
In the magic square research was find some of them with other characteristic that make them more interesting. One are the kind that I call double square.
Definition
Give a normal magic square of order N, it is a double square if the square
that we obtain from it when we apply a function to all its element is still magic
(with a different magic constant)
In this case I want to find a prime magic square with its corresponding non prime square still magic. So I want a double square with the function to apply is the P() that we see in the prime analysis.
I choose the order 12 because it is the low, and then I generate many prime squares to see if the non prime was near to the magic state. Well, only 4/5 rows or columns where still magic, so I change method: I build it.
Let me call A the prime square and B the corresponding not prime square.
Suppose that A is magic and we want to make the i rows of B to become
magic: for doing this we must choose the values from row i and from one
row j that give the magic constant for B and that the same shifts don't changes
the sum of the value for row i of square A (that was magic).
In other hand we must give another controls to the procedure that find the magic
state of one row of A (when we build it): it may choose the changes that make even
A and B rows magic.
Ok, if this work ok, what we do for the diagonal?
I think that this was a big problem, so I generate prime square until one had the
corresponding not prime with the diagonals ok; if we add another control that
don't allow changes of value in the diagonals for the procedure that give one
row magic, we have solve the problem.
Yes, but even if I find that square, the procedure didn't find the right solution, because the available changes are very little if we don't allow diagonal elements to changes.
So, we must think for diagonals solution when rows and columns are magic.
Sorry, even if I allow diagonal element to changes, the procedure that give rows
to magic don't make all rows magic.
But this is not a problem, let the procedure to pike up value in other manner, not
only from 1° and 2° rows or 1° and 3° rows, ...
Yes, but I want to find one square, so I take the best square that the automatic
procedure give with rows and columns magic and manually indicate the rows to changes.
This works ok and I have only to give diagonals magic.
So I take the diagonal procedure and use it for making the A square magic, and then look if even B become with magic diagonals value. The important rule is to not stop the diagonal procedure when it find A magic, but let it search for other solution (until B become magic).
With less than 5 hour in the 286 I find this double square:
Normal: 88 9 104 85 15 70 58 49 42 141 106 103 144 54 132 114 40 46 118 26 37 102 10 47 128 84 4 87 140 67 71 108 32 69 50 30 12 83 51 86 31 119 36 77 55 143 142 35 29 14 96 98 125 5 91 38 136 44 64 130 72 112 62 121 23 25 109 27 63 7 127 122 43 45 134 94 34 105 124 139 3 60 24 65 66 33 97 6 90 19 131 41 126 75 48 138 8 81 95 68 133 89 39 20 129 73 22 113 99 137 11 92 82 135 16 120 78 18 61 21 80 111 28 2 57 74 76 110 117 79 123 13 101 107 56 17 100 116 1 115 52 59 93 53 Prime: 457 23 569 439 47 349 271 227 181 811 577 563 827 251 743 619 173 199 647 101 157 557 29 211 719 433 7 449 809 331 353 593 131 347 229 113 37 431 233 443 127 653 151 389 257 823 821 149 109 43 503 521 691 11 467 163 769 193 311 733 359 613 293 661 83 97 599 103 307 17 709 673 191 197 757 491 139 571 683 797 5 281 89 313 317 137 509 13 463 67 739 179 701 379 223 787 19 419 499 337 751 461 167 71 727 367 79 617 523 773 31 479 421 761 53 659 397 61 283 73 409 607 107 3 269 373 383 601 643 401 677 41 547 587 263 59 541 641 1 631 239 277 487 241
Magic Square  Tognon Stefano Research 