Office Applications and Entertainment, Magic Cubes  
Index  About the Author 
7.0 Special Cubes, Prime Numbers
7.8 Perfect Concentric Magic Cubes (6 x 6 x 6)
The Perfect Concentric Magic Cubes as constructed by Walter Trump (2003) and Mitsutoshi Nakamura (2004) for consecutive integers,
are based on order 4 Plane Symmetrical Center Cubes with Horizontal Magic Planes, as
7.8.2 Border Construction (General)
This section describes the construction of order 6 Prime Number Perfect Concentric Magic Cubes based on the application of:
The construction method, based on this principle, can be summarized as follows:
The relation between the top and bottom plane can be represented as follows:






Plane 1 (Top)

a(5) = s6 / 3  a(35) a(4) = s6 / 3  a(34) a(3) = s6 / 3  a(33) a(2) = s6 / 3  a(32) 
a(25) = s6 / 3  a(30) a(19) = s6 / 3  a(24) a(13) = s6 / 3  a(18) a(7) = s6 / 3  a(12) 

= 


= 

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
Based on the equations listed above, guessing routines can be written to generate Prime Number Perfect Concentric Magic Cubes of order 6 within a reasonable time
(PrimeCubes78).
Comparable with order 6 Magic Squares, Perfect Magic Cubes of order 6 might be subject to following transformations:
Note: Secondary properties, like the applied symmetry, are not invariant to the transformations described above.
The obtained results regarding the miscellaneous Prime Number Perfect Concentric Magic Cubes as deducted and discussed in previous sections are summarized in following table: 
Order
Main Characteristics
Subroutine
Results




6
Perfect Magic Cubes, Concentric
Perfect Magic Cubes, Plane Permutations





Following sections will describe and illustrate how Prime Number (Simple) Magic Cubes can be constructed based on the sum of three suitable selected Latin Cubes.

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