Office Applications and Entertainment, Magic Squares  
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14.0 Special Magic Squares, Prime Numbers
14.7.1 Magic Squares (9 x 9), Overlapping Subsquares
Prime Number Magic Squares of order 9 with a Magic Sum s1 can be composed out of:
as illustrated below:
In previous section a procedure has been applied to generate sets of 4^{th} order Prime Number Pan Magic Squares with 16 different Prime Numbers for a Magic Sum MC4 (Priem4c).
14.7.2 Magic Squares (9 x 9), Overlapping Subsquares, Partly Symmetric
Composed Squares, as described in Section 14.7.1 above, can't be Center Symmetric because of the two 4^{th} order Pan Magic Corner Squares.
Attachment 14.7.42 shows for miscellaneous Magic Sums the first occurring 9^{th} order Partial Symmetric Prime Number Magic Square with Overlapping Sub Squares.
14.7.3 Magic Squares (9 x 9), Overlapping Subsquares, Associated
Finally, Prime Number Associated Magic Squares of order 9 can be composed out of:
A comparable procedure (Priem4f2) can be used:
Attachment 14.7.49 shows one Associated Prime Number Magic Square for each of the applicable Magic Sums.
14.7.4 Concentric Magic Squares (9 x 9)
A 9^{th} order Prime Number Concentric Magic Square consists of a Prime Number Concentric Magic Square of the 7^{th} order, as discussed in Section 14.5.1, with a border around it.
a(73) = s1  a(74)  a(75)  a(76)  a(77)  a(78)  a(79)  a(80)  a(81) a(64) = 2 * s1/9  a(72) a(55) = 2 * s1/9  a(63) a(46) = 2 * s1/9  a(54) a(37) = 2 * s1/9  a(45) a(28) = 2 * s1/9  a(36) a(19) = 2 * s1/9  a(27) a(18) = 7 * s1/9  a(27)  a(36)  a(45)  a(54)  a(63)  a(72) + a(73)  a(81) a(10) = 2 * s1/9  a(18) a( 9) = 2 * s1/9  a(73) a( 8) = 2 * s1/9  a(80) a( 7) = 2 * s1/9  a(79) a( 6) = 2 * s1/9  a(78) a( 5) = 2 * s1/9  a(77) a( 4) = 2 * s1/9  a(76) a( 3) = 2 * s1/9  a(75) a( 2) = 2 * s1/9  a(74) a( 1) = 2 * s1/9  a(81)
a routine can be written to generate Prime Number Concentric Magic Squares of order 9 (ref. Priem9a1).
Note:
This results in following alternative border equations:
which enable the development of a much faster routine to generate Prime Number Concentric Magic Squares of order 7 (ref. Priem9a2).
14.7.5 Bordered Magic Squares (9 x 9), Miscellaneous Inlays
Based on the collections of 7^{th} order Inlaid and Ultra Magic Squares, as discussed in Section 14.5.2, 14.5.6 and 14.5.8, also following 9^{th} order Bordered Magic Squares can be generated with routine Priem9a1 or Priem9a2:
Each square shown corresponds with numerous squares for the same Magic Sum.
14.7.6 Bordered Magic Squares (9 x 9), Split Border
Alternatively a 9^{th} order Bordered Magic Square with Magic Sum s9 can be constructed based on:
as illustrated below:
Based on the principles described in Section 14.7.1 above, a fast procedure (Priem9c) can be developed:
Attachment 14.7.9 shows one Prime Number Bordered Magic Square for some of the occurring Magic Sums.
14.7.7 Eccentric Magic Squares (9 x 9)
Also for Prime Number Eccentric Magic Squares of order 9 it is convenient to split the supplementary rows and columns into three equal parts each summing to s1 = s9 / 3:
This enables, based on the same principles, the development of a fast procedure (ref. Priem9b):
Attachment 14.7.7 shows one Prime Number Eccentric Magic Square for some of the occurring Magic Sums.
14.7.8 Eccentric Magic Squares (9 x 9) Prime Number Eccentric Magic Squares of order 9 with a Magic Sum s1 might contain:
as illustrated below:
Based on this definition a dedicated procedure (Priem9g1) can be used:
Attachment 14.7.31 shows for miscellaneous Magic Sums the first occurring order 9 Prime Number Eccentric Magic Square with Overlapping Sub Squares.
14.7.9 Composed Magic Squares (9 x 9) The order 9 Magic Square shown below, with magic Sum s1, is composed out of:
Based on this definition a dedicated procedure (ref. Priem9g2) can be used:
Attachment 14.7.32 shows for miscellaneous Magic Sums the first occurring Prime Number Composed Magic Square.
14.7.10 Simple Magic Squares (9 x 9) Composed of (Semi) Magic Sub Squares (3 x 3)
Comparable with the method discussed in Section 14.4.10, Prime Number Magic Squares of order 9, with Magic Sum 3 * s1, can be composed out of one Magic Center Square and eight Semi Magic Border Squares of order 3 with Magic Sum s1.
The 8 border squares can be arranged in 8! ways around the center square, resulting in 8! * 8 * 12^{4} * 24^{4} = 2,22 10^{15} Magic Squares of the 9^{th} order for subject Magic Sum.
14.7.11 Simple Magic Squares (9 x 9) Composed of (Semi) Magic Sub Squares (3 x 3)
Comparable with the method discussed in Section 14.7.10 above, Prime Number Magic Squares of order 9, with Magic Sum 3 * s1, can be composed out of:
of order 3 with Magic Sum s1.
14.7.12 Associated Magic Squares (9 x 9) Composed of (Semi) Magic Sub Squares (3 x 3)
Comparable with the method discussed in Section 14.7.11 above, Associated Prime Number Magic Squares of order 9, with Magic Sum 3 * s1, can be composed out of:
of order 3 with Magic Sum s1.
14.7.13 Associated Magic Squares (9 x 9) with Associated Square Inlays Order 4 and 5
Associated Magic Squares of order 9 with Square Inlays of order 4 and 5 can be obtained by means of a transformation of order 9 Composed Magic Squares as illustrated below: 
MC = 13023
2887 1873 421 607 2857 2851 733 181 613 337 691 2143 2617 547 97 1033 2767 2791 277 751 2203 2557 103 127 1861 2797 2347 2287 2473 1021 7 2281 2713 2161 43 37 2833 223 211 2521 673 457 1723 2011 2371 1327 1201 1597 1663 1801 1741 2293 757 643 1471 1831 1063 1423 1987 2017 1447 877 907 1231 1297 1693 1567 2251 2137 601 1153 1093 373 2683 2671 61 523 883 1171 2437 2221 = > MC = 13023
673 2833 457 223 1723 211 2011 2521 2371 2857 2887 2851 1873 733 421 181 607 613 1801 1327 1741 1201 2293 1597 757 1663 643 547 337 97 691 1033 2143 2767 2617 2791 1987 1471 2017 1831 1447 1063 877 1423 907 103 277 127 751 1861 2203 2797 2557 2347 2251 1231 2137 1297 601 1693 1153 1567 1093 2281 2287 2713 2473 2161 1021 43 7 37 523 373 883 2683 1171 2671 2437 61 2221
The Magic Square shown at the left side above is composed out of:
Based on this definition a routine can be developed to generate subject Composed Magic Squares (ref. Priem9f3).
14.7.14 Associated Magic Squares (9 x 9) with Associated Diamond Inlays Order 4 and 5
Associated Magic Squares of order 9 with Associated Diamond Inlays of order 4 and 5
can be constructed as follows:
It is convenient to split the two bottom rows and right columns into parts summing to s3 = s9 / 3 and s6 = 2 * s9 / 3, which results in following border equations:
a9( 4) = 2 * s9/9  a9(78) a9(76) = s9  a9( 4)  a9(13)  a9(22)  a9(31)  a9(40)  a9(49)  a9(58)  a9(67) a9(75) = s9  a9( 3)  a9(12)  a9(21)  a9(30)  a9(39)  a9(48)  a9(57)  a9(66) a9(55) = s9  a9(56)  a9(57)  a9(58)  a9(59)  a9(60)  a9(61)  a9(62)  a9(63) a9(64) = s9  a9(65)  a9(66)  a9(67)  a9(68)  a9(69)  a9(70)  a9(71)  a9(72) a9(74) = s9  a9( 2)  a9(11)  a9(20)  a9(29)  a9(38)  a9(47)  a9(56)  a9(65) a9(73) = s9  a9(74)  a9(75)  a9(76)  a9(77)  a9(78)  a9(79)  a9(80)  a9(81) a9(54) =(7 * s9  4 * a9(55)  5 * a9(56)  8 * a9(64)  9 * a9(65)  10 * a9(66)  10 * a9(70) +  a9(74)  2 * a9(75)  3 * a9(76) + 12 * a9(77)  5 * a9(78)  10 * a9(79)) / 8 a9(46) = s9  a9(47)  a9(48)  a9(49)  a9(50)  a9(51)  a9(52)  a9(53)  a9(54)
with a9(81), a9(80), a9(72), a9(71), a9(78), a9(66), a9(56) and a9(65) the independent variables.
14.7.15 Inlaid Magic Squares (9 x 9),
Square Inlays Order 3 and 4 (Overlapping)
The 9^{th} order (Simple) Inlaid Magic Square shown below:
contains following inlays:
The relation between the Magic Sums s(1), s(2), s(3) and s(4) is: s(1) = 8 * s9 / 9  s(4) s(2) = 6 * s9 / 9  s(3)
With s9 the Magic Sum of the 9^{th} order Inlaid Magic Square.
a(76) =  s9 / 9 + a(78)  s(3) + s(4) a(75) =  s9 / 9 + a(79)  s(3) + s(4) a(73) = 12 * s9 / 9  a(77)  2 * a(78)  2 * a(79)  2 * a(80)  a(81) + 3 * s(3)  3 * s(4) a(64) = s9  a(72)  s(3)  s(4) a(55) = s9  a(63)  s(3)  s(4) a(46) = s9  a(54)  s(3)  s(4) a(45) = 40 * s9 / 9  2*a(54)  2*a(63)  2*a(72)  a(77)  2*a(78)  2*a(79)  2*a(80)  2*a(81)  6*s(4)
a procedure can be developed
Attachment 14.7.21
shows for miscellaneous Magic Sums the first occurring Prime Number Inlaid Magic Square of order 9.
The obtained results regarding the miscellaneous types of order 9 Prime Number Magic Squares as deducted and discussed in previous sections are summarized in following table: 
Type
Characteristics
Subroutine
Results
Composed (1)
Overlapping Sub Squares
Overlapping Sub Squares, Partly Symmetric
Overlapping Sub Squares, Associated
Associated Corner Squares Order 3 and 6
Associated Corner Squares Order 4 and 5
Concentric

Bordered
Miscellaneous Types
Split Border Lines, Center Square order 5
Eccentric
Split Border Lines
Overlapping Sub Squares
Associated Corner Squares
Composed (2)
Eight Semi Magic Border Squares
One Magic Center SquareFour Semi Magic Border Squares
Four Semi Magic Corner Squares
One Magic Center SquareFour Semi Magic Anti Symmetric Border Squares
Four Semi Magic Anti Symmetric Corner Squares
One Magic Center SquareAssociated
Square Inlays Order 4 and 5

Diamond Inlays Order 4 and 5
Inlaid
Square Inlays Order 3 and 4 (Overlapping)
Comparable routines as listed above, can be used to generate Prime Number Magic Squares of order 10, which will be described in following sections.

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