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14.0    Special Magic Squares, Prime Numbers

14.6    Magic Squares (8 x 8), Part II

14.6.15 Magic Squares (8 x 8), Composed of Magic Sub Squares (4 x 4)
Containing Embedded Magic Squares (4 x 4)

The following 8th order Magic Square (Magic Sum = s1) is composed out of four 4th order Magic Sub Squares and contains - in addition to this - five 4th order Embedded Magic Squares.

For all Magic Sub Squares the (Main) Bent Diagonals sum to s1/2.

 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16) a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24) a(25) a(26) a(27) a(28) a(29) a(30) a(31) a(32) a(33) a(34) a(35) a(36) a(37) a(38) a(39) a(40) a(41) a(42) a(43) a(44) a(45) a(46) a(47) a(48) a(49) a(50) a(51) a(52) a(53) a(54) a(55) a(56) a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64)

The properties described above result in following linear equations:

```a(61) =  0.5  * s1 - a(62) - a(63) - a(64)
a(59) =  0.5  * s1 - a(60) - a(61) - a(62)
a(57) =            - a(58) + a(61) + a(62)
a(56) = -0.25 * s1 + a(61) + a(62) + a(64)
a(55) =  0.25 * s1 - a(64)
a(54) =  0.25 * s1 - a(61)
a(53) =  0.25 * s1 - a(62)
a(52) = -0.25 * s1 + a(60) + a(61) + a(62)
a(51) =  0.25 * s1 - a(60)
a(50) =  0.25 * s1 + a(58) - a(61) - a(62)
a(49) =  0.25 * s1 - a(58)
a(47) = -0.25 * s1 + a(48) + a(56) + a(64)
a(46) =  0.25 * s1 - a(48) - a(56) + a(61)
a(45) =  0.5  * s1 - a(48) - a(61) - a(64)
a(44) =(-0.25 * s1 + 2 * a(48) + a(56) - 2 * a(60) + a(63) + 2 * a(64))/2
a(43) = -0.5  * s1 + a(44) + 2 * a(60) + a(61) + a(62)
a(42) =            - a(43) - a(58) + a(60) + a(61) + a(62)
a(41) =              a(42) + 2 * a(58) - a(61) - a(62)
a(40) =  0.5  * s1 - a(48) - a(56) - a(64)
a(39) =  0.25 * s1 - a(48)
a(38) = -0.25 * s1 + a(48) + a(61) + a(64)
a(37) =  0.5  * s1 - a(38) - a(39) - a(40)
a(36) =  0.25 * s1 + a(41) - a(58) - a(60)
a(35) =  0.5  * s1 - a(36) - a(37) - a(38)
a(34) =            - a(35) - a(58) + a(60) + a(61) + a(62)
a(33) =  0.5  * s1 - a(34) - a(35) - a(36)
a(31) =  0.25 * s1 + a(32) - a(56) - a(64)
a(30) =  0.25 * s1 - a(32) + a(56) - a(61)
a(29) =  0.5  * s1 - a(30) - a(31) - a(32)'
a(28) =(-0.25 * s1 + 2 * a(32) - a(56) + 2 * a(60) + a(61) + a(62) - a(64))/2
a(27) =  0.5  * s1 - a(28) - a(29) - a(30)
a(26) =            - a(28) + a(58) + a(60)
a(25) =  0.5  * s1 - a(26) - a(27) - a(28)
a(24) =            - a(32) + a(56) + a(64)
a(23) =  0.25 * s1 - a(32)
a(22) =              a(30) + 2 * a(32) - a(56) - a(64)
a(21) =  0.5  * s1 - a(22) - a(23) - a(24)
a(20) = -0.25 * s1 + a(25) + a(58) + a(60)
a(19) =  0.5  * s1 - a(20) - a(21) - a(22)
a(18) =            - a(20) + a(58) + a(60)
a(17) =  0.5  * s1 - a(18) - a(19) - a(20)
a(15) = -0.25 * s1 + a(16) + a(56) + a(64)
a(14) =  0.25 * s1 - a(16) + a(21) - a(32)
a(13) =  0.5  * s1 - a(14) - a(15) - a(16)
a(12) =  0.25 * s1 + a(16) - a(17) + a(32) - a(58) - a(60)
a(11) =  0.5  * s1 - a(12) - a(13) - a(14)
a(10) =            - a(11) - a(58) + a(60) + a(61) + a(62)
a( 9) =  0.5  * s1 - a(10) - a(11) - a(12)
a( 8) =  0.5  * s1 - a(16) - a(24) - a(32)
a( 7) =  0.5  * s1 - a(16) - a(21) - a(30)
a( 6) =  0.5  * s1 - a(13) - a(24) - a(31)
a( 5) =  0.5  * s1 - a( 6) - a( 7) - a( 8)
a( 4) =  0.5  * s1 - a(12) - a(20) - a(28)
a( 3) =  0.5  * s1 - a( 4) - a( 5) - a( 6)
a( 2) =  0.5  * s1 - a( 9) - a(20) - a(27)
a( 1) =  0.5  * s1 - a( 2) - a( 3) - a( 4)
```

which can be incorporated in a routine to generate subject Prime Number Composed Squares of order 8 (ref. Priem8d2).

Attachment 14.6.41 shows for miscellaneous Magic Sums the first occurring Prime Number Composed Square of order 8.

Each square shown corresponds with numerous squares for the same Magic Sum.

14.6.16 Franklin Squares (8 x 8)

The original Franklin Square (distinct consecutive integers), as constructed by Benjamin Franklin, was based on following defining properties (s1 = Magic Sum):

1. The numbers of every half-row and half-column sum to s1/2;
3. The numbers of the main bent diagonals and all the bent diagonals parallel to it sum to s1;
4. Every 2 × 2 sub square sums to s1/2 (Compact).

The properties described above result in following linear equations:

a(61) =  0.5 * s1 - a(62) - a(63) - a(64)
a(58) =           - a(60) + a(62) + a(64)
a(57) =  0.5 * s1 - a(59) - a(62) - a(64)
a(55) =  0.5 * s1 - a(56) - a(63) - a(64)
a(54) =             a(56) - a(62) + a(64)
a(53) =           - a(56) + a(62) + a(63)
a(52) =             a(56) - a(60) + a(64)
a(51) =  0.5 * s1 - a(56) - a(59) - a(64)
a(50) =             a(56) + a(60) - a(62)
a(49) =           - a(56) + a(59) + a(62)
a(47) =           - a(48) + a(63) + a(64)
a(46) =             a(48) + a(62) - a(64)
a(45) =  0.5 * s1 - a(48) - a(62) - a(63)
a(44) =             a(48) + a(60) - a(64)
a(43) =           - a(48) + a(59) + a(64)
a(42) =             a(48) - a(60) + a(62)
a(41) =  0.5 * s1 - a(48) - a(59) - a(62)
a(40) =  0.5 * s1 - a(48) - a(56) - a(64)
a(39) =             a(48) + a(56) - a(63)
a(38) =  0.5 * s1 - a(48) - a(56) - a(62)
a(37) = -0.5 * s1 + a(48) + a(56) + a(62) + a(63) + a(64)
a(36) =  0.5 * s1 - a(48) - a(56) - a(60)
a(35) =             a(48) + a(56) - a(59)
a(34) =  0.5 * s1 - a(48) - a(56) + a(60) - a(62) - a(64)
a(33) = -0.5 * s1 + a(48) + a(56) + a(59) + a(62) + a(64)
a(31) =           - a(32) + a(63) + a(64)
a(30) =             a(32) + a(62) - a(64)
a(29) =  0.5 * s1 - a(32) - a(62) - a(63)
a(28) =             a(32) + a(60) - a(64)
a(27) =           - a(32) + a(59) + a(64)
a(26) =             a(32) - a(60) + a(62)
a(25) =  0.5 * s1 - a(32) - a(59) - a(62)
a(23) =  0.5 * s1 - a(24) - a(63) - a(64)
a(22) =             a(24) - a(62) + a(64)
a(21) =           - a(24) + a(62) + a(63)
a(20) =             a(24) - a(60) + a(64)
a(19) =  0.5 * s1 - a(24) - a(59) - a(64)
a(18) =             a(24) + a(60) - a(62)
a(17) =           - a(24) + a(59) + a(62)
a(16) =           - a(32) + a(48) + a(64)
a(15) =             a(32) - a(48) + a(63)
a(14) =           - a(32) + a(48) + a(62)
a(13) =  0.5 * s1 + a(32) - a(48) - a(62) - a(63) - a(64)
a(12) =           - a(32) + a(48) + a(60)
a(11) =             a(32) - a(48) + a(59)
a(10) =           - a(32) + a(48) - a(60) + a(62) + a(64)
a( 9) =  0.5 * s1 + a(32) - a(48) - a(59) - a(62) - a(64)
a( 8) =  0.5 * s1 - a(24) - a(48) - a(64)
a( 7) =             a(24) + a(48) - a(63)
a( 6) =  0.5 * s1 - a(24) - a(48) - a(62)
a( 5) = -0.5 * s1 + a(24) + a(48) + a(62) + a(63) + a(64)
a( 4) =  0.5 * s1 - a(24) - a(48) - a(60)
a( 3) =             a(24) + a(48) - a(59)
a( 2) =  0.5 * s1 - a(24) - a(48) + a(60) - a(62) - a(64)
a( 1) = -0.5 * s1 + a(24) + a(48) + a(59) + a(62) + a(64)

which can be incorporated in a routine to generate Prime Number Franklin Squares of order 8 (ref. Priem8f).

Attachment 14.6.43 shows for miscellaneous Magic Sums the first occurring Prime Number Franklin Square of order 8.

Each square shown corresponds with numerous squares for the same Magic Sum.

14.6.17 Pan Magic Squares (8 x 8), Composed of Magic Sub Squares (4 x 4)

Composed Associated Magic Squares, as shown in Attachment 14.6.6d, can be transformed to Composed Pan Magic and Complete Magic Squares as illustrated below (Euler):

Associated (MC = 22440)
 1583 3001 5399 1237 3229 89 2731 5171 3449 3187 2063 2521 2953 3221 2749 2297 607 2339 3631 4643 4211 2473 2777 1759 5581 2693 127 2819 827 5437 2963 1993 3617 2647 173 4783 2791 5483 2917 29 3851 2833 3137 1399 967 1979 3271 5003 3313 2861 2389 2657 3089 3547 2423 2161 439 2879 5521 2381 4373 211 2609 4027
Pan Magic (MC = 22440)
 1583 3001 5399 1237 5171 2731 89 3229 3449 3187 2063 2521 2297 2749 3221 2953 607 2339 3631 4643 1759 2777 2473 4211 5581 2693 127 2819 1993 2963 5437 827 439 2879 5521 2381 4027 2609 211 4373 3313 2861 2389 2657 2161 2423 3547 3089 3851 2833 3137 1399 5003 3271 1979 967 3617 2647 173 4783 29 2917 5483 2791

Attachment 14.6.42 shows the Pan Magic Squares, which can be obtained by transformation of the Composed Magic Squares as shown in Attachment 14.6.6d.

14.6.18 Pan Magic Squares (8 x 8), Pan Magic Square Inlays (4 x 4)

Prime Number Magic Squares composed of Pan Magic Sub Squares, as discussed in Section 14.6.1, can be transformed to Pan Magic Squares as illustrated below:

Composed (MC8 = 3360)
 71 89 733 787 83 97 739 761 691 829 29 131 673 827 17 163 107 53 769 751 101 79 757 743 811 709 149 11 823 677 167 13 227 193 599 661 293 337 443 607 439 821 67 353 349 701 199 431 241 179 613 647 397 233 547 503 773 487 401 19 641 409 491 139
Pan Magic (MC8 = 3360)
 71 83 89 97 733 739 787 761 227 293 193 337 599 443 661 607 691 673 829 827 29 17 131 163 439 349 821 701 67 199 353 431 107 101 53 79 769 757 751 743 241 397 179 233 613 547 647 503 811 823 709 677 149 167 11 13 773 641 487 409 401 491 19 139

The resulting Pan Magic Square is Complete, 4 x 4 Compact and Four Way V type Zig Zag (ref. Section 14.15.5) and has following additional properties:

1. The corner points of all 3 x 3 sub squares (64 ea) sum to half the Magic Sum;
2. The corner points of all 5 x 5 sub squares (16 ea) sum to half the Magic Sum.

Attachment 14.6.45 shows the Pan Magic Squares, which can be obtained by transformation of the Composed Magic Squares as shown in Attachment 14.6.44.

Each square shown corresponds with numerous Pan Magic Squares with the same Magic Sums and variable values {ai},

14.6.19 Summary

The obtained results regarding the miscellaneous types of order 8 Prime Number Magic Squares as deducted and discussed in previous sections are summarized in following table:

 Type Characteristics Subroutine Results Composed Magic, Embedded Magic Squares Pan Magic (1) Euler Pan Magic (2) Alternative Franklin Half Rows and Half Columns sum to s1/2 All Bent Diagonals sum to s1, Compact - - - -
 Comparable routines as listed above, can be used to generate Prime Number Magic Squares of order 9, which will be described in following sections.