Office Applications and Entertainment, Magic Squares

14.0    Special Magic Squares, Prime Numbers

14.13   Consecutive Primes (1)

Prime Number Magic Squares with Consecutive Primes can be generated with comparable routines as described in Section 14, however based on consecutive prime variable values {a_i} with i = 1 ... n.

14.13.1 Magic Squares (4 x 4)

Based on the equations defining a Magic Square of the fourth order:

a(13) =  s1 - a(14) - a(15) - a(16)
a( 9) =  s1 - a(10) - a(11) - a(12)
a( 7) =       a( 8) - a(10) + a(12) - a(13) + a(16)
a( 6) =  s1 - a( 8) - a(11) - a(12) + a(13) - a(16)
a( 5) =     - a( 8) + a(10) + a(11)
a( 4) =  s1 - a( 7) - a(10) - a(13)
a( 3) = -s1 - a( 8) + a( 9) + 2 * a(10) +         2 * a(13) + a(14)
a( 2) =       a( 8) - a( 9) - 2 * a(10) + a(15) + 2 * a(16)
a( 1) =       a( 8) + a(12) - a(13)

and preselected ranges of consecutive primes, routine Priem4b2 can be used to generate order 4 Prime Number Magic Squares with Consecutive Prime Numbers.

Prime Number Magic Squares with Consecutive Prime Numbers for MC = 258 (32 ea) and MC = 276 (64 ea) were already found by means of routine Priem4 and shown in Attachment 14.2.1.

Attachment 14.13.1 shows for a few higher Magic Sums the first occurring Prime Number Magic Square with Consecutive Prime Numbers.

Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum and variable values.

14.13.2 Magic Squares (5 x 5)

Based on the equations defining a Magic Square of the fifth order:

a(21) =  s1 - a(22) - a(23) - a(24) - a(25)
a(16) =  s1 - a(17) - a(18) - a(19) - a(20)
a(11) =  s1 - a(12) - a(13) - a(14) - a(15)
a( 9) =       a(10) - a(13) + a(15) - a(17) + a(20) - a(21) + a(25)
a( 7) = (s1 - a( 8) - a( 9) - a(10) + a(11) - a(13) + a(16) - a(19) + a(21) - a(25))/2
a( 6) =  s1 - a( 7) - a( 8) - a( 9) - a(10)
a( 5) =  s1 - a( 9) - a(13) - a(17) - a(21)   
a( 4) =  s1 - a( 9) - a(14) - a(19) - a(24)
a( 3) =  s1 - a( 8) - a(13) - a(18) - a(23)
a( 2) =  s1 - a( 7) - a(12) - a(17) - a(22)
a( 1) =  s1 - a( 2) - a( 3) - a( 4) - a( 5)

and preselected ranges of consecutive primes, routine Priem5b2 can be used to generate order 5 Prime Number Magic Squares with Consecutive Prime Numbers.

Attachment 14.13.2 shows for miscellaneous Magic Sums the first occurring Prime Number Magic Square with Consecutive Prime Numbers.

Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum and variable values.

One of the possible aspects of the square with the smallest Magic Sum (MC = 313) was previously published by Max Alekseyey (2009).

14.13.3 Pan Magic Squares (6 x 6)

Based on the equations defining a Pan Magic Square of the sixth order:

a(31) =        s1 - a(32) - a(33) - a(34) - a(35) - a(36)
a(25) =        s1 - a(26) - a(27) - a(28) - a(29) - a(30)
a(19) =        s1 - a(20) - a(21) - a(22) - a(23) - a(24)
a(17) =  2/3 * s1 - a(18) + a(20) + a(21) - a(23) - a(24) + a(26) + a(27) - a(29) - a(30) - a(35) - a(36)
a(16) =    2 * s1 + a(18) - a(20) - 2*a(21) - 2*a(22) - a(23) - a(26) - 2*a(27) - 2*a(28) - a(29) - a(34) + a(36)
a(15) =  2/3 * s1 - a(18) - a(33) - a(36)
a(14) =             a(18) - a(20) - a(21) + a(23) + a(24) - a(26) - a(27) + a(29) + a(30) - a(32) + a(36)
a(13) = -4/3 * s1 - a(18) + a(20) + 2*a(21) + 2*a(22) + a(23) + a(26) + 2*a(27) + 2*a(28) + a(29) - a(31) - a(36)
a(12) = 11/6 * s1 - a(20) - a(21) - a(22) - a(26) - 2*a(27) - a(28) - a(32) - a(33) - a(34)
a(11) = -7/6 * s1 + a(22) + a(23) + a(24) - a(26) + a(28) + a(29) + a(30) + a(34) + a(35) + a(36)
a(10) = -7/6 * s1 + a(21) + a(22) + a(23) - a(25) + a(27) + a(28) + a(29) + a(33) + a(34) + a(35)
a(9)  = -7/6 * s1 + a(20) + a(21) + a(22) + a(26) + a(27) + a(28) - a(30) + a(32) + a(33) + a(34)
a(8)  = 11/6 * s1 - a(22) - a(23) - a(24) - a(28) - 2*a(29) - a(30) - a(34) - a(35)-a(36)
a(7)  = 11/6 * s1 - a(21) - a(22) - a(23) - a(27) - 2*a(28) - a(29) - a(33) - a(34)-a(35)
a(6)  = -5/6 * s1 - a(18) + a(20) + a(21) + a(22) - a(24) + a(26) + 2*a(27) + a(28)-a(30)+a(32)+a(33)+a(34)-a(36)
a(5)  =  1/2 * s1 + a(18) + a(19) + a(24) - a(27) - a(28) - a(29) - a(34) - a(35)
a(4)  =  1/6 * s1 - a(18) + a(20) + a(21) - a(28) - a(29) - a(30) + a(31) + a(32)
a(3)  =  1/2 * s1 + a(18) - a(20) - 2*a(21) - a(22) - a(26) - 2*a(27) - a(28) + a(30) + a(31) + a(35) + 2*a(36)
a(2)  =  1/6 * s1 - a(18) + a(21) + a(22) - a(25) - a(26) - a(30) + a(34) + a(35)
a(1)  =  1/2 * s1 + a(18) + a(23) + a(24) - a(25) - a(26) - a(27) - a(31) - a(32)

and preselected ranges of consecutive primes, routine Priem6b1 can be used to generate order 6 Prime Number Magic Squares with Consecutive Prime Numbers.

Although the first range of 36 Consecutive Primes with a valid Magic Sum MC6 = 930 can be determined, subject routine is not very feasible due to the high number of independent variables (16 ea).

In spite of the above following solution for MC6 = 930 was found by A. W. Johnson Jr (1981/82):

Pan Magic Square

67	193	71	251	109	239
139	233	113	181	157	107
241	97	191	89	163	149
73	167	131	229	151	179
199	103	227	101	127	173
211	137	197	79	223	83

Subject square corresponds with numerous Prime Number Pan Magic Squares with the same Magic Sum and variable values.

14.13.4 Simple Magic Squares (6 x 6)

Prime Number Simple Magic Squares of order 6 can be constructed very efficiently with the Generator Principle, as applied for the construction of Bimagic Squares.

The Generator Method, as applied for Consecutive Prime Numbers can be summarised as follows:

• Generate the Magic Series for the applicable 36 Consecutive Prime Numbers and the related Magic Constant (ref. MgcLns6);
• Construct Generators with 6 Magic Rows, based on the Magic Series obtained above (ref. CnstrGen01);
• Construct Semi Magic Squares based on the Generators obtained above, by permutating the numbers within the rows and determine the number of related Magic Squares (ref. CnstrSqrs6);
• Permutate the rows and columns within the Semi Magic Squares, in order to obtain Magic Squares (if possible).

For the Consecutive Prime Numbers used in Section 14.13.3 above (MC6 = 930), 12980 Magic Series could be generated, resulting in numerous Generators.

In order to limit the collection a little bit, 585 Generators were selected based on the application of 8 Unique Magic Series for each Generator.

The first Generator resulted already in 566 Semi Magic Squares with 671 related Simple Magic Squares.

The first occurring Semi Magic Square and the resulting Simple Magic Square - after row and column permutation(s) - are shown below:

Semi Magic Square 67 79 233 191 181 179 71 229 101 197 193 139 73 239 103 199 149 167 227 83 173 127 163 157 241 89 223 109 131 137 251 211 97 107 113 151

Simple Magic Square 101 197 193 139 71 229 103 199 149 167 73 239 173 127 163 157 227 83 223 109 131 137 241 89 97 107 113 151 251 211 233 191 181 179 67 79

The method described above has been applied for miscellaneous Magic Sums, for which the first occurring Simple Magic Squares are shown in Attachment 14.13.4.

Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum and variable values.

14.13.5a Simple Magic Squares (7 x 7)

Prime Number Simple Magic Squares of order 7 can be constructed with the Generator Principle, as discussed in Section 14.13.4 above.

A possible Semi Magic Square and resulting Simple Magic Square are shown below, for the smallest consecutive prime numbers {7 ... 239} for which an Order 7 Simple Magic Square exists (MC7 = 797):

Semi Magic Square 7 17 193 167 163 137 113 11 19 97 191 179 149 151 13 23 211 181 109 157 103 67 197 199 127 53 71 83 227 223 31 43 61 73 139 233 89 29 41 173 131 101 239 229 37 47 59 79 107

Simple Magic Square 13 23 109 103 181 211 157 227 223 61 139 43 31 73 7 17 163 113 167 193 137 239 229 59 107 47 37 79 11 19 179 151 191 97 149 233 89 173 101 41 29 131 67 197 53 83 127 199 71

The Generator Method has been applied for miscellaneous Magic Sums, for which the first occurring Simple Magic Squares are shown in Attachment 14.13.5.

Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum and variable values.

14.13.5b Inlaid Magic Squares (7 x 7)

Occasionally Prime Number Inlaid Magic Squares of order 7 with Diamond Inlays can be constructed.

An example is shown below, for the consecutive prime numbers {7 ... 239} with the related Magic Sum s7 = 797.

Inlaid Magic Square

199	11	19	107	227	53	181
211	109	151	71	61	157	37
47	103	233	73	149	13	179
43	89	41	131	97	173	223
127	229	113	193	29	83	23
163	59	101	191	67	79	137
7	197	139	31	167	239	17

The Inlaid Magic Square shown above contains:

• One 3^th order Magic Diamond Inlay with Magic Sum s3 = 393
• One 4^th order Magic Diamond Inlay with Magic Sum s4 = 404

and is unique for the applied Magic Sums and variable values.

14.13.6  Simple Magic Squares (8 x 8)

Prime Number Simple Magic Squares of order 8 can be constructed with the Generator Principle, as applied in previous sections.

A possible Semi Magic Square and resulting Simple Magic Square are shown below, for the smallest consecutive prime numbers {79 ... 439} for which an Order 8 Simple Magic Square exists (MC8 = 2016):

Semi Magic Square 439 433 137 157 179 191 229 251 431 421 131 151 277 181 197 227 419 389 127 281 163 193 211 233 379 353 113 149 337 199 223 263 97 109 409 383 173 349 257 239 89 103 401 139 373 307 311 293 83 101 331 359 347 283 271 241 79 107 367 397 167 313 317 269

Simple Magic Square 229 439 433 137 191 179 157 251 197 431 421 131 181 277 151 227 311 89 103 401 307 373 139 293 271 83 101 331 283 347 359 241 223 379 353 113 199 337 149 263 257 97 109 409 349 173 383 239 211 419 389 127 193 163 281 233 317 79 107 367 313 167 397 269

The Generator Method has been applied for miscellaneous Magic Sums, for which the first occurring Simple Magic Squares are shown in Attachment 14.13.6.

Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum and variable values.

14.13.7a Simple Magic Squares (9 x 9)

Prime Number Simple Magic Squares of order 9 can be constructed with the Generator Principle, as applied in previous sections.

A possible Semi Magic Square and resulting Simple Magic Square are shown below, for the smallest consecutive prime numbers {37 ... 479} for which an Order 9 Simple Magic Square exists (MC9 = 2211):

Semi Magic Square 37 41 43 59 463 307 373 409 479 67 83 113 47 359 353 421 389 379 157 127 179 71 439 367 397 191 283 211 151 181 337 227 347 103 383 271 233 263 251 401 223 269 229 149 193 241 311 277 433 73 281 293 163 139 349 331 317 457 313 101 109 97 137 449 443 419 239 53 107 197 173 131 467 461 431 167 61 79 89 257 199

Simple Magic Square 83 359 389 47 67 379 353 113 421 127 439 191 71 157 283 367 179 397 151 227 383 337 211 271 347 181 103 263 223 149 401 233 193 269 251 229 41 463 409 59 37 479 307 43 373 311 73 163 433 241 139 281 277 293 331 313 97 457 349 137 101 317 109 461 61 257 167 467 199 79 431 89 443 53 173 239 449 131 107 419 197

The Magic Square shown above is essential different from the corresponding Magic Square as previously published by A. Susuki (1957).

The Generator Method has been applied for miscellaneous Magic Sums, for which the first occurring Simple Magic Squares are shown in Attachment 14.13.7.

Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum and variable values.

14.13.7b Composed Magic Squares (9 x 9)

Prime Number Composed Magic Squares of order 9 can be constructed when the Magic Sum s9 is a multiple of 9.

An example is shown below, for the consecutive prime numbers {601 ... 1151} with the related Magic Sum s9 = 7821.

Composed Magic Square

1103	619	1021	733	701	1013	853	691	1087
727	1063	653	1033	773	941	811	997	823
607	1051	709	1109	1049	881	751	757	907
1039	743	1093	601	953	641	1061	1031	659
937	827	829	883	613	983	1151	967	631
859	839	769	1009	1097	617	919	643	1069
877	821	991	787	857	977	863	929	719
911	971	947	647	661	1091	673	1123	797
761	887	809	1019	1117	677	739	683	1129

The Magic Square shown above is composed of:

• One 4^th order Magic Corner Square with Magic Sum s4 = 4 * s9 / 9 = 3476 (top/left)
• One 5^th order Magic Corner Square with Magic Sum s5 = 5 * s9 / 9 = 4345 (bottom/right)
• Two Magic Rectangles order 4 x 5 with s4 = 3476 and s5 = 4345

Subject Magic Squares can be transformed to:

• Order 9 Inlaid Magic Squares with order 4 and 5 Square Inlays
• Order 9 Bordered Magic Squares with order 5 Magic Center Square

As illustrated below:

Inlaid Magic Square 613 937 983 827 1151 829 967 883 631 701 1103 1013 619 853 1021 691 733 1087 1097 859 617 839 919 769 643 1009 1069 773 727 941 1063 811 653 997 1033 823 857 877 977 821 863 991 929 787 719 1049 607 881 1051 751 709 757 1109 907 661 911 1091 971 673 947 1123 647 797 953 1039 641 743 1061 1093 1031 601 659 1117 761 677 887 739 809 683 1019 1129

Bordered Magic Square 1103 619 701 1013 853 691 1087 1021 733 727 1063 773 941 811 997 823 653 1033 937 827 613 983 1151 967 631 829 883 859 839 1097 617 919 643 1069 769 1009 877 821 857 977 863 929 719 991 787 911 971 661 1091 673 1123 797 947 647 761 887 1117 677 739 683 1129 809 1019 607 1051 1049 881 751 757 907 709 1109 1039 743 953 641 1061 1031 659 1093 601

It should be noted that the reversed transformation is not necessarily possible because of the bottom-left / top-right Main Diagonal.

Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum(s) and variable values.

14.13.7c Inlaid Magic Squares (9 x 9)
         Diamond Inlays Order 4 and 5

Prime Number Inlaid Magic Squares of order 9 with Diamond Inlays can be constructed when the Magic Sum s9 is a multiple of 9.

An example is shown below, for the consecutive prime numbers {601 ... 1151} with the related Magic Sum s9 = 7821.

Inlaid Magic Square

701	1061	971	769	631	1049	821	827	991
839	829	911	1069	727	967	787	809	883
941	659	719	1021	643	1039	1151	887	761
881	797	619	929	653	919	1103	983	937
1129	1109	1123	1051	863	709	617	607	613
691	683	733	673	1063	977	1093	1097	811
1013	953	739	601	1091	743	857	877	947
773	823	997	677	1033	661	751	1087	1019
853	907	1009	1031	1117	757	641	647	859

The Inlaid Magic Square shown above contains:

• One 4^th order Magic Diamond Inlay with Magic Sum s4 = 4 * s9 / 9 = 3476
• One 5^th order Magic Diamond Inlay with Magic Sum s5 = 5 * s9 / 9 = 4345

and corresponds with miscellaneous Prime Number Magic Squares with the same Magic Sum(s) and variable values.

14.13.8 Summary

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

Following sections will describe how Order 10 Prime Number Magic Squares with Consecutive Primes can be found with comparable routines as described in previous sections.