Office Applications and Entertainment, Magic Squares

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14.0    Special Magic Squares, Prime Numbers

14.21   Magic Squares, Higher Order, Inlaid (1)

14.21.1 Magic Squares (8 x 8)

Examples of Prime Number Inlaid Magic Squares of order 8, with four 3th order Embedded Semi Magic Squares of the same Magic Sum and corresponding Border, are shown in:

Prime Number Inlaid Magic Squares of order 8, with four 3th order Embedded Magic Squares with different Magic Sums and an Associated Border, have been described in Section 14.6.12.

Examples of such Prime Number Inlaid Magic Squares of order 8 are shown in Attachment 14.6.40.

14.21.2 Magic Squares (9 x 9)

Examples of Prime Number Magic Squares of order 9, composed out of eight Prime Number Semi Magic Squares and one Magic Center Square of order 3 with the same Magic Sum, have been discussed in Section 14.11.8.

The 9th order Prime Number Inlaid Magic Square shown below (s9 = 34227), is composed out of nine each 3th order Simple Magic Squares with different Magic Sums s(1) ... s(9).

7919 449 4049
269 4139 8009
4229 7829 359
7187 311 3533
23 3677 7331
3821 7043 167
7013 239 3527
107 3593 7079
3659 6947 173
6197 383 3191
251 3257 6263
3323 6131 317
6173 2819 2417
47 3803 7559
5189 4787 1433
7877 881 4289
761 4349 7937
4409 7817 821
6803 2213 3023
233 4013 7793
5003 5813 1223
6599 1571 3617
947 3929 6911
4241 6287 1259
6257 1283 2861
71 3467 6863
4073 5651 677

The Magic Sums s(1) thru s(9) of the Magic Sub Squares comply with the equations defining a Magic Square of order 3 (ref. Section 14.1.1).

The Prime Number Inlaid Magic Square shown above can be constructed by selecting:

  • An order 3 Magic Main Square containing possible Center Elements for the Sub Squares
  • Nine order 3 Magic Sub Squares based on the corresponding Magic Sums (ref. Priem9k)

Attachment 14.6.54 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 9.

Each square shown corresponds with numerous solutions, which can be obtained by rotation/reflection of the main square or the nine inlays (8 * 89 = 1,07 109).

Prime Number Inlaid Magic Squares of order 9, with following inlays:

  • two each 4th order Simple Magic Squares - Magic Sums s(1) and s(4) - with the center element in common,
  • two each 3th order Simple Magic Squares with Magic Sums s(2) and s(3)

and an Associated Border, have been described in Section 14.7.15.

Examples of such Prime Number Inlaid Magic Squares of order 9 are shown in Attachment 14.7.21.

14.21.3 Magic Squares (10 x 10)

Examples of Prime Number Inlaid Magic Squares of order 10, with four 4th order Embedded (Pan) Magic Squares of the same Magic Sum and corresponding Border, are shown in:

The 10th order Inlaid Magic Square shown below, is composed out of an Associated Border and four each 4th order Embedded (Pan) Magic Squares with different Magic Sums s(1), s(2), s(3) and s(4).

a(1)

a(2)

a(3)

a(4)

a(5)

a(6)

a(7)

a(8)

a(9)

a(10)

a(11)

a(12)

a13)

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)

a(65)

a(66)

a(67)

a(68)

a(69)

a(70)

a(71)

a(72)

a(73)

a(74)

a(75)

a(76)

a(77)

a(78)

a(79)

a(80)

a(81)

a(82)

a(83)

a(84)

a(85)

a(86)

a(87)

a(88)

a(89)

a(90)

a(91)

a(92)

a(93)

a(94)

a(95)

a(96)

a(97)

a(98)

a(99)

a(100)

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:

s(1) = 4 * s10 / 5 - s(4)
s(2) = 4 * s10 / 5 - s(3)

With s10 the Magic Sum of the 10th order Inlaid Magic Square.

Based on the equations describing the Associated Border:

a(95) =           a(96) - s(3) + s(4)
a(94) =           a(97) - s(3) + s(4)
a(93) =           a(98) - s(3) + s(4)
a(92) =           a(99) - s(3) + s(4)
a(91) =   s10 - 2*a(96) - 2 * a(97) - 2 * a(98) - 2 * a(99) - a(100) + 4 * s(3) - 4 * s(4)
a(81) =   s10   - a(90) - s(3) - s(4)
a(71) =   s10   - a(80) - s(3) - s(4)
a(61) =   s10   - a(70) - s(3) - s(4)
a(60) = 5*s10/2 - a(70) - a(80) - a(90) - a(96) - a(97) - a(98) - a(99) - a(100) - 4 * s(4)
a(51) =   s10   - a(60) - s(3) - s(4)

a(50) = s10/5 - a(51)
a(41) = s10/5 - a(60)
a(40) = s10/5 - a(61)
a(31) = s10/5 - a(70)

a(30) = s10/5 - a(71)
a(21) = s10/5 - a(80)
a(20) = s10/5 - a(81)
a(11) = s10/5 - a(90)

a(10) = s10/5 - a(91)
a( 9) = s10/5 - a(92)
a( 8) = s10/5 - a(93)
a( 7) = s10/5 - a(94)
a( 6) = s10/5 - a(95)

a(5) = s10/5 - a( 96)
a(4) = s10/5 - a( 97)
a(3) = s10/5 - a( 98)
a(2) = s10/5 - a( 99)
a(1) = s10/5 - a(100)

a procedure can be developed:

  • to collect previously generated (Pan) Magic Squares of the fourth order (ref. Section 14.2.2) and
  • to complete subject 10th order Prime Number Inlaid Magic Squares (ref. Priem10k).

Attachment 14.6.42 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 10.

More unique solutions per Magic Sum s10 are possible, by variation of the type (Associated, Pan Magic, Simple) or the Magic Sums s(1) ... s(4) of the Embedded Magic Squares.

Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the four inlays and variation of the border.

14.21.4 Magic Squares, Bordered (12 x 12)

Prime Number Inlaid Magic Squares of order 12, with four 5th order Embedded Magic Squares of the same Magic Sum and corresponding Border, can be constructed with a procedure (Priem12a) which:

  • reads previously generated 10th order Composed (Associated) Magic Squares (ref. Attachment 14.8.66) and
  • completes the 12th order Prime Number Inlaid Magic Squares with a suitable border.

Attachment 14.6.51 shows for miscellaneous Magic Sums subject Prime Number Inlaid Magic Squares of order 12.

The 12th order Inlaid Magic Square shown below, is composed out of an Associated Border and four each 5th order Embedded (Ultra) Magic Squares with different Magic Sums s(1), s(2), s(3) and s(4).

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) a(65) a(66) a(67) a(68) a(69) a(70) a(71) a(72)
a(73) a(74) a(75) a(76) a(77) a(78) a(79) a(80) a(81) a(82) a(83) a(84)
a(85) a(86) a(87) a(88) a(89) a(90) a(91) a(92) a(93) a(94) a(95) a(96)
a(97) a(98) a(99) a(100) a(101) a(102) a(103) a(104) a(105) a(106) a(107) a(108)
a(109) a(110) a(111) a(112) a(113) a(114) a(115) a(116) a(117) a(118) a(119) a(120)
a(121) a(122) a(123) a(124) a(125) a(126) a(127) a(128) a(129) a(130) a(131) a(132)
a(133) a(134) a(135) a(136) a(137) a(138) a(139) a(140) a(141) a(142) a(143) a(144)

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:

s(1) = 5 * s12 / 6 - s(4)
s(2) = 5 * s12 / 6 - s(3)

With s12 the Magic Sum of the 12th order Inlaid Magic Square.

Based on the equations describing the Associated Border:

a(138) =           a(139) - s(3) + s(4)
a(137) =           a(140) - s(3) + s(4)
a(136) =           a(141) - s(3) + s(4)
a(135) =           a(142) - s(3) + s(4)
a(134) =           a(143) - s(3) + s(4)
a(133) =   s12 - 2*a(139) - 2*a(140) - 2*a(141) - 2*a(142) - 2*a(143) - a(144) + 5*s(3) - 5*s(4)
a(121) =   s12 -   a(132) - s(3) - s(4)
a(109) =   s12 -   a(120) - s(3) - s(4)
a( 97) =   s12 -   a(108) - s(3) - s(4)
a( 85) =   s12 -   a( 96) - s(3) - s(4)
a( 84) = 3*s12 -   a( 96) - a(108) - a(120) - a(132) - a(139) - a(140) - a(141) +
                                                                       - a(142) - a(143) - a(144) - 5*s(4)
a( 73) =   s12 -   a( 84) - s(3) - s(4)

a(72) = s12/6 - a(73)
a(61) = s12/6 - a(84)
a(60) = s12/6 - a(85)
a(49) = s12/6 - a(96)
a(48) = s12/6 - a(97)

a(37) = s12/6 - a(108)
a(36) = s12/6 - a(109)
a(25) = s12/6 - a(120)
a(24) = s12/6 - a(121)
a(13) = s12/6 - a(132)

a(12) = s12/6 - a(133)
a(11) = s12/6 - a(134)
a(10) = s12/6 - a(135)
a( 9) = s12/6 - a(136)
a( 8) = s12/6 - a(137)
a( 7) = s12/6 - a(138)

a(6) = s12/6 - a(139)
a(5) = s12/6 - a(140)
a(4) = s12/6 - a(141)
a(3) = s12/6 - a(142)
a(2) = s12/6 - a(143)
a(1) = s12/6 - a(144)

a procedure can be developed:

  • to collect previously generated (Ultra) Magic Squares of the fifth order (ref. Section 14.3.5) and
  • to complete subject 12th order Prime Number Inlaid Magic Squares (ref. Priem12k).

Attachment 14.6.50 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 12.

More unique solutions per Magic Sum s12 are possible, by variation of the type (Associated, Pan Magic, Simple) or the Magic Sums s(1) ... s(4) of the Embedded Magic Squares.

Attachment 14.6.52 contains a few additional Prime Number Inlaid Magic Square of order 12 based on:

  • Embedded Concentric Magic Sub Squares                     (ref. Section 14.3.2)
  • Embedded Associated Magic Sub Squares with Diamond Inlay  (ref. Section 14.3.6)

Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the four inlays and variation of the border.

14.21.5 Magic Squares, Composed (12 x 12)

An example of a Prime Number Magic Square of order 12, composed out of 9 Prime Number Pan Magic Squares of order 4 with the same Magic Sum, has been provided in Section 14.11.2.

The 12th order Inlaid Magic Square shown below (s12 = 33468), is composed out of nine each 4th order Pan Magic Squares with different Magic Sums s(1) ... s(9).

557 1721 4091 5507
3671 5927 137 2141
1847 431 5381 4217
5801 3797 2267 11
383 821 4643 5261
4373 5531 113 1091
911 293 5171 4733
5441 4463 1181 23
449 1013 4049 4973
3833 5189 233 1229
1193 269 4793 4229
5009 4013 1409 53
461 1163 3659 4481
3389 4751 191 1433
1223 401 4421 3719
4691 3449 1493 131
761 1877 3581 4937
2999 5519 179 2459
1997 641 4817 3701
5399 3119 2579 59
701 887 5303 5657
4703 6257 101 1487
971 617 5573 5387
6173 4787 1571 17
827 1913 3851 5237
3221 5867 197 2543
2063 677 5087 4001
5717 3371 2693 47
1151 2393 3011 4649
2741 4919 881 2663
2591 953 4451 3209
4721 2939 2861 683
1439 2309 2777 3911
2399 4289 1061 2687
2441 1307 3779 2909
4157 2531 2819 929

The Magic Sums s(1) thru s(9) of the Pan Magic Sub Squares comply with the equations defining a Magic Square of order 3 (ref. Section 14.1.1).

The Prime Number Inlaid Magic Square shown above can be constructed by selecting:

  • An order 3 Magic Main Square containing possible Order 4 Magic Sums
  • Nine order 4 Pan Magic Sub Squares based on these Magic Sums and the equations deducted in Section 14.2.2

Attachment 14.6.55 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 12.

Each square shown corresponds with numerous solutions, which can be obtained by rotation/reflection of the main square or selecting other aspects of the nine inlays (8 * 3849 = 1,45 1024).

Alternatively Order 12 Prime Number Inlaid Magic Squares can be constructed by selecting:

  • An order 4 Pan Magic Main Square containing possible Center Elements for the Sub Squares
  • Sixteen order 3 Magic Sub Squares based on the corresponding Magic Sums (ref. Priem12c)

Attachment 14.6.56 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 12.

Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or rotation/reflection of the sixteen inlays (384 * 812 = 2,64 1013).

14.21.6 Magic Squares, Composed (15 x 15)

An example of an Associated Prime Number Magic Square of order 15, composed out of 9 Prime Number Magic Squares of order 5 with the same Magic Sum, has been provided in Section 14.11.11.

The 15th order Inlaid Magic Square shown below (s15 = 285015), is composed out of nine each 5th order Ultra Magic Squares with different Magic Sums s(1) ... s(9).

41189 13043 15053 31481 3089
8081 24509 17789 25733 27743
2333 40433 20771 1109 39209
13799 15809 23753 17033 33461
38453 10061 26489 28499 353
34439 13469 5279 26297 8111
4649 25667 12791 26339 18149
4691 31019 17519 4019 30347
16889 8699 22247 9371 30389
26927 8741 29759 21569 599
36923 16673 4493 21419 14057
2543 19469 18047 32843 20663
13967 36833 18713 593 23459
16763 4583 19379 17957 34883
23369 16007 32933 20753 503
32303 11273 11159 25523 4457
7253 21617 14033 20963 20849
2693 30539 16943 3347 31193
13037 12923 19853 12269 26633
29429 8363 22727 22613 1583
36809 18743 1613 22901 14939
1451 22739 15359 36293 19163
14843 36713 19001 1289 23159
18839 1709 22643 15263 36551
23063 15101 36389 19259 1193
42101 12107 16319 27239 7529
8969 19889 23831 24197 28409
5927 40499 21059 1619 36191
13709 17921 18287 22229 33149
34589 14879 25799 30011 17
38189 14669 8837 30509 4241
5009 26681 12689 28949 23117
3449 37397 19289 1181 35129
15461 9629 25889 11897 33569
34337 8069 29741 23909 389
40853 12323 10499 31793 6947
8273 29567 17333 24533 22709
1013 34919 20483 6047 39953
18257 16433 23633 11399 32693
34019 9173 30467 28643 113
32939 12941 6983 28619 4673
5813 27449 10133 24359 18401
1553 29819 17231 4643 32909
16061 10103 24329 7013 28649
29789 5843 27479 21521 1523

The Magic Sums s(1) thru s(9) of the Concentric Magic Sub Squares comply with the equations defining a Magic Square of order 3 (ref. Section 14.1.1).

The Prime Number Inlaid Magic Square shown above can be constructed by selecting:

  • An order 3 Magic Main Square containing possible Center Elements for the Sub Squares
  • Nine order 5 Ultra Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.3.5

Attachment 14.6.53 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 15, composed of:

  • Ultra      Magic Sub Squares                     (ref. Section 14.3.5)
  • Concentric Magic Sub Squares                     (ref. Section 14.3.2)
  • Associated Magic Sub Squares with Diamond Inlay  (ref. Section 14.3.6)

Each square shown corresponds with numerous solutions, which can be obtained by rotation/reflection of the main square or selecting other aspects of the nine inlays.

Alternatively Order 15 Prime Number Inlaid Magic Squares can be constructed by selecting:

  • An order 5 Symmetric Magic Main Square containing possible Center Elements for Order 3 Sub Squares
  • Twenty five order 3 Simple Magic Sub Squares based on the corresponding Magic Sums (ref. Priem15k4)

Attachment 14.6.57 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 15, based on Associated Main Squares with Diamond Inlay.

Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or rotation/reflection of the twenty five inlays.

14.21.7 Magic Squares, Composed (16 x 16)

An example of a Prime Number Magic Square of order 16, composed out of 16 Prime Number Pan Magic Squares of order 4 with the same Magic Sum, has been provided in Section 14.11.3.

The 16th order Inlaid Magic Square shown below (s16 = 63840), is composed out of sixteen each 4th order Pan Magic Squares with different Magic Sums s(1) ... s(16).

359 947 6653 7253
6317 7589 23 1283
953 353 7247 6659
7583 6323 1289 17
823 1531 6067 6967
5347 7687 103 2251
1627 727 6871 6163
7591 5443 2347 7
691 457 7489 7759
7069 8179 271 877
709 439 7507 7741
7927 7321 1129 19
599 1523 6803 7919
6353 8369 149 1973
1619 503 7823 6899
8273 6449 2069 53
1321 1993 5737 7177
4813 8101 397 2917
2377 937 6793 6121
7717 5197 3301 13
677 2999 5297 8039
4889 8447 269 3407
3209 467 7829 5507
8237 5099 3617 59
443 1031 6359 7211
6089 7481 173 1301
1163 311 7079 6491
7349 6221 1433 41
571 1087 6661 7237
6199 7699 109 1549
1117 541 7207 6691
7669 6229 1579 79
1373 1709 5861 6581
4751 7691 263 2819
1901 1181 6389 6053
7499 4943 3011 71
1237 1009 6211 6619
5521 7309 547 1699
1327 919 6301 6529
6991 5839 2017 229
1471 1297 6607 7333
5623 8317 487 2281
1747 1021 6883 7057
7867 6073 2731 37
593 3257 4799 7883
4463 8219 257 3593
3467 383 7673 5009
8009 4673 3803 47
2617 2689 5119 6451
3259 8311 757 4549
3319 1987 5821 5749
7681 3889 5179 127
1613 2399 5393 6959
4259 8093 479 3533
2789 1223 6569 5783
7703 4649 3923 89
2273 2609 5003 5807
2969 7841 239 4643
2843 2039 5573 5237
7607 3203 4877 5
1663 2221 4663 6361
3631 7393 631 3253
2791 1093 5791 5233
6823 4201 3823 61

The Magic Sums s(1) thru s(16) of the Pan Magic Sub Squares comply with the equations defining a Pan Magic Square of order 4 (ref. Section 14.2.2).

The Prime Number Inlaid Magic Square shown above can be constructed by selecting:

  • An order 4 Pan Magic Main Square containing possible Order 4 Magic Sums
  • Sixteen order 4 Pan Magic Sub Squares based on these Magic Sums and the equations deducted in Section 14.2.2

Attachment 14.6.70 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 16.

Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the sixteen inlays (384 * 38416 = 8,58 1043).

14.21.8 Magic Squares, Composed (20 x 20)

An example of a Prime Number Magic Square of order 20, composed out of 25 Prime Number Pan Magic Squares of order 4 with the same Magic Sum, has been provided in Section 14.11.4.

Examples of 20th order Inlaid Magic Squares, composed out of twenty five each 4th order Pan Magic Squares with different Magic Sums s(1) ... s(25), are shown in Attachment 14.6.82.

The Magic Sums s(1) thru s(25) of the Pan Magic Sub Squares comply with the equations defining an Associated Magic Square of order 5 with Diamond Inlay (ref. Section 14.3.6).

The Prime Number Inlaid Magic Squares described above can be constructed by selecting:

  • An order 5 Magic Main Square containing possible Order 4 Magic Sums
  • Twenty five order 4 Pan Magic Sub Squares based on these Magic Sums and the equations deducted in Section 14.2.2

Attachment 14.6.82 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 20, based on Associated Main Squares with Diamond Inlay.

Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the twentfive inlays.

Alternatively Order 20 Prime Number Inlaid Magic Squares can be constructed by selecting:

  • An order 4 Pan Magic Main Square containing possible Center Elements for the Sub Squares
  • Sixteen order 5 Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.3.6

Attachment 14.6.81 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 20.

Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the sixteen inlays.

14.21.9 Magic Squares, Composed (25 x 25)

Attachment 14.6.90 shows an example of a 25th order Inlaid Magic Square, composed out of twenty five each 5th order Magic Squares with different Magic Sums s(1) ... s(25).

The Magic Sums s(1) thru s(25) of the Magic Sub Squares comply with the equations defining an Associated Magic Square of order 5 with Diamond Inlay (ref. Section 14.3.6).

The Prime Number Inlaid Magic Square described above can be constructed by selecting:

  • An order 5 Magic Main Square containing possible Center Elements for the Sub Squares
  • Twenty five order 5 Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.3.6

The square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the twenty five inlays.

14.21.10 Summary

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

Type

Characteristics

Subroutine

Results

Order 9

Sub Squares Order 3, with Different Magic Sums

Priem9k

Attachment 14.6.54

Order 10

Sub Squares Order 4, with Different Magic Sums

Priem10k

Attachment 14.6.42

Order 12

Sub Squares Order 5, with Identical Magic Sum

Priem12a

Attachment 14.6.51

Sub Squares Order 5, with Different Magic Sums
                     Miscellaneous Inlays

Priem12k

Attachment 14.6.50
Attachment 14.6.52

Order 12

Sub Squares Order 4, with Different Magic Sums

Priem12b

Attachment 14.6.55

Sub Squares Order 3, with Different Magic Sums

Priem12c

Attachment 14.6.56

Order 15

Sub Squares Order 5, with Different Magic Sums

Priem15k2

Attachment 14.6.53

Sub Squares Order 3, with Different Magic Sums

Priem15k4

Attachment 14.6.57

Order 16

Sub Squares Order 4, with Different Magic Sums

Priem16k

Attachment 14.6.70

Order 20

Sub Squares Order 5, with Different Magic Sums

Priem20k1

Attachment 14.6.81

Sub Squares Order 4, with Different Magic Sums

Priem20k2

Attachment 14.6.82

Order 25

Sub Squares Order 5, with Different Magic Sums

Priem25k

Attachment 14.6.90

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-

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Following sections will describe how Inlaid Magic Squares with Concentric Main and/or Sub Squares can be generated with comparable routines as described in previous sections.


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