4(16) a3(14) a3(15) a3(16) a4(13)
=
c1 c2 c3 c4 c5 c6 c7 c8
c65 c66 c67 c68 c69 c70 c71 c72
c129 c130 c131 c132 c133 c134 c135 c136
c193 c194 c195 c196 c197 c198 c199 c200
c257 c258 c259 c260 c261 c262 c263 c264
c321 c322 c323 c324 c325 c326 c327 c328
c385 c386 c387 c388 c389 c390 c391 c392
c449 c450 c451 c452 c453 c454 c455 c456

with aj(i) independent for i = 6, 7, 8, 10, 11, 12 and j = 1 ... 4
and  aj(i) defined     for i = 13 ... 16           and j = 1 ... 4

Based on following substitution:

a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8)
a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16)
a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24)
a(25) a(26) a(27) a(28) a(29) a(30) a(31) a(32)
a(33) a(34) a(35) a(36) a(37) a(38) a(39) a(40)
a(41) a(42) a(43) a(44) a(45) a(46) a(47) a(48)
a(49) a(50) a(51) a(52) a(53) a(54) a(55) a(56)
a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64)
=
c1 c9 c17 c25 c33 c41 c49 c57
c65 c73 c81 c89 c97 c105 c113 c121
c129 c137 c145 c153 c161 c169 c177 c185
c193 c201 c209 c217 c225 c233 c241 c249
c257 c265 c273 c281 c289 c297 c305 c313
c321 c329 c337 c345 c353 c361 c369 c377
c385 c393 c401 c409 c417 c425 c433 c441
c449 c457 c465 c473 c481 c489 c497 c505

the defining equations of the Left Square (Semi Magic, Broken Rows) can be written as:

a(12) = s4 - a(11) - a(10) - a(9)
a(20) = s4 - a(19) - a(18) - a(17)
a(26) = s4 - a(18) - a(10) - a(2)
a(27) = s4 - a(19) - a(11) - a(3)
a(28) =      a(19) + a(18) + a(17) - a(12) - a(4)

a(13) = s4 - a(14) - a(15) - a(16)
a(21) = s4 - a(22) - a(23) - a(24)
a(29) = s4 - a(30) - a(31) - a(32)
a(34) = s4 - a(42) - a(50) - a(58)
a(35) = s4 - a(43) - a(51) - a(59)
a(36) = s4 - a(44) - a(52) - a(60) 

a(37) = s8 - a(13) - a(21) - a(29) - a(45) - a(53) - a( 5) - a(61)
a(38) = s8 - a(14) - a(22) - a(30) - a(46) - a(54) - a( 6) - a(62)
a(39) = s8 - a(15) - a(23) - a(31) - a(47) - a(55) - a( 7) - a(63)
a(45) = s8 - a(46) - a(47) - a(42) - a(43) - a(44) - a(41) - a(48)
a(53) = s8 - a(54) - a(55) - a(56) - a(49) - a(50) - a(51) - a(52)

with a(i) independent for i = 10, 11, 14, 15, 18, 19, 22, 23, 30, 31, 42, 43, 44, 46, 47, 50, 51, 52, 54, 55
and  a(i) defined     for i = 1 ... 8, 9, 16, 17, 24, 25, 32, 33, 40, 41, 48, 49, 56, 57 ... 64

Based on the equations listed above, guessing routines can be written to generate Prime Number Bordered Magic Cubes of order 8 within a reasonable time (PrimeCubes8).

Attachment 7.6.1 shows, for a few Magic Sums, the first occurring 8th order Prime Number Bordered Magic Cubes with Semi Magic Surface Planes.

7.6.3 Summary

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

Order

Main Characteristics

Subroutine

Results

5

Semi Magic Surface Planes (Makarova)

-

Attachment 7.3.1

     Magic Top/Bottom Planes

PrimeCubes5a

Attachment 7.3.3

     Magic Top/Bottom and Back/Front Planes

PrimeCubes5b

Attachment 7.3.4

     Magic Surface Planes (s-Magic)

PrimeCubes5c

Attachment 7.3.5

6

Semi Magic Surface Planes

PrimeCubes6

Ref. Sect. 7.4.2

     Magic Top/Bottom Planes

PrimeCubes6c

Attachment 7.4.8

     Magic Surface Planes (s-Magic)

PrimeCubes61

Attachment 7.4.9

7

Semi Magic Surface Planes

PrimeCubes7

Attachment 7.5.2

     Magic Surface Planes (s-Magic)

PrimeCubes71

Attachment 7.5.4

8

Semi Magic Surface Planes

PrimeCubes8

Attachment 7.6.1

-

-

-

-

Comparable routines as listed above, can be used to generate Associated Prime Number Magic Cubes, which will be described in following sections.

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