Office Applications and Entertainment, Magic Cubes

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 d_i = a_i + b_i + c_i, i = 1 ... 64 as illustrated below:

The key to possible solutions is to find Correlated Magic Lines {a_i, i = 1 ... 4}, {b_j, j = 1 ... 4} and {c_k, k = 1 ... 4} such that d_ijk = a_i + b_j + c_k (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:

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 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)

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 {a_i, i = 1 ... 4}, {b_j, j = 1 ... 4} and {c_k, k = 1 ... 4};
• ensuring that for the resulting cube D the numbers d_ijk = a_i + b_j + c_k (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 4^th 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 {a_i, i = 1 ... 4}, {b_j, j = 1 ... 4} and {c_k, k = 1 ... 4};
• ensuring that for the resulting cube D the numbers d_ijk = a_i + b_j + c_k (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 4^th 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 {a_i, i = 1 ... 4} and {b_j, j = 1 ... 4}:

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

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:

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.