Office Applications and Entertainment, Magic Cubes

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8.0†††Magic Cubes (6 x 6 x 6)

8.1†††Historical Background


The historical development, from the first order 6 Simple Magic Cube to the order 6 Perfect Magic Cube and later the Pantriagonal Associated Magic Cube, can be summarised as follows:

Type

Author

Year

Simple Magic Cube

W. Firth

1889

Magic Center Planes

John Worthington

1910

Simple Magic Cube, Associated

Ir. Weidemann

1922

Pantriagonal Complete

Gahuko Abe

1948

Magic Border Planes (s-Magic)

Walter Trump

2003, Sept

Perfect Magic Cube

Walter Trump

2003, Sept.

Perfect Magic Cube

Mitsutoshi Nakamura

2004, July

Pantriagonal Associated

Mitsutoshi Nakamura

2008

The Magic Cubes listed above are shown in Attachment 8.1.1 and Attachment 8.1.2.

Following sections will describe and illustrate how comparable cubes can be constructed or generated.

8.2†††Simple Magic Cubes

An efficient method to generate Simple Magic Cubes of order 6 is described in Section 6.5.2.

Miscellaneous examples are shown in Attachment 6.5.2.

8.3†††Associated Magic Cubes

An efficient method to generate Associated Magic Cubes of order 6 is described in Section 6.5.3.

Miscellaneous examples are shown in Attachment 6.5.4.

8.4†††Magic Cubes with Magic Border Planes (s-Magic)

The first Magic Cube with Magic Border Planes, as constructed by Walter Trump (2003), was based on John Worthingtonís Magic Cube with Magic Center Planes (1910).

Section 6.5.4 describes an efficient method to generate Associated Magic Cubes with Magic Center Planes.

Attachment 6.5.41 contains miscellaneous examples of Associated Magic Cubes with Magic Center Planes.

Attachment 6.5.42 contains the corresponding Associated Magic Cubes with Magic Border Planes (s-Magic).

8.5†††Bordered Magic Cubes

8.5.1 Border Construction

Comparable with the method discussed in Section 4.3b, order 6 Bordered Magic Cubes can be constructed based on Complementary Anti Symmetric Magic Squares of order 6.

Examples of such squares, which can be used as top squares for Bordered Magic Cubes, are shown in Attachment 8.5.1.

The relation between opposite surface squares 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.

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 Magic Square 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(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 Magic Back Square 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, a guessing routine can be written to generate Bordered Magic Cubes of order 6 within a reasonable time (MgcCube6i).

Attachment 8.5.2 shows, for the Anti Symmetric Magic Squares enclosed in Attachment 8.5.1, the first occurring border.

For each suitable top square numerous borders can be generated (n6, dependent from the integers applied in subject top square).

Moreover, each border corresponds with:

  • 48 borders which can be obtained by means of rotation and reflection;
  • 4 borders which can be obtained by interchanging plane n with plane (7 - n) for n = 2, 3;
  • 2 borders which can be obtained by permutating plane 2, 3 and simultaneously plane 5, 4.

Consequently each border corresponds with 2 * 4 * 48 * n6 = 384 * n6 suitable borders.

Attachment 8.5.3 shows for Border C002 the eight borders which can be obtained from each other by plane exchange.

8.5.2 Center Cubes

Any of the following order 4 Magic Cubes, based on the integers 77, 78 ... 140 = (1, 2 ... 64) + 76, can be used as Center Cube for the borders deducted in previous section:

  • Simple, Associated
  • Simple, Associated and 3D-Compact
  • Simple, Associated with Horizontal Magic Planes
  • Simple, Horizontal Associated Magic Planes
  • Simple, Horizontal Pan Magic Planes (3D-Compact)
  • Simple, Plane Symmetrical
  • Simple, Plane Symmetrical with Horizontal Magic Planes

  • Pantriagonal, Complete
  • Pantriagonal, Complete with Horizontal Magic Planes
  • Pantriagonal, 2D-Compact
  • Pantriagonal, 2D-Compact and Plane Symmetrical
  • Pantriagonal, 2D Compact and Complete
  • Pantriagonal, Associated

In general the resulting Bordered Magic Cube will be s-Magic. For Center Cubes with Horzontal Magic Planes, the six horizontal planes will be magic.

It should be noted that order 4 Almost Perfect Magic Cubes are not suitable, as the Space Diagonals don't sum to the Magic Sum s4 (ref. Section 3.3).

However order 4 Plane Symmetrical Cubes with Horizontal Magic Planes can be used for the construction of Concentric Perfect Magic Cubes, which will be discussed in Section 8.6.

8.6†††Perfect Magic Cubes

8.6.1 Center Cubes

As mentioned above order 4 Almost Perfect Magic Cubes are not suitable for the construction of Concentric Magic Cubes.

This would require a border with corner pairs which are non symmetric over the diagonals, which is not possible (ref. Exhibit VIIIa).

The Perfect Concentric Magic Cubes as constructed by Walter Trump (2003) and Mitsutoshi Nakamura (2004) are based on order 4 Plane Symmetrical Cubes with Horizontal Magic Planes (ref. Attachment 8.1.1).

These center cubes have the additional property that the vertical plane diagonals sum to 2 * s4 per plane, which facilitates the border construction.

8.6.2 Border Construction

Mitsutoshi Nakamura applied a border for which the top and bottom square are concentric, with exception of the corner points which are symmetric over the space diagonals.

This border type allows for a construction comparable with the border described in Section 8.5.1 above (ref. Exhibit VIIIb).

Attachment 8.6.2 shows some additional order 6 Perfect Concentric Magic Cubes, based on a selection of order 4 Plane Symmetrical Cubes.

Attachment 8.6.3 shows some additional order 6 Perfect Concentric Magic Cubes, based on miscellaneous top squares.

8.6.3 Transformations

Comparable with order 6 Magic Squares (ref. 'Magic Squares' Section 6.3), Perfect Magic Cubes of order 6 might be subject to following transformations:

  • Any plane n can be interchanged with plane (7 - n), as well as the combination of these permutations.
    The possible number of unique transformations is 23 / 2 = 4.

  • Any permutation can be applied to the planes 1, 2, 3 provided that the same permutation is applied to the planes 6, 5, 4. The possible number of transformations is 3! = 6.

  • Combination of abovementioned transformations will result in 24 unique solutions, which are shown in Attachment 8.6.4.

Note: Secondary properties, like the applied symmetry, are not invariant to the transformations described above.

Based on these 24 transformations and the 48 cubes which can be found by means of rotation and/or reflection any 6th order (Perfect) Magic Cube corresponds with a Class of 24 * 48 = 1152 (Perfect) Magic Cubes.

8.6.4 Enumeration (Partial)

Although a complete enumeration of order 6 Perfect Concentric Magic Cubes is beyond the scope of this section, a partial enumeration can be made based on the results of previous sections.

The number of surface planes which can be generated with the variables of the edge constant are:

  • 256 Top††Squares based on the 32 remaining integers of the top and bottom squares;
  • ††2 Left Squares based on the 32 remaining integers of the left and right squares;
  • ††1 Back Square††based on the 32 remaining integers of the back and front squares.

The number of suitable Plane Symmetrical Center Cubes which can be generated with the edge constant is 128.

This results in 2 * 256 * 128 = 65536 Perfect Concentric Magic Cubes, not counting rotation, reflection or transformation as discussed in Section 8.6.3 above.

8.6.5 Higher Order Perfect Concentric Magic Cubes

Mitsutoshi Nakamura has proven that Perfect Concentric Magic Cubes can be constructed for any even order higher than 4, and provides on his website examples of such cubes for order 6 to 40.

8.7†††Pantriagonal Magic Cubes

Mitsutoshi Nakamura provides on his website, amongst others, algorithms to construct Pantriagonal Magic Cubes of order m = 4x + 2 for m >= 6.

8.7.1 Pantriagonal and Complete

The algorithm to construct a Pantriagonal Complete Magic Cube of order m = 6 has been incorporated in procedure CnstrPntr6. The resulting cube is shown below:

Plane 1
1 44 132 207 164 103
41 129 7 167 106 201
135 4 38 100 204 170
198 155 121 28 71 78
158 124 192 68 75 34
118 195 161 81 31 65
Plane 2
42 127 8 166 108 200
133 5 39 102 203 169
2 45 130 206 163 105
157 126 191 69 73 35
120 194 160 79 32 66
197 154 123 29 72 76
Plane 3
134 6 37 101 202 171
3 43 131 205 165 104
40 128 9 168 107 199
119 193 162 80 33 64
196 156 122 30 70 77
159 125 190 67 74 36
Plane 4
189 146 139 19 62 96
149 142 183 59 93 25
136 186 152 99 22 56
10 53 114 216 173 85
50 111 16 176 88 210
117 13 47 82 213 179
Plane 5
148 144 182 60 91 26
138 185 151 97 23 57
188 145 141 20 63 94
51 109 17 175 90 209
115 14 48 84 212 178
11 54 112 215 172 87
Plane 6
137 184 153 98 24 55
187 147 140 21 61 95
150 143 181 58 92 27
116 15 46 83 211 180
12 52 113 214 174 86
49 110 18 177 89 208

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

Consequently a Pantriagonal Magic Cube belongs to a collection of 63 * 48 = 10368 elements which can be found by means of rotation, reflection or planar shifts.

The Class of 48 elements which can be obtained by rotation/reflection of a Pantriagonal Magic Cube is shown in Attachment 8.7.2.

The Class of 216 elements which can be obtained by planar shifts of a Pantriagonal Magic Cube is shown in Attachment 8.7.1. Each cube of Attachment 8.7.2 can be used as a Base for Attachment 8.7.1.

It should be noted that the planar shifts are from right to left (L1 ... L5), from front to back (B1 ... B5) and from bottom to top (T1 ... T5).

The cube shown above is essential different from the cube of Gahuko Abe as it canít be obtained by any of the operations described above (rotation, reflection, planar shifts).

8.7.2 Pantriagonal and Associated

The algorithm to construct a Pantriagonal Associated Magic Cube of order m = 6 has been incorporated in procedure AssPntr6. The resulting cube is shown below:

Plane 1
1 79 191 80 192 108
102 171 47 62 100 169
193 103 194 77 6 78
105 60 104 167 49 166
196 73 8 101 198 75
54 165 107 164 106 55
Plane 2
36 129 143 182 34 127
160 190 161 2 135 3
141 132 32 131 31 184
130 4 158 5 159 195
30 189 29 134 136 133
154 7 128 197 156 9
Plane 3
199 97 200 71 12 72
39 180 92 179 91 70
202 67 14 95 204 69
96 177 41 68 94 175
16 64 206 65 207 93
99 66 98 173 43 172
Plane 4
45 174 44 119 151 118
124 10 152 11 153 201
42 123 149 176 40 121
148 13 122 203 150 15
147 126 38 125 37 178
145 205 146 17 120 18
Plane 5
208 61 20 89 210 63
84 81 83 188 28 187
22 58 212 59 213 87
33 186 86 185 85 76
214 82 215 56 27 57
90 183 35 74 88 181
Plane 6
162 111 53 110 52 163
142 19 116 209 144 21
51 168 50 113 157 112
139 211 140 23 114 24
48 117 155 170 46 115
109 25 137 26 138 216

Although other Pantriagonal Magic Cubes can be constructed by means of rotation, reflection and/or planar shifts, the associated property is not invariant to planar shifts.

Attachment 8.7.3 shows some additional Pantriagonal Associated Magic Cubes which could be found with the method described in Exhibit VIIIc.

8.8†††Summary

The obtained results regarding the miscellaneous types of order 6 Magic Cubes as deducted and discussed in previous sections are summarized in following table:

Type

Characteristics

Subroutine

Results

Simple

Classic

MgcCube6a

Attachment 6.5.2

Associated

Classic

MgcCube6b

Attachment 6.5.4

Magic Center Planes

MgcCube6c

Attachment 6.5.41

Magic Border Planes (s-Magic)

-

Attachment 6.5.42

Bordered

Symmetrical Edges

MgcCube6i

Attachment 8.5.2
Attachment 8.5.3

Perfect

Concentric, Miscellaneous Center Cubes

-

Attachment 8.6.2

Concentric, Miscellaneous Top Squares

-

Attachment 8.6.3

Plane Permutations

-

Attachment 8.6.4

Pantriagonal

Complete

CnstrPntr6

Attachment 8.7.1
Attachment 8.7.2

Associated

AssPntr6

Attachment 8.7.3

Next section will provide some methods for the construction and generation of order 7 Magic Cubes.


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