Office Applications and Entertainment, Magic Squares | ||
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14.0 Special Magic Squares, Prime Numbers
14.8 Magic Squares (10 x 10), Part I
14.8.1 Magic Squares (10 x 10), Composed (1)
Prime Number Magic Squares of order 10 with a Magic Sum s10 can be composed out of:
as illustrated below:
Based on the principles described in previous section a comparable procedure (Priem10a1) can be developed:
Following attachments show the first occurring 10th order Prime Number Composed Magic Square for miscellaneous 6th order Corner Squares and Magic Sums:
Each square shown corresponds with numerous squares for the same Magic Sum.
14.8.2 Magic Squares (10 x 10), Composed (2)
Alternatively Prime Number Magic Squares of order 10 - composed out of Sub Squares as described in Section 14.8.1 - can be arranged as shown below:
Based on the principles described in Section 14.8.1 above, a comparable procedure
(Priem10a2) can be developed.
Following attachments show the first occurring 10th order Prime Number Composed Magic Square for miscellaneous 6th order Corner Squares and Magic Sums:
Each square shown corresponds with numerous squares for the same Magic Sum.
14.8.3 Magic Squares (10 x 10), Composed (3)
Alternatively Prime Number Magic Squares of order 10 with a Magic Sum s10, can be composed out of:
and arranged as illustrated below:
Based on the principles described above a comparable procedure (Priem10b) can be developed:
Attachment 14.8.56 shows for miscellaneous Magic Sums the first occurring 10th order Prime Number Composed Magic Square, based on 6th order Eccentric Magic Squares (ref. Attachment 14.4.5).
Other Collections of Sub Squares D might be used, provided that the key condition d1 + d8 = d2 + d7
= s10 / 5 is fulfilled.
The elements of both collections can be obtained by moving the row and columns such that the original 2 x 2 Center Square becomes the Top-Left Corner Square. The resulting Magic Squares are still (Partly) Compact and Pan Diagonal.
Each square shown corresponds with numerous squares for the same Magic Sum.
14.8.4 Magic Squares (10 x 10), Composed (4)
Alternatively Prime Number Magic Squares of order 10 with a Magic Sum s10, can be composed out of:
and arranged as illustrated below:
The Center Cross might be constructed with the method of Al Antaki (10th century):
A procedure (Priem10g) can be developed to:
Attachment 14.8.41 shows the first occurring 10th order Prime Number Composed Magic Square for a few Magic Sums.
Each Center Cross corresponds with (8!) * (8!) = 1,6 109 Center Crosses,
which can be obtained by permutation of the horizontal and vertical pairs.
With the Center Cross fixed, each square corresponds with 4! * 3844 = 0,5 1012
squares.
14.8.5 Magic Squares (10 x 10) Alternatively Prime Number Magic Squares of order 10 with a Magic Sum s10, can be composed out of:
as illustrated below:
Based on this definition a comparable procedure (ref. Priem10c2) can be developed:
Following attachments show the first occurring 10th order Prime Number Composed Magic Square for miscellaneous 6th order Corner Squares and Magic Sums:
Each square shown corresponds with numerous squares for the same Magic Sum.
14.8.7 Concentric Magic Squares (10 x 10)
A 10th order Prime Number Concentric Magic Square consists of a Prime Number Concentric Magic Square of the 8th order with a border around it.
This results in following border equations: |
a(10) = s5 - a( 9) - a( 8) - a( 7) - a(6)
a(50) = s5 - a(40) - a(30) - a(20) -a(10)
a(91) = s10/2 - a(10)
a(96) = s10/2 - a( 6)
a(97) = s10/2 - a( 7)
a(98) = s10/2 - a( 8)
a(99) = s10/2 - a( 9)
a(11) = s10/2 - a(20)
a(21) = s10/2 - a(30)
a(31) = s10/2 - a(40)
a(41) = s10/2 - a(50)a( 1) = s5 - a( 2) - a( 3) - a( 4) - a(5)
a( 81) = s5 - a(71) - a(61) - a(51) - a(1)
a(100) = s10/2 - a( 1)
a( 92) = s10/2 - a( 2)
a( 93) = s10/2 - a( 3)
a( 94) = s10/2 - a( 4)
a( 95) = s10/2 - a( 5)
a( 60) = s10/2 - a(51)
a( 70) = s10/2 - a(61)
a( 80) = s10/2 - a(71)
a( 90) = s10/2 - a(81)
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which enable the development of a fast procedure to generate Prime Number Concentric Magic Squares of order 10 (ref. Priem10c).
Each square shown corresponds with numerous squares for the same Magic Sum.
14.8.8 Bordered Magic Squares (10 x 10), Miscellaneous Inlays
Based on the collections of 8th order Magic Squares, as deducted in Section 14.6.1 thru 14.6.3, Section 14.6.5, and Section 14.6.7 thru 14.6.10, also following Bordered Magic Squares can be generated with routine Priem10c:
It should be noted that the Attachments listed above contain only those solutions which could be found within 100 seconds.
14.8.9 Bordered Magic Squares (10 x 10), Split Border
Alternatively a 10th order Bordered Magic Square with Magic Sum s10 can be constructed based on:
as illustrated below:
As the first border - as specified above - occurs for s10 = 10850, it is convenient to construct the border first and the 6th order Magic Center Square later (based on the remainder of the available pairs).
The Magic Center Square can be added with a separate routine e.g. Priem10e2 for Concentric, Partly Compact, Magic Center Squares as discussed in Section 14.4.4.
14.8.10 Bordered Magic Squares (10 x 10), Composed Border
Another type of order 10 Bordered Magic Squares with Magic Sum s10 can be constructed based on:
as illustrated below:
It is convenient to construct the Composed Border first and the 4th order Magic Center Square later (based on the remainder of the available pairs).
The Magic Center Square can be added with a separate routine e.g. Priem10f2 for Associated Magic Center Squares as discussed in Section 14.2.3.
Note:
the 10th order Composed Magic Square will be associated, if the opposite Magic Rectangles (3 x 4) are Anti Symmetric and Complementary as well.
14.8.11 Eccentric Magic Squares (10 x 10)
Also for Prime Number Eccentric Magic Squares of order 10 it is convenient to split the supplementary rows and columns into: two parts summing to s3 = 3 * s10 / 10 and one part summing to s4 = 4 * s10 / 10.
This enables, based on the same principles, the development of a set of fast procedures (ref. Priem10d):
Attachment 14.8.82 shows,
based on the 8th order Eccentric Magic Squares as discussed in Section 14.6.6,
one Prime Number Eccentric Magic Square for some of the occurring Magic Sums.
The obtained results regarding the miscellaneous types of order 10 Prime Number Magic Squares as deducted and discussed in previous sections are summarized in following table: |
Type
Characteristics
Subroutine
Results
Composed
Sub Sqrs Order 4 (3 ea) and 6 (1 ea)
Sub Sqrs Order 4 (3 ea) and 6 (1 ea)
Crnr Sqrs Order 4 (3 ea) and 6 (1 ea)
Crnr Sqrs Order 4 (4 ea)
Ass Rect (2 ea), Crnr Sqrs Order 4 and 6
Concentric
Split Border Lines
Bordered
Split Border Lines, Miscellaneous Types
Split Border Lines, Center Square order 6
Composed Border, Center Square order 4
Associated
Composed Border, Center Square order 4
Eccentric
Split Border Lines
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-
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-
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Comparable routines as listed above, can be used to generate miscellaneous other types of order 10 Prime Number Magic Squares, which will be described in following sections.
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