Office Applications and Entertainment, Magic Squares
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Construction of Prime Number Magic Squares:

Pan Magic Square (4 x 4)

A1, (MC = 100) 7 1 49 43 43 49 1 7 1 7 43 49 49 43 7 1

B1, (MC = 140) 60 10 40 30 40 30 60 10 30 40 10 60 10 60 30 40

C, (MC = 240) 67 11 89 73 83 79 61 17 31 47 53 109 59 103 37 41

Requirements Pan Magic Squares:

• The series {a_i, i = 1 ... 4} and {b_j, j = 1 ... 4} should be balanced:
  a₁ + a₄ = a₂ + a₃ and b₁ + b₄ = b₂ + b₃;
• The Latin Squares A1 and B1 should be Pan Magic, which condition is met by the symmetry applied above:
  A1: top pairs are identical to bottom pairs; B1: left pairs are identical to right pairs.

Ultra Magic Square (5 x 5)

A1, (MC = 2250) 900 450 0 840 60 0 840 60 900 450 60 900 450 0 840 450 0 840 60 900 840 60 900 450 0

B1, (MC = 1255) 461 41 389 251 113 251 113 461 41 389 41 389 251 113 461 113 461 41 389 251 389 251 113 461 41

C, (MC = 3505) 1361 491 389 1091 173 251 953 521 941 839 101 1289 701 113 1301 563 461 881 449 1151 1229 311 1013 911 41

Requirements Ultra Magic Squares:

• The series {a_i, i = 1 ... 5} and {b_j, j = 1 ... 5} should be balanced:
  a₁ + a₅ = a₂ + a₄ = 2 * a₃ and b₁ + b₅ = b₂ + b₄ = 2 * b₃;
• The Latin Pan Magic Squares A1 and B1 should be Associated, which condition is applied above:
  A1: center element a₃; B1: center element b₃.

Magic Square (6 x 6)

General Scheme:

A1 a1 a6 a6 a6 a1 a1 a5 a2 a5 a2 a2 a5 a3 a4 a3 a3 a4 a4 a4 a3 a4 a4 a3 a3 a2 a5 a2 a5 a5 a2 a6 a1 a1 a1 a6 a6

B1 b6 b2 b4 b3 b5 b1 b1 b5 b3 b4 b2 b6 b1 b2 b4 b3 b5 b6 b1 b5 b4 b3 b2 b6 b6 b5 b3 b4 b2 b1 b6 b2 b3 b4 b5 b1

C a1 + b6 a6 + b2 a6 + b4 a6 + b3 a1 + b5 a1 + b1 a5 + b1 a2 + b5 a5 + b3 a2 + b4 a2 + b2 a5 + b6 a3 + b1 a4 + b2 a3 + b4 a3 + b3 a4 + b5 a4 + b6 a4 + b1 a3 + b5 a4 + b4 a4 + b3 a3 + b2 a3 + b6 a2 + b6 a5 + b5 a2 + b3 a5 + b4 a5 + b2 a2 + b1 a6 + b6 a1 + b2 a1 + b3 a1 + b4 a6 + b5 a6 + b1

Balanced series {a_i, i = 1 ... 6} and {b_j, j = 1 ... 6} will definitively return 6^th order Magic Squares
- with Symmetrical Main Diagonals - when applied on Squares A1 and B1.

Example Prime Numbers

A1, (MC = 4500) 47 1453 1453 1453 47 47 1321 179 1321 179 179 1321 199 1301 199 199 1301 1301 1301 199 1301 1301 199 199 179 1321 179 1321 1321 179 1453 47 47 47 1453 1453

B1, (MC = 3150) 1050 210 630 420 840 0 0 840 420 630 210 1050 0 210 630 420 840 1050 0 840 630 420 210 1050 1050 840 420 630 210 0 1050 210 420 630 840 0

C, (MC = 7650) 1097 1663 2083 1873 887 47 1321 1019 1741 809 389 2371 199 1511 829 619 2141 2351 1301 1039 1931 1721 409 1249 1229 2161 599 1951 1531 179 2503 257 467 677 2293 1453

Ultra Magic Square (7 x 7)

A1, (MC = 54271) 5857 9649 6793 7717 7753 7789 8713 7789 8713 5857 9649 6793 7717 7753 7717 7753 7789 8713 5857 9649 6793 9649 6793 7717 7753 7789 8713 5857 8713 5857 9649 6793 7717 7753 7789 7753 7789 8713 5857 9649 6793 7717 6793 7717 7753 7789 8713 5857 9649

B1, (MC = 127050) 0 24780 11520 36300 30330 18150 5970 36300 30330 18150 5970 0 24780 11520 5970 0 24780 11520 36300 30330 18150 11520 36300 30330 18150 5970 0 24780 18150 5970 0 24780 11520 36300 30330 24780 11520 36300 30330 18150 5970 0 30330 18150 5970 0 24780 11520 36300

C, (MC = 181321) 5857 34429 18313 44017 38083 25939 14683 44089 39043 24007 15619 6793 32497 19273 13687 7753 32569 20233 42157 39979 24943 21169 43093 38047 25903 13759 8713 30637 26863 11827 9649 31573 19237 44053 38119 32533 19309 45013 36187 27799 12763 7717 37123 25867 13723 7789 33493 17377 45949

Requirements Ultra Magic Squares:

• The series {a_i, i = 1 ... 7} and {b_j, j = 1 ... 7} should be balanced:
  a₁ + a₇ = a₂ + a₆ = a₃ + a₅ = 2 * a₄ and b₁ + b₇ = b₂ + b₆ = b₃ + b₅ = 2 * b₄
• The Latin Pan Magic Squares A1 and B1 should be Associated, which condition is applied above:
  A1: center element a₄; B1: center element b₄.

Most Perfect Pan Magic Square (8 x 8)

A1, (MC = 19248) 19 83 1019 1583 4793 4729 3793 3229 4793 4729 3793 3229 19 83 1019 1583 19 83 1019 1583 4793 4729 3793 3229 4793 4729 3793 3229 19 83 1019 1583 19 83 1019 1583 4793 4729 3793 3229 4793 4729 3793 3229 19 83 1019 1583 19 83 1019 1583 4793 4729 3793 3229 4793 4729 3793 3229 19 83 1019 1583	B1, (MC = 4776) 0 1194 0 1194 0 1194 0 1194 84 1110 84 1110 84 1110 84 1110 480 714 480 714 480 714 480 714 504 690 504 690 504 690 504 690 1194 0 1194 0 1194 0 1194 0 1110 84 1110 84 1110 84 1110 84 714 480 714 480 714 480 714 480 690 504 690 504 690 504 690 504
C, (MC = 24024) 19 1277 1019 2777 4793 5923 3793 4423 4877 5839 3877 4339 103 1193 1103 2693 499 797 1499 2297 5273 5443 4273 3943 5297 5419 4297 3919 523 773 1523 2273 1213 83 2213 1583 5987 4729 4987 3229 5903 4813 4903 3313 1129 167 2129 1667 733 563 1733 2063 5507 5209 4507 3709 5483 5233 4483 3733 709 587 1709 2087

Requirements Most Perfect Pan Magic Squares:

• The series {a_i, i = 1 ... 8} and {b_j, j = 1 ... 8} should be balanced:
  a₁ + a₈ = a₂ + a₇ = a₃ + a₆ = a₄ + a₅ and b₁ + b₈ = b₂ + b₇ = b₃ + b₆ = b₄ + b₅;
• The orientation of Square B1 should be 90^o rotated compared with Square A1.

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