Crea sito

Pandiagonal Squares of Consecutive Primes

Competition #2

This competition is organized by Макарова Наталия (Natalia Makarova)

Definitions

Definition 1: Pandiagonal magic squares

Magic square is called pandiagonal, if the sum of the numbers in each of the broken diagonal is the magic constant of the square.

For example – pandiagonal square of order 8 of prime numbers:

  5   13  463  293  443  283   53   31
313  379   71   73   89   79  191  389
 23  211  167  331  199  353  149  151
449  239   41   97   59  127  349  223
 19   47  439  269  457  317   29    7
241  383  109  103   17   83  229  419
101  139  181  311  277  281  163  131
433  173  113  107   43   61  421  233

Magic constant of the square S = 1584

The Contest

In the contest is required to construct pandiagonal squares of consecutive primes.
It is necessary to solve the problem for the orders n = 4 – 10.

Rule:

Basic rule: solutions with known magic constants are not accepted.
n=4: Required to find solutions with magic constant S > 682775764735680
n=6: Required to find solutions with magic constant S > 930

Known solutions

  1. n = 4 (minimal, author Max Alekseyev)
    170693941183817+
    0     116  132  164 
    162 134   30    86 
    74   42     206  90 
    176 120   44    72
    
    S=682775764735680
    

    See
    http://oeis.org/A245721
    http://dxdy.ru/post891839.html#p891839

  2. n=4 (authors J. Wroblewski and J. K. Andersen)
    320572022166380833+
    0   88  16  84
    76 24  60 28
    78 10  94 6
    34 66 18  70
    
    S = 1282288088665523520
    

    See
    http://dxdy.ru/post751928.html#p751928
    http://www.primepuzzles.net/conjectures/conj_042.htm

    Required to find solutions with magic constant S > 682775764735680.

  3. n = 6 (minimal)
    67+
    0 126 4 184 42 172
    72 166 46 114 90 40
    174 30 124 22 96 82
    6 100 64 162 84 112
    132 36 160 34 60 106
    144 70 130 12 156 16
    
    S=930
    

    See
    http://oeis.org/A073523

    Required to find solutions with magic constant S > 930.

  4. For the orders n = 5, 7 – 10 unknown solutions.

Format of solution

Solutions should be introduced in a normalized form, plus dimension.
The first line must contains the dimension of problem (4, 5..10)
The second the normalized form of solution.
For example:

6
67: 0,126,4,184,42,172,72,166,46,114,90,40,174,30,124,22,96,82,6,100,64,162,84,112,132,36,160,34,60,106,144,70,130,12,156,16

Scoring

Contestant receives one point for each new decision.
Solutions with the same magic constant are considered equal, even if they are not isomorphic.

The Prize

Instituted a price to the participant who has won first place – $ 100 USA.
In cases of more people will have the same final score for the first positons, the price goes to the one that makes that score before the other.
If the winner will be the contestant from Russia, he will receive a prize in rubles at the official exchange rate on the last day of the competition.

The competition was taken from 23/09/2014 to 23/12/2014 but due to no winners (no one introduce a valid solution), it is now extended just for fun and not for price.

Help links

  1. https://ru.wikipedia.org/wiki/Магический_квадрат
  2. https://en.wikipedia.org/wiki/Magic_square
  3. B. Rosser and R. J. Walker. The algebraic theory of diabolic magic squares
    http://yadi.sk/d/tl-_Ab-o5AYhS
  4. N. Makarova. Unconventional pandiagonal squares of primes
    http://www.natalimak1.narod.ru/panpr.htm
    http://www.natalimak1.narod.ru/pannetr.htm
    http://www.natalimak1.narod.ru/pannetr2.htm
  5. Contest “Pandiagonal Magic Squares of Prime Numbers” (from Al Zimmermann)
    http://www.azspcs.net/Contest/PandiagonalMagicSquares/FinalReport
  6. The smallest magic constant for any n x n magic square made from consecutive primes
    http://oeis.org/A073520
  7. The smallest magic constant of pan-diagonal magic squares which consist of distinct prime numbers
    http://oeis.org/A179440
1 vote

Leave a Response