Office Applications and Entertainment, Magic Cubes  
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6.0 Construction Methods (Higher Order)
In previous sections several procedures were developed for sequential generation of Magic Cubes of order 3, 4 and 5, based on the linear equations describing subject Magic Cubes.
6.2 Composition by means of Sub Cubes
Comparable with the method discussed in Section 9.8 of 'Magic Squares'
it is possible to compose a 9^{th} order Simple Magic Cube out of 27 Magic Cubes of order 3, each with 27 consecutive integers and corresponding Magic Sum.
An example obtained by the method described above is shown in Attachment 6.2.1,
together with Base Cube C_{i}
and Multiplier Cube B.
With 192 possible cubes for both B and C_{i}
(i = 1 ... 27), the resulting number of 9^{th} order Magic Cubes with Magic Sum 3285 will be
192 * 192^{27} = 8,56 10^{63}.
6.3 Composition by means of Factor Cubes
The consecutive integers c_{i} (i = 1 ... m^{3}) of a Magic Cube C of order m with the values 1 ... m^{3} can be written as:
(c_{i}  1) = b_{1} +
m * b_{2}
+ m^{2} * b_{3}
with b_{j} = 0, 1, 2, ... m  1 for j = 1, 2, 3
Consequently any Magic Cube C of order m with the numbers 1 ... m^{3} can be written as:
C =
B1 +
m * B2 +
m^{2} * B3 +
[1]
where the matrices B1,
B2 and
B3
 further referred to as Factor Cubes  contain only the integers 0, 1, 2, 3 ... m  1.
6.3.2 Pantriagonal Magic Cubes
Associated Pantriagonal Magic Cubes (m = 2x + 1, m >= 5, m Mod 3 ≠ 0)
Based on a decomposition (ref. Attachment 6.3.1) of John Hendricks 7^{th} order Associated Pantriagonal Cube (1973), procedure AssPntr21 could be built, which generates the Factor Cubes
B1,
B2,
B3 and the resulting cube
C.
Associated Pantriagonal Magic Cubes (m = 2x + 1, m >= 5, m Mod 3 = 0)
To deal with orders m for which m Mod 3 = 0, Mitsutoshi Nakamura applies a transformation S_{m,3}(x), which
transforms the Factor Cubes
B1,
B2 and
B3
as generated by procedure AssPntr21, into Factor Cubes
S_{m,3}(B1),
S_{m,3}(B2) and
S_{m,3}(B3)
which result in Associated Pantriagonal Magic Cubes.
Based on comparable principles, Mitsutoshi Nakamura developed  amongst others  methods to construct following types of Magic Cubes:
Associated Pantriagonal Magic Cubes (m = 4x)
This method has been incorporated in procedure AssPntr23 which generates the Factor Cubes
and resulting Cube for the defined order.
Nonassociated Pantriagonal Magic Cubes, 2Dcompact and Complete (m = 4x)
This method has been incorporated in procedure CnstrPntr4x which generates the Factor Cubes
and resulting Cube for the defined order.
More algorithms for higher order (single even) Associated Pantriagonal Magic Cubes are available on
Mitsutoshi Nakamura's website regarding the subject.
6.3.3 Pandiagonal Magic Cubes (Odd)
Associated Pandiagonal Magic Cubes (m = 2x+1, m >= 7, gcd(m, 3 x 5) = 1)
This amazing simple method has been incorporated in procedure AssPanDia21 which generates the Factor Cubes
and resulting Cube for the defined order.
Associated Pandiagonal Magic Cubes (m = 2x+1, m >= 7, 1 < gcd(m, 3 x 5) < m)
To deal with orders m for which either m Mod 3 = 0 or m Mod 5 = 0 a transformation S_{m,q}(x), with q = gcd(m, 3 x 5),
has been applied which transforms the Factor Cubes
B1,
B2 and
B3
as generated by procedure AssPanDia21, into Factor Cubes
S_{m,q}(B1),
S_{m,q}(B2) and
S_{m,q}(B3)
which result in Associated Pandiagonal Magic Cubes.
Associated Pandiagonal Magic Cubes (m = 15)
For this particular case a transformation
S_{15}(x) = R_{3,5}(x mod 3, x mod 5)  1
has been applied, which transforms the Factor Cubes
B1,
B2 and
B3
as generated by procedure AssPanDia21, into Factor Cubes
S_{15}(B1),
S_{15}(B2) and
S_{15}(B3)
which result in an Associated Pandiagonal Magic Cube of order 15.
6.3.5 Pandiagonal Pantriagonal Magic Cubes (Nasik, m = odd)
Nasik, Associated (m = 2x+1, m >= 9, gcd(m, 3x5x7) = 1)
This method has been incorporated in procedure AssNasik21 which generates the Factor Cubes
and resulting Cube for the defined order.
Nasik, Associated (m = 2x+1, m >= 9, 1 < gcd(m, 3x5x7) < m)
This method applying a transformation S_{m,q}(x), with q = gcd(m, 3 x 5 x 7),
which transforms the Factor Cubes as generated by procedure AssNasik21
into Factor Cubes
S_{m,q}(B1),
S_{m,q}(B2) and
S_{m,q}(B3),
has been incorporated in procedure AssNasik22 which generates the Factor Cubes
and resulting Associated Nasik Magic Cubes for the defined order.
Nasik, Associated (m = 2x+1, m >= 9, and gcd(m, 3x5x7) = m )
In this case m = 15, m = 21, m = 35 or m = 105 a
transformation S_{m}(x)
has to be applied to transform the Factor Cubes
B1,
B2 and
B3
as generated by procedure AssNasik21, into Factor Cubes
S_{m}(B1),
S_{m}(B2) and
S_{m}(B3)
which result in Associated Nasik Magic Cubes of order m.
More algorithms for higher order even (Associated and/or 3DCompact) Nasik Magic Cubes are available on Mitsutoshi Nakamura's website regarding the subject.
6.4 Knight Jump Method (m = prime, m >= 7)
Based on Euler's knight jump method Yoshi Tamori developed following algorithm to construct Pantriagonal Magic Cubes for prime orders m >= 7:
The algorithm described above has been applied in following Interactive Solution (Java Script):
Procedure:
The Script will only work when your browser supports Scripts and when the Script support option(s) is (are) enabled. You can view the results for m = 7, m = 11 and m = 13 in Attachment 6.4.1.

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