Problems & Puzzles: Puzzles

Puzzle 689 Pandiagonal and twin primes Radko Nachev sent one example of Pandiagonal Magic Squares composed only by twin primes:   2657 50591 11117 56531   2659 50593 11119 56533 11717 55931 3257 49991   11719 55933 3259 49993 49331 3917 57791 9857   49333 3919 57793 9859 57191 10457 48731 4517   57193 10459 48733 4519     Magical sum 120896       Magical sum 120904 Q1. Is this the 4x4 example with the smallest magical sum? Q2. Find examples for 5x5 & 6x6 (if the smallest magical sum, the better)

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Contribution came from Natalia Makarova and S. Belyaev.

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Natalia wrote:

My program found:

The minimal pandiagonal 4x4 square of twin primes (the first number):

239 5279 1871 4787
4421 2237 2789 2729
4217 1301 5849 809
3299 3359 1667 3851

Magic constant is 12176.

The minimal pandiagonal 4x4 square of twin primes (the second number):

241 5281 1873 4789
4423 2239 2791 2731
4219 1303 5851 811
3301 3361 1669 3853

Magic constant is 12184.

The minimal pandiagonal 5x5 square of twin primes (the first number):

41 2729 2141 599 1049
1721 197 1031 2129 1481
3119 881 1061 1319 179
659 1301 2267 1871 461
1019 1451 59 641 3389

Magic constant is 6559.

The minimal pandiagonal 5x5 square of twin primes (the second number):

43 2731 2143 601 1051
1723 199 1033 2131 1483
3121 883 1063 1321 181
661 1303 2269 1873 463
1021 1453 61 643 3391

Magic constant is 6569.

The program S. Belyaev's found the following pandiagonal 6x6 square of twin primes.

The first number:
5 101 1427 1931 71 419
881 1319 569 29 347 809
179 107 521 59 2027 1061
269 2237 599 431 227 191
239 149 641 1487 821 617
2381 41 197 17 461 857

Magic constant is 3954.

The second number:
7 103 1429 1933 73 421
883 1321 571 31 349 811
181 109 523 61 2029 1063
271 2239 601 433 229 193
241 151 643 1489 823 619
2383 43 199 19 463 859

Magic constant is 3966.
 

I suppose that this square is not minimal.
 

I can work more, to find the best result.

...

She also sent the following object a kind of related to the original question by Radko:

I have pandiagonal square of order 4 of twin primes and it's minimal:

17 191 31 181
151 61 137 71
179 29 193 19
73 139 59 149

Magic constant is equal to 420.
Used a pair of twins:

17,19
29,31
59,61
71,73
137,139
149,151
179,181
191,193

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On June 7, 2013, Natalia Added:

The program S. Belyaev's found new pandiagonal 6x6 square of twin primes.

The first number:

5 431 59 641 197 2081
239 419 1487 101 1019 149
1151 659 569 227 191 617
311 599 29 1319 1049 107
1667 17 461 269 821 179
41 1289 809 857 137 281

Magic constant is 3414.

The second number:

7 433 61 643 199 2083
241 421 1489 103 1021 151
1153 661 571 229 193 619
313 601 31 1321 1051 109
1669 19 463 271 823 181
43 1291 811 859 139 283

Magic constant is 3426.

I suppose that this square is not minimal.

My program found:
the non-minimal pandiagonal 7x7 square of twin primes (the first number):

17 5279 7589 37361 3371 44069087 17189
34031 44066819 6197 13679 4019 1319 13829
17681 59 7559 10499 44097479 3929 2687
44073947 34589 419 6689 13721 6299 4229
2729 19961 2969 44067677 11057 31079 4421
4787 7547 35081 461 8969 16631 44066417
6701 5639 44080079 3527 1277 11549 31121

Magic constant is 44139893.

The non-minimal pandiagonal 7x7 square of twin primes (the second number):

19 5281 7591 37363 3373 44069089 17191
34033 44066821 6199 13681 4021 1321 13831
17683 61 7561 10501 44099481 3931 2689
44073949 34591 421 6691 13723 6301 4231
2731 19963 2971 44067679 11059 31081 4423
4789 7549 35083 463 8971 16633 44066419
6703 5641 44080081 3529 1279 11551 31123

Magic constant is 44139907.

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Jos Luyendijk wrote on Aug 25, 2020:

Although a little bit late, you might like this solution.

MC7a = 359585 Rows
419	3851	6299	49367	56531	87539	155579
33749	132857	52709	59669	71261	2129	7211
54419	63029	98711	85619	3389	5879	48539
70841	1301	5099	9239	75989	137909	59207
88589	5417	48119	53591	60917	74201	28751
51479	81041	144407	70379	881	4271	7127
60089	72089	4241	31721	90617	47657	53171
MC7b = 359599 Rows
421	3853	6301	49369	56533	87541	155581
33751	132859	52711	59671	71263	2131	7213
54421	63031	98713	85621	3391	5881	48541
70843	1303	5101	9241	75991	137911	59209
88591	5419	48121	53593	60919	74203	28753
51481	81043	144409	70381	883	4273	7129
60091	72091	4243	31723	90619	47659	53173