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 7.0    Special Cubes, Prime Numbers 7.10   Magic Cubes, Sum of Latin Cubes (1) 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 for each set of Correlated Magic Lines shown above, an example of the resulting Prime Number Magic Cubes with the related Magic Sums.

7.10.3 Simple Latin Cubes (4 x 4 x 4)
Horizontal Latin Diagonal Planes

It can be noticed in the example above that the horizontal planes are magic as well, due to the applied Latin Cubes (Horizontal Latin Diagonal Planes).

Simple Latin Cubes with Horizontal Latin Diagonal Planes, can be obtained by applying following set of linear equations:

 Plane 1 (Top) 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(56) - a(59) - a(60) + a(61) - a(64) 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) Plane 2 a(45) = s1 - a(46) - a(47) - a(48) a(41) = s1 - a(42) - a(43) - a(44) a(39) = a(40) - a(42) + a(44) - a(45) + a(48) a(38) = s1 - a(40) - a(43) - a(44) + a(45) - a(48) a(37) = s1 - a(38) - a(39) - a(40) a(36) = s1 - a(40) - a(44) - a(48) 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)

Plane 3

```a(29) =    s1 - a(30) - a(31) - a(32)
a(27) = -2*s1 + a(32) + a(40) + a(43) + a(44) - a(45) + 2 * a(48) + a(54) + a(59) + 2 * a(64)
a(26) =         a(29) - a(40) + a(42) - a(44) + 2 * a(45) - a(48) + a(56) + a(60) - a(62) - a(63)
a(25) =    s1 - a(26) - a(27) - a(28)
a(24) =    s1 - a(28) + a(29) - a(32) - a(40) - a(44) + a(45) - a(48)
a(23) =    s1 - a(29) - a(42) - a(45) + a(54) + a(59) - 2 * a(61)
a(22) =    s1 - a(23) - a(26) - a(27)
a(21) =    s1 - a(22) - a(23) - a(24)
a(20) =    s1 - a(24) - a(28) - a(32)
a(19) =    s1 - a(23) - a(27) - a(31)
a(18) =    s1 - a(22) - a(26) - a(30)
a(17) =    s1 - a(18) - a(19) - a(20)
```
 Plane 4 a(8) = s1 - a(24) - a(40) - a(56) a(7) = s1 - a(23) - a(39) - a(55) a(6) = s1 - a(22) - a(38) - a(54) a(5) = s1 - a(21) - a(37) - a(53) a(4) = s1 - a(20) - a(36) - a(52) a(3) = s1 - a(19) - a(35) - a(51) a(2) = s1 - a(18) - a(34) - a(50) a(1) = s1 - a(17) - a(33) - a(49) 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) = s1 - a(29) - a(45) - a(61) 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)

For Latin Cubes based on the integers {0, 1, 2, 3} the related Magic Sum s1 = 6.

The equations for Plane 1 and 2 cover (standard) Latin Diagonal Squares (48 ea) which can be read from an external source (ref. Attachment 7.10.5).

Based on this collection and the above listed equations for Plane 3 and 4, a fast routine can be written to generate the defined Simple Latin Cubes of order 4 (ref. LtnCbs4a).

Subject routine produced 96 Simple Latin Cubes with Horizontal Latin Diagonal Planes within 0,82 seconds, which are shown in Attachment 7.10.6.

Prime Number Simple Magic Cubes D can be generated by selecting combinations of Latin Cubes (A, B, C) while:

• substituting the integers {0, 1, 2, 3} by {ai, i = 1 ... 4}, {bj, j = 1 ... 4} and {ck, k = 1 ... 4};
• ensuring that for the resulting cube D the numbers dijk = ai + bj + ck (i,j,k = 1 ... 4) are distinct prime numbers (64 ea).

This can be achieved with routine CnstrCbs4b, which counted 331776 of subject Prime Number Simple Magic Cubes of the 4th order within 1600 seconds (per magic sum).

Attachment 7.10.2 shows for each set of Correlated Magic Lines shown in Section 7.10.2 above, an example of the resulting Prime Number Magic Cubes with Horizontal Magic Planes.

7.10.4 Simple Latin Cubes (4 x 4 x 4)
Latin Planes, Generalised

More general Simple Latin Cubes can be based on Latin Planes, for which Plane 1 and 2 can be read from an external source with the 576 (standard) Latin Squares (ref. Attachment 7.10.7).

The defining equations for Plane 3 can be written as:

```a(29) = s1 - a(30) - a(31) - a(32)
a(27) =      a(32) - a(38) + a(48) + a(53) + a(57) - a(62) - a(63)
a(26) =      a(29) - a(39) + a(45) + a(56) + a(60) - a(62) - a(63)
a(25) = s1 - a(26) - a(27) - a(28)
a(23) = s1 - a(24) - a(28) - a(32) + a(36) - a(42) - a(56) - a(60) + a(62) + a(63)
a(21) = s1 - a(22) - a(23) - a(24)
a(20) = s1 - a(24) - a(28) - a(32)
a(19) = s1 - a(23) - a(27) - a(31)
a(18) = s1 - a(22) - a(26) - a(30)
a(17) = s1 - a(18) - a(19) - a(20)
```

The defining equations for Plane 4 are identical to those shown in Section 7.10.3 above.

For Latin Cubes based on the integers {0, 1, 2, 3} the related Magic Sum s1 = 6.

Based on the collection of Latin Squares and the equations for Plane 3 and 4, a fast routine can be written to generate the defined Simple Latin Cubes of order 4 (ref. LtnCbs4b).

Subject routine produced 8160 Simple Latin Cubes within 86 seconds, of which the first 48 are shown in Attachment 7.10.8.

Prime Number Simple Magic Cubes D can be generated by selecting combinations of Latin Cubes (A, B, C) while:

• substituting the integers {0, 1, 2, 3} by {ai, i = 1 ... 4}, {bj, j = 1 ... 4} and {ck, k = 1 ... 4};
• ensuring that for the resulting cube D the numbers dijk = ai + bj + ck (i,j,k = 1 ... 4) are distinct prime numbers (64 ea).

This can be achieved with routine CnstrCbs4b, which generated, with A (Cube 1) and B (Cube 48) constant, 96 Prime Number Simple Magic Cubes of the 4th order within 21 seconds.

Numerous Prime Number Simple Magic Cubes can be generated by a careful (pre) selection of A and B based on the results of Section 7.10.3 above.

7.10.5 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 7.10.7 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.

7.10.6 Combined Magic Squares (8 x 8)

An order 4 Magic Square C2 can be constructed based on Latin Diagonal 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 = A + B
 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 Latin Diagonal Square of order 4, based on the third Magic Line {ck, k = 1 ... 4} as illustrated 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. 7.10.7 Summary The obtained results regarding the miscellaneous Prime Number Magic Cubes and - Squares as deducted and discussed in previous sections are summarized in following table:
 Type Main Characteristics Subroutine Results - - - - Magic Cubes Simple, Horizontal Magic Planes Latin Cubes Simple, Hor Latin Diagonal Planes Simple, Latin Planes Magic Squares Order 8, Composed - Order 8, Combined - - - - -
 Following sections will describe and illustrate how Prime Number (Simple) Magic Cubes can be constructed based on the sum of two suitable selected Latin Cubes.