Office Applications and Entertainment, Magic Squares

20.0   Special Magic Squares, Prime Numbers, Diamond Inlays

This section describes how Prime Number Magic Squares with Diamond Inlays can be constructed for miscellaneous orders.

20.1.1 Concentric Magic Squares (5 x 5)
       Diamond Inlay Order 3

The defining equations for fifth order Concentric Magic Squares with third order Diamond Inlays are:

a(21) =        s1 - a(22) - a(23) - a(24) - a(25)
a(18) = -0.2 * s1 + 2 * a(23)
a(17) =  0.8 * s1 - a(19) - 2 * a(23)
a(15) =  0.6 * s1 - a(19) - a(23)
a(14) =        s1 - 2 * a(19) - 2 * a(23)
a(10) =        s1 + a(19) - a(20) - a(22) - a(24) - 2 * a(25)
a(13) =  0.2 * s1

with a(19), a(20), a(22) ... a(25) the independent variables, s1 the Magic Sum and p2 = 2 * s1 / 5.

The equations shown above can be incorporated in a guessing routine to generate subject Concentric Magic Squares (ref. ConcDia5), of which an example is shown below:

Mc5 = 7265

2887	619	823	2833	103
109	2383	193	1783	2797
1153	853	1453	2053	1753
313	1123	2713	523	2593
2803	2287	2083	73	19

Attachment 20.1.1 shows one Prime Number Concentric Magic Square with Diamond Inlay for miscellaneous Magic Sums.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

20.1.2 Concentric Magic Squares (7 x 7)
       Diamond Inlay Order 4

It can be proven that order 7 (Prime Number) Concentric Magic Squares with 4 x 4 Diamond Inlay can’t exist for distinct integers.

20.1.3 Concentric Magic Squares (9 x 9)
       Diamond Inlay Order 5

The defining equations for 5 x 5 Diamond Inlays suitable for order 9 Concentric Magic Squares are

a(58) = -  s1/9 - a(59) - a(60) + a(67) + a(69) + 2 * a(77)
a(57) =  6*s1/9 - a(61) - a(67) - a(69) - 2 * a(77)
a(50) = -3*s1/9 + 2 * a(59) + a(67) + a(69)
a(49) =  6*s1/9 - a(51) - 2 * a(59) - a(67) - a(69)
a(45) =  5*s1/9 - a(53) - a(61) - a(69) - a(77)
a(43) =  5*s1/9 - a(51) - a(53) - a(59) - a(69)
a(42) =  7*s1/9 - 2 * a(51) - 2 * a(59) - a(67) - a(69)
a(35) =           a(53) - a(67) + a(69)
a(34) =  4*s1/9 + a(51) - a(52) + a(53) + a(59) - 2 * a(61) - a(67) - 2 * a(77)
a(41) =    s1/9

with a(44), a(51), a(52), a(53), a(59), a(60), a(61), a(67), a(68), a(69), a(77) the independent variables,
s1 the Magic Sum and p2 = 2 * s1 / 9.

The equations shown above can be incorporated in a guessing routine to generate subject diamonds, of which an example is shown below:

Mc5 = 6865

o	o	o	o	5	o	o	o	o
o	o	o	2693	47	2729	o	o	o
o	o	2087	773	677	2609	719	o	o
o	1223	269	1109	2657	353	2477	1523	o
857	2687	1163	617	1373	2129	1583	59	1889
o	1187	1319	2393	89	1637	1427	1559	o
o	o	2027	1973	2069	137	659	o	o
o	o	o	53	2699	17	o	o	o
o	o	o	o	2741	o	o	o	o

Prime Number Concentric Magic Squares with order 5 Diamond Inlays can be constructed as follows:

• Generate order 5 Diamond Inlays;
• Complete the order 7 border (4 x 3 corner numbers), based on a selection from the order 5 Diamond Inlays;
• Complete the order 9 border, based on a selection from the order 7 border / diamond combinations.

Attachment 20.3.1 shows miscellaneous suitable order 5 Diamond Inlays (ref. Diamond5).

Attachment 20.3.2 shows the first occurring order 7 border / diamond combination with 4 x 3 corner numbers, for each of the order 5 Diamond Inlays shown in Attachment 20.3.1 (ref. Priem7e).

Attachment Attachment 20.3.3 shows the first occurring order 9 Concentric Magic Squares for each of the order 7 border / diamond combinations shown in Attachment 20.3.2 (ref. ConcDia9).

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

Note:
It can be proven that order 9 (Prime Number) Concentric Magic Squares with both 4 x 4 and 5 x 5 Diamond Inlays can’t exist for distinct integers.

20.1.4 Concentric Magic Squares (11 x 11)
       Diamond Inlay Order 6

The defining equations for a 6 x 6 Diamond Inlay suitable for order 11 Concentric Magic Squares are

a(92) =   6*s1/11 - a(96) - a(104) - a(106) - 2 * a(116)
a(91) =     s1/11 - a(93) - a(94) - a(95) - a(97) + a(104) + a(106) + 2 * a(116)
a(82) =   6*s1/11 - a(84) - 2 * a(94) - a(104) - a(106)
a(81) =  -  s1/11 - a(83) - a(85) + 2 * a(94) + a(104) + a(106)
a(73) =  -  s1/11 + 0.5 * a(74) + 0.5 * a(84) + 0.5 * a(86) + 0.5 * a(96)
a(72) =  -3*s1/11 + a(94) + a(104) + a(106) + a(116)
a(71) =   7*s1/11 - 0.5 * a(74) - 0.5 * a(84) - 0.5 * a(86) - a(94) - 0.5 * a(96) - a(104) - a(106) - a(116)
a(66) =   6*s1/11 - a(76) - a(86) - a(96) - a(106) - a(116)
a(63) =     p2    + a(64) - a(74) + a(76) - a(83) - a(84) - 2 * a(85) + a(94) + a(106)
a(62) =   9*s1/11 - a(74) - a(84) - a(86) - a(94) - a(96) - a(104) - a(106) - a(116)
a(54) =   6*s1/11 - a(64) - a(74) - a(84) - a(94) - a(104)
a(53) =  17*s1/11 - 2 * a(64) - a(74) - a(75) - a(76) - a(84) - 2*a(86) + a(91) +
                                                      - a(94) - 2*a(96) - a(97) - a(104) - 2*a(106) - 2*a(116)
a(52) =           - a(64) - a(76) + a(84) + a(94) + a(104)
a(42) = -12*s1/11 + a(64) + a(74) + a(76) + a(84) + a(86) + a(94) + 2 * a(96) + a(104) + 2 * a(106) + 2 * a(116)
a(61) =     s1/11

with the independent variables:

       a(i) for i = 64, 65, 74, 75, 76, 83 ... 86, 93 ... 97, 104, 105, 106, 116

       s1 the Magic Sum and p2 = 2 * s1 / 11.

The equations shown above can be incorporated in a guessing routine to generate subject diamonds, of which an example is shown below:

Mc6 = 15342

o	o	o	o	o	7	o	o	o	o	o
o	o	o	o	5011	37	5101	o	o	o	o
o	o	2053	283	4957	523	4987	4933	163	o	o
o	o	3733	4651	1021	1567	3163	2383	1381	o	o
o	2281	457	3943	1933	2971	2767	1171	4657	2833	o
4027	4801	3691	787	3391	2557	1723	4327	1423	313	1087
o	1063	2803	673	2347	2143	3181	4441	2311	4051	o
o	o	211	2731	4093	3547	1951	463	4903	o	o
o	o	4951	4831	157	4591	127	181	3061	o	o
o	o	o	o	103	5077	13	o	o	o	o
o	o	o	o	o	5107	o	o	o	o	o

Prime Number Concentric Magic Squares with order 6 Diamond Inlays can be constructed as follows:

• Generate order 6 Diamond Inlays,
  together with the four corner points of the related order 7 concentric square;
• Complete the order 9 border (4 x 5 corner numbers),
  based on a selection from the Order 7 concentric square / diamond combinations;
• Complete the order 11 border, based on a selection from the order 9 concentric square / diamond combinations.

Attachment 20.4.1 shows miscellaneous suitable order 6 Diamond Inlays and the four corner points of the order 7 concentric squares (ref. Diamond6).

Attachment 20.4.2 shows the first occurring order 9 border / diamond combination with 4 x 5 corner numbers, for each of the order 6 Diamond Inlays shown in Attachment 20.4.1 (ref. Priem9e).

Attachment Attachment 20.4.3 shows the first occurring order 11 Concentric Magic Squares for each of the order 9 border / diamond combinations shown in Attachment 20.4.2 (ref. ConcDia11).

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

20.1.5 Concentric Magic Squares (13 x 13)
       Diamond Inlay Order 7

The defining equations for a 7 x 7 Diamond Inlay suitable for order 13 Concentric Magic Squares are

a(121) =  8 * s1/13 - a(127) - a(135) - a(139) - a(149) - a(151) - 2 * a(163)
a(122) = -    s1/13 - a(123) - a(124) - a(125) - a(126) + a(135) + a(139) + a(149) + a(151) + 2 * a(163)
a(109) =  8 * s1/13 - a(113) - a(123) - a(125) - 2 * a(137) - a(149) - a(151)
a(110) = -3 * s1/13 - a(111) - a(112) + a(123) + a(125) + 2 * a(137) + a(149) + a(151)
a( 98) =  3 * s1/13 - a(109) + 2 * a(111) - a(113) + a(135) - 2 * a(137) + a(139) - a(149) - a(151)
a( 91) =  7 * s1/13 - a(103) - a(115) - a(127) - a(139) - a(151) - a(163)
a( 97) =  8 * s1/13 - a( 99) - 2 * a(111) - a(123) - a(125) - a(135) - a(139)
a( 87) = 15 * s1/13 - a( 99) - a(101) - a(111) - a(115) - a(121) - a(125) - a(127) - a(135) - 2 * a(139) +
                                                                                   - a(149) - a(151) - 2*a(163)
a( 86) =  9 * s1/13 - 2 * a(99) - 2 * a(111) - a(123) - a(125) - a(135) - a(139)
a( 74) =  4 * s1/13 + a( 99) - a(100) + a(101) + a(111) - 2 * a(113) + a(115) - a(123) - 2 * a(137) +
                                                                                  + a(139) - a(149) - a(151)
a( 77) =  7 * s1/13 - a( 89) - a(101) - a(113) - a(125) - a(137) - a(149)
a( 75) =  7 * s1/13 - a( 89) - a(103) - a(113) - a(123) - a(137) - a(151)
a( 63) = -8 * s1/13 + a( 89) + a(101) + a(103) + a(113) + a(115) - a(122) - a(123) - a(124) - a(126) +
                                                         + a(137) + 2 * a(139) + a(149) + 2 * a(151) + 2*a(163)
a( 62) = 12 * s1/13 - a( 75) - a( 88) - a(101) - a(114) - a(122) - a(123) - a(124) - a(125) - a(126) - 2*a(127)

With the independent variables:

   a(i) for i = 76, 89, 90, 91, 99 ... 103, 111 ... 115, 123 ... 127, 135 ... 139, 149, 150, 151, 163

   s1 the Magic Sum and p2 = 2 * s1 / 13.

The equations shown above can be incorporated in a guessing routine to generate subject diamonds, of which an example is shown below:

Mc7 = 59759

o	o	o	o	o	o	17033	o	o	o	o	o	o
o	o	o	o	o	8573	53	47	o	o	o	o	o
o	o	o	o	191	16901	8291	16937	131	o	o	o	o
o	o	o	9491	311	383	401	16553	16823	15797	o	o	o
o	o	6983	3833	16097	587	15287	653	10061	13241	10091	o	o
o	1301	16427	11597	881	12527	5147	7937	16193	5477	647	15773	o
16187	16433	6737	14723	12791	3947	8537	13127	4283	2351	10337	641	887
o	16607	16481	2207	5903	9137	11927	4547	11171	14867	593	467	o
o	o	263	16631	7013	16487	1787	16421	977	443	16811	o	o
o	o	o	1277	16763	16691	16673	521	251	7583	o	o	o
o	o	o	o	16883	173	8783	137	16943	o	o	o	o
o	o	o	o	o	8501	17021	17027	o	o	o	o	o
o	o	o	o	o	o	41	o	o	o	o	o	o

Prime Number Concentric Magic Squares with order 7 Diamond Inlays can be constructed as follows:

• Generate order 7 Diamond Inlays;
• Complete the order 9 border (4 x 3 corner numbers), based on a selection from the order 7 Diamond Inlays;
• Complete the order 11 border (4 x 7 corner numbers), based on a selection from the order 9 border / diamond combinations;
• Complete the order 13 border, based on a selection from the order 11 border / diamond combinations.

Attachment 18.7.7 shows miscellaneous suitable order 7 Diamond Inlays (ref. Diamond7).

Attachment 18.7.8 shows the first occurring order 9 border / diamond combination with 4 x 3 corner numbers, for each of the order 7 Diamond Inlays shown in Attachment 18.7.7 (ref. Priem9f).

Attachment 18.7.9 shows the first occurring order 11 border / diamond combination with 4 x 7 corner numbers, for each of the order 7 Diamond Inlays shown in Attachment 18.7.8 (ref. Priem11f).

Attachment Attachment 18.7.10 shows the first occurring order 13 Concentric Magic Squares for each of the order 11 border / diamond combinations shown in Attachment 18.7.9 (ref. MgcSqr13a).

20.2.1 Associated Magic Squares (5 x 5)
       Diamond Inlay Order 3

The defining equations for fifth order Associated Magic Squares with third order Diamond Inlays are:

a(21) =        s1 - a(22) - a(23) - a(24) - a(25)
a(18) =  0.8 * s1 + a(19) - 2 * a(20) - a(22) + 2 * a(23) - a(24) - 2 * a(25)
a(17) =  0.8 * s1 - a(19) - 2 * a(23)
a(16) = -0.6 * s1 - a(19) + a(20) + a(22) + a(24) + 2 * a(25)
a(15) =  0.6 * s1 - a(19) - a(23)
a(14) =        s1 - 2 * a(19) + a(22) - 2 * a(23) - a(24)
a(13) =  0.2 * s1

with a(19), a(20), a(22) ... a(25) the independent variables, s1 the Magic Sum and p2 = 2 * s1 / 5.

The equations shown above can be incorporated in a guessing routine to generate subject Associated Magic Squares (ref. AssDia5), of which an example is shown below:

Mc5 = 1255

461	233	239	11	311
149	83	101	443	479
431	389	251	113	71
23	59	401	419	353
191	491	263	269	41

Attachment 20.5.1 shows one Prime Number Associated Magic Square with Diamond Inlay for miscellaneous Magic Sums.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

20.2.2 Associated Magic Squares (7 x 7)
       Diamond Inlays Order 3 and 4

The defining equations for Order 7 Associated Magic Squares with Order 3 and 4 Diamond Inlays are:

a(46) =   4 * s1/7 - a(40) - a(34) - a(28)
a(45) =       s1/7 + a(47) - 2 * a(27) + a(26) - a(20) + a(46) + a(40) - a(28)
a(43) =       s1   - a(44) - a(45) - a(47) - a(48) - a(49) - a(46)
a(39) =   3 * s1/7 - a(33) - a(27)
a(38) =              a(20) - a(46) + a(28)
a(37) = - 3 * s1/7 + a(41) - a(44) + a(48) + a(27) + a(20) + a(34)
a(36) =       s1   - 2 * a(41) - a(42) + a(44) - a(48) + a(33) - 2 * a(20) + a(46) - a(40) - a(34) - a(28)
a(35) = (17 * s1/7 - 2 * a(41) - 2*a(42) - 2*a(47) - 2 * a(48) - 2 * a(49) - a(33) + 2 * a(20) +
                                                           - 2 * a(46) - 2 * a(40) - 2 * a(34)) / 2
a(32) =   4 * s1/7 - a(26) - 2 * a(20) + a(46) - a(28)
a(29) =   3 * s1/7 - a(35) - 2 * a(33) - 2 * a(27) + a(26) + 3 * a(20) - a(46) - a(34) + a(28)
a(26) =   4 * s1/7 - a(20) -     a(34) -     a(28)
a(19) =   4 * s1/7 - a(33) - 2 * a(27)
a(25) =       s1/7

with the independent variables

    a(i) for i = 20, 27, 28, 33, 34, 40, 41, 42, 44, 47, 48, 49

    s1 the Magic Sum and p2 = 2 * s1 / 7.

The equations shown above can be incorporated in a guessing routine to generate subject Associated Magic Squares (ref. AssDia7a), of which an example is shown below:

Mc7 = 4781

509	1307	1277	389	149	797	353
647	383	137	1193	1109	503	809
317	887	263	1097	83	1187	947
1319	593	347	683	1019	773	47
419	179	1283	269	1103	479	1049
557	863	257	173	1229	983	719
1013	569	1217	977	89	59	857

Attachment 20.6.1 shows one Prime Number Associated Magic Square with Diamond Inlays for miscellaneous Magic Sums.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

Note:
For larger Magic Sums a more efficient Border Routine has been developed which is incorporated in AssDia7b.

20.2.3 Associated Magic Squares (9 x 9)
       Diamond Inlays Order 4 and 5

Order 9 Associated Magic Squares with Order 4 and 5 Diamond Inlays can be constructed as follows:

• Read previously generated Order 5 Associated - or Ultra Magic Diamonds with Magic Sum s5 = 5*s9/9;
• Generate Order 4 Associated Magic Diamonds with Magic Sum s4 = 4*s9/9;
• Complete the Order 9 Associated Magic Squares with the remaining Border Pairs.

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(79) =     s3   - a9(80) - a9(81)
a9(70) =     s3   - a9(71) - a9(72)
a9(63) =     s3   - a9(72) - a9(81)
a9(62) =     s3   - a9(71) - a9(80)
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 the independent border variables

    a9(i) for i = 56, 65, 66, 71, 72, 78, 80, 81

    s9 the Magic Sum, s3 = s9 / 3 and p2 = 2 * s9 / 9.

The equations shown above can be incorporated in a guessing routine to complete subject Associated Magic Squares (ref. AssDia9), of which an example is shown below:

Mc9 = 28719

6269	2153	1151	5639	2579	599	239	3989	6101
1931	1439	6203	3413	569	3581	3539	2141	5903
1373	5981	1973	863	3719	6311	4493	3923	83
269	2111	4973	4049	5021	1193	5861	4799	443
5879	6359	3023	911	3191	5471	3359	23	503
5939	1583	521	5189	1361	2333	1409	4271	6113
6299	2459	1889	71	2663	5519	4409	401	5009
479	4241	2843	2801	5813	2969	179	4943	4451
281	2393	6143	5783	3803	743	5231	4229	113

Attachment 20.7.1 shows one Prime Number Associated Magic Square with Diamond Inlays for miscellaneous Magic Sums.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

20.2.4 Associated Magic Squares (11 x 11)
       Diamond Inlays Order 5 and 6

Order 11 Associated Magic Squares with Order 5 and 6 Diamond Inlays can be constructed as follows:

• Read previously generated order 6 Associated Magic Diamonds with Magic Sum s6 = 6*s1/11;
• Generate order 5 Associated Magic Diamonds with Magic Sum s5 = 5*s1/11;
• Complete the Order 11 Associated Magic Squares with the remaining Border Pairs.

It is convenient to split the bottom row and right column into parts summing to s4 = 4*s1/11 and s7 = 7*s1/11, which results in following border equations:

a(115) =   s1 - a(  5) - a( 16) - a( 27) - a( 38) - a( 49) - a( 60) - a( 71) - a( 82) - a( 93) - a(104)
a( 55) =   s1 - a( 45) - a( 46) - a( 47) - a( 48) - a( 49) - a( 50) - a( 51) - a( 52) - a( 53) - a( 54)
a(118) =   s4 - a(119) - a(120) - a(121)
a(103) =   s1 - a(  4) - a( 15) - a( 26) - a( 37) - a( 48) - a( 59) - a( 70) - a( 81) - a( 92) - a(114)
a( 88) =   s4 - a( 99) - a(110) - a(121)
a( 87) =   s1 - a( 78) - a( 79) - a( 80) - a( 81) - a( 82) - a( 83) - a( 84) - a( 85) - a( 86) - a( 88)
a(113) =   s1 - a(111) - a(112) - a(114) - a(115) - a(116) - a(117) - a(118) - a(119) - a(120) - a(121)
a( 89) =   s1 - a(  1) - a( 12) - a( 23) - a( 34) - a( 45) - a( 56) - a( 67) - a( 78) - a(100) - a(111)
a(101) =   s1 - a(100) - a(102) - a(103) - a(104) - a(105) - a(106) - a(107) - a(108) - a(109) - a(110)
a( 90) =   s1 - a(  2) - a( 13) - a( 24) - a( 35) - a( 46) - a( 57) - a( 68) - a( 79) - a(101) - a(112)
a( 97) =(2*s1/11 - a(9)- a( 20) + a( 23) + a( 24) + a( 26) + a( 27) + a( 28) + a( 29) + a( 30) + a( 32) + 
                                         + a(33) - a(42) - a(53) - a(64) - a(75) - a(86) - a(108) - a(119)) / 2
a( 91) =   s1 - a( 97) - a( 89) - a( 90) - a( 92) - a( 93) - a( 94) - a( 95) - a( 96) - a( 98) - a( 99)

with the independent border variables:

   a(i) for i = 43, 44, 77, 98, 99, 100, 102, 107 ... 112, 114, 117, 119, 120, 121

   s1 the Magic Sum, p2 = 2 * s1 / 11 and s4 = 2 * p2.

The equations shown above can be incorporated in a guessing routine to generate subject Associated Magic Squares (ref. AssDia11), of which an example is shown below:

Mc11 = 36619

6653	5897	677	89	461	5171	6389	6521	857	947	2957
5507	2579	2531	6269	4679	1637	1907	4451	647	1361	5051
809	719	6101	5039	4421	1559	3929	1427	2309	4217	6089
347	3881	401	2711	2939	3617	4241	5351	5861	5399	1871
6551	167	2399	4937	3167	2999	1511	1787	4547	3467	5087
4517	5477	5441	3821	5639	3329	1019	2837	1217	1181	2141
1571	3191	2111	4871	5147	3659	3491	1721	4259	6491	107
4787	1259	797	1307	2417	3041	3719	3947	6257	2777	6311
569	2441	4349	5231	2729	5099	2237	1619	557	5939	5849
1607	5297	6011	2207	4751	5021	1979	389	4127	4079	1151
3701	5711	5801	137	269	1487	6197	6569	5981	761	5

Attachment 20.8.1 shows one Prime Number Associated Magic Square with Diamond Inlays for miscellaneous Magic Sums.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

20.2.5 Associated Magic Squares (13 x 13)
       Diamond Inlays Order 6 and 7

Order 13 Associated Magic Squares with Order 6 and 7 Diamond Inlays can be constructed as follows:

• Read previously generated order 7 Associated Magic Diamonds with Magic Sum s7 = 7*s1/13;
• Generate order 6 Associated Magic Diamonds with Magic Sum s6 = 6*s1/13;
• Complete the Order 13 Associated Magic Squares with the remaining Border Pairs.

It is convenient to split the two bottom rows and right columns into parts summing to s4 = 4*s1/13 and s9 = 9*s1/13, which results in following border equations:

a(162) = s1 - a(  6) - a( 19) - a( 32) - a( 45)-a( 58)-a( 71)-a( 84)-a( 97) - a(110) - a(123) - a(136) - a(149)
a(161) = s5 - a(162) - a(163) - a(164) - a(165)
a(148) = s1 - a(  5) - a( 18) - a( 31) - a( 44)-a( 57)-a( 70)-a( 83)-a( 96) - a(109) - a(122) - a(135) - a(161)
a( 78) = s1 - 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(116) = s1 - a(105) - a(106) - a(107) - a(108)-a(109)-a(110)-a(111)-a(112) - a(113) - a(114) - a(115) - a(117)
a(166) = s4 - a(167) - a(168) - a(169)
a(130) = s4 - a(143) - a(156) - a(169)
a(153) = s4 - a(154) - a(155) - a(156)
a(129) = s4 - a(142) - a(155) - a(168)
a(160) = s4 - a(159) - a(158) - a(157)
a(118) = s1 - a(  1) - a( 14) - a( 27) - a( 40)-a( 53)-a( 66)-a( 79)-a( 92) - a(105) - a(131) - a(144) - a(157)
a(147) = s1 - a(144) - a(145) - a(146) - a(148)-a(149)-a(150)-a(151)-a(152) - a(153) - a(154) - a(155) - a(156)
a(119) = s1 - a(  2) - a( 15) - a( 28) - a( 41)-a( 54)-a( 67)-a( 80)-a( 93) - a(106) - a(132) - a(145) - a(158)
a( 50) = s1 - a( 40) - a( 41) - a( 42) - a( 43)-a( 44)-a( 45)-a( 46)-a( 47) - a( 48) - a( 49) - a( 51) - a( 52)
a(133) = s1 - a(  3) - a( 16) - a( 29) - a( 42)-a( 55)-a( 68)-a( 81)-a( 94) - a(107) - a(120) - a(146) - a(159)

a(140) =(p2 - a(10) - a(23) + a(27) + a(28) + a(29) + a(31) + a( 32) + a( 33) + a( 34) + a( 35) + a( 37) + 
            + a(38) + a(39) - a(49) - a(62) - a(75) - a(88) - a(101) - a(114) - a(127) - a(153) - a(166)) / 2

a(134) = s1 - a(140) - a(131) - a(132) - a(133)-a(135)-a(136)-a(137)-a(138) - a(139) - a(141) - a(142) - a(143)

with the independent border variables:

   a(i) for i = 64, 65, 104, 117, 128, 131, 132, 141 ... 146, 152, 154 ... 159, 164, 165, 167, 168 and 169

   s1 the Magic Sum, p2 = 2 * s1 / 13, s4 = 2 * p2 and s5 = 5 * s1 / 13.

The equations shown above can be incorporated in a guessing routine to generate subject Associated Magic Squares (ref. AssDia13), of which an example is shown below:

Mc13 = 113711

467	16931	15383	2207	2441	7727	16553	16067	947	1571	16007	16217	1193
16673	887	14717	2711	14633	8237	101	10301	15683	1493	4391	7691	16193
16127	14087	1913	13421	13121	17489	7817	11	2357	2903	6047	3191	15227
1721	3083	4751	1523	7457	1733	17387	13577	6221	15077	15671	16187	9323
17093	1847	1061	9791	2543	17183	2957	12413	16811	11351	2837	593	17231
13913	16481	17471	6323	557	7331	461	13337	5003	12161	17477	863	2333
4253	173	15881	17327	15017	14753	8747	2741	2477	167	1613	17321	13241
15161	16631	17	5333	12491	4157	17033	10163	16937	11171	23	1013	3581
263	16901	14657	6143	683	5081	14537	311	14951	7703	16433	15647	401
8171	1307	1823	2417	11273	3917	107	15761	10037	15971	12743	14411	15773
2267	14303	11447	14591	15137	17483	9677	5	4373	4073	15581	3407	1367
1301	9803	13103	16001	1811	7193	17393	9257	2861	14783	2777	16607	821
16301	1277	1487	15923	16547	1427	941	9767	15053	15287	2111	563	17027

It can be noticed that the order 7 Diamond Inlay of the square shown above contains an order 3 and an order 4 Diamond Inlay as well.

Attachment 20.8.2 shows one Prime Number Associated Magic Square with Diamond Inlays for miscellaneous Magic Sums.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

20.3   Summary

The obtained results regarding the miscellaneous Magic Squares with Diamond Inlays as deducted and discussed in previous sections are summarized in following table:

Following sections will describe and illustrate how Prime Number (Pan) Magic Squares can be constructed based on the sum of suitable selected Latin Squares.