Office Applications and Entertainment, Magic Cubes

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7.0    Special Cubes, Prime Numbers

7.10   Magic Cubes, Sum of Latin Cubes

7.10.1 Introduction

Comparable with Prime Number Magic Squares (ref. Section 14.12), Prime Number Magic Cubes can be constructed based on suitable selected Latin Cubes.

7.10.2 Simple Magic Cubes (4 x 4 x 4)

The elements of three Latin Cubes A, B and C, with latin space diagoanls, might result in a Simple Prime Number Magic Cube D with elements di = ai + bi + ci, i = 1 ... 64 as illustrated below:

A
1650 2868 168 390
168 390 1650 2868
390 168 2868 1650
2868 1650 390 168
B
1061 659 4091 2999
2999 4091 659 1061
659 1061 2999 4091
4091 2999 1061 659
C
2 0 3192 5252
5252 3192 0 2
0 2 5252 3192
3192 5252 2 0
MC = 22332
2713 3527 7451 8641
8419 7673 2309 3931
1049 1231 11119 8933
10151 9901 1453 827

2868 1650 390 168
390 168 2868 1650
168 390 1650 2868
1650 2868 168 390

659 1061 2999 4091
4091 2999 1061 659
1061 659 4091 2999
2999 4091 659 1061

5252 3192 0 2
2 0 3192 5252
3192 5252 2 0
0 2 5252 3192

8779 5903 3389 4261
4483 3167 7121 7561
4421 6301 5743 5867
4649 6961 6079 4643

168 390 1650 2868
1650 2868 168 390
2868 1650 390 168
390 168 2868 1650

4091 2999 1061 659
659 1061 2999 4091
2999 4091 659 1061
1061 659 4091 2999

0 2 5252 3192
3192 5252 2 0
2 0 3192 5252
5252 3192 0 2

4259 3391 7963 6719
5501 9181 3169 4481
5869 5741 4241 6481
6703 4019 6959 4651

390 168 2868 1650
2868 1650 390 168
1650 2868 168 390
168 390 1650 2868

2999 4091 659 1061
1061 659 4091 2999
4091 2999 1061 659
659 1061 2999 4091

3192 5252 2 0
0 2 5252 3192
5252 3192 0 2
2 0 3192 5252

6581 9511 3529 2711
3929 2311 9733 6359
10993 9059 1229 1051
829 1451 7841 12211

The key to possible solutions is to find Correlated Magic Lines {ai, i = 1 ... 4}, {bj, j = 1 ... 4} and {ck, k = 1 ... 4} such that dijk = ai + bj + ck (i,j,k = 1 ... 4) are distinct prime numbers (64 ea).

A few of such sets of Correlated Magic Lines are shown in following table:

a1 a2 a3 a4 - b1 b2 b3 b4 - c1 c2 c3 c4 - MC4
0 42 1470 1932 - 17 149 197 2549 - 0 2 780 1170 - 8308
0 138 1260 2928 - 41 1289 2591 5501 - 0 2 72 572 - 14394
0 54 264 1302 - 17 1667 4217 4967 - 0 2 2562 4160 - 19212
168 390 1650 2868 - 659 1061 2999 4091 - 0 2 3192 5252 - 22332
0 30 402 2550 - 239 2969 4019 13679 - 0 2 222 3302 - 27414
0 138 768 2508 - 41 281 5501 25031 - 0 2 572 1848 - 36690
0 30 2310 3948 - 71 281 1019 19961 - 0 2 782 9240 - 37644
0 48 168 3948 - 101 179 13709 17789 - 0 2 882 1430 - 38256

Attachment 7.10.2 shows the resulting Prime Number Magic Cubes (8 ea) with the related Magic Sums.

Note: It should be noted that, due to the selected Latin Cubes, the Horizontal Planes are magic as well, as shown in the example above.

7.10.3 Related Magic Squares (8 x 8)

Based on the Prime Number Magic Cubes, as discussed in Section 7.10.2 above, following 8 x 8 Prime Number Magic Squares can be constructed:

  • Composed Magic Squares, based on the horizontal Magic Planes of subject cubes;
  • Combined Magic Squares, based on an alternative combination of the Correlated Magic Lines,
    which will be illustrated in Section 7.10.4 below.

Attachment 7.10.3 shows the Composed Magic Squares (8 ea) with the related Magic Sums, based on the Magic Cubes enclosed in Attachment 7.10.2.

7.10.4 Combined Magic Squares (8 x 8)

An order 4 Magic Square C2 can be constructed based on Diagonal Latin Squares A and B, defined by two of the three Correlated Magic Lines, say {ai, i = 1 ... 4} and {bj, j = 1 ... 4}:

A
1650 2868 168 390
168 390 1650 2868
390 168 2868 1650
2868 1650 390 168
B
1061 659 4091 2999
2999 4091 659 1061
659 1061 2999 4091
4091 2999 1061 659
C2
2711 3527 4259 3389
3167 4481 2309 3929
1049 1229 5867 5741
6959 4649 1451 827

The resulting square C2 can be combined with 4 ea Diagonal Latin Squares of order 4, based on the third Magic Line {ck, k = 1 ... 4} as shown below:

Latin Squares (4 ea)
2 0 3192 5252 2 0 3192 5252
5252 3192 0 2 5252 3192 0 2
0 2 5252 3192 0 2 5252 3192
3192 5252 2 0 3192 5252 2 0
2 0 3192 5252 2 0 3192 5252
5252 3192 0 2 5252 3192 0 2
0 2 5252 3192 0 2 5252 3192
3192 5252 2 0 3192 5252 2 0
Combined Square (8 x 8)
2713 2711 6719 8779 4261 4259 6581 8641
7963 5903 3527 3529 9511 7451 3389 3391
3167 3169 9733 7673 2309 2311 9181 7121
6359 8419 4483 4481 5501 7561 3931 3929
1051 1049 4421 6481 5869 5867 8933 10993
6301 4241 1229 1231 11119 9059 5741 5743
6959 6961 9901 7841 1451 1453 6079 4019
10151 12211 4651 4649 4643 6703 829 827

Attachment 7.10.4 shows Combined Magic Squares (8 ea) with the related Magic Sums, based on the Magic Cubes enclosed in Attachment 7.10.2.

Each square shown corresponds, for a given square C2, with numerous Prime Number Combined Magic Squares (order of magnitude 484 = 5308416).


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