Archivi tag: Prime numbers

Result for Ultra Magic Squares of prime numbers

The competition Ultra Magic Squares of prime numbers is over.

Thanks to all people that have partecipate.

The result is in this table:

Task
Dim.
User
Magic
Assoc.
Result
0
7
primesmagicgames
4613
1318
227 617 677 431 1217 1307 137 1259 827 1061 509 521 167 269 347 929 1187 17 557 719 857 89 479 29 659 1289 839 1229 461 599 761 1301 131 389 971 1049 1151 797 809 257 491 59 1181 11 101 887 641 701 1091
0
8
Natalia Makarova
2040
510
241 199 409 467 47 79 359 239
421 137 7 53 487 179 317 439
31 281 347 353 227 277 127 397
449 197 109 379 491 337 11 67
443 499 173 19 131 401 313 61
113 383 233 283 157 163 229 479
71 193 331 23 457 503 373 89
271 151 431 463 43 101 311 269
0
9
Natalia Makarova
13059
2902
2843 149 1973 2039 971 1031 2141 293 1619
2063 563 1811 113 2549 1601 2633 1721 5
2393 503 1613 2381 1193 41 2411 101 2423
173 2711 2879 773 1583 1493 461 443 2543
569 83 821 311 1451 2591 2081 2819 2333
359 2459 2441 1409 1319 2129 23 191 2729
479 2801 491 2861 1709 521 1289 2399 509
2897 1181 269 1301 353 2789 1091 2339 839
1283 2609 761 1871 1931 863 929 2753 59
0
10
Natalia Makarova
46150
9230
9133 2017 1069 1669 3583 4999 8629 1489 6343 7219 5209 4219 5101 6793 43 6841 7951 2683 5557 1753 7603 7369 6883 8059 8863 919 1471 769 4111 103 163 8179 4723 4243 4663 5869 1741 6553 1723 8293 2833 4051 709 1021 7177 5701 1993 6991 8101 7573 1657 1129 2239 7237 3529 2053 8209 8521 5179 6397 937 7507 2677 7489 3361 4567 4987 4507 1051 9067 9127 5119 8461 7759 8311 367 1171 2347 1861 1627 7477 3673 6547 1279 2389 9187 2437 4129 5011 4021 2011 2887 7741 601 4231 5647 7561 8161 7213 97

Here are the best solution arrived after the ending of the competition:

Task
Dim.
User
Magic
Assoc.
Result

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

Magic Cubes of Prime Numbers

Competition #1

This competition is organized by Макарова Наталия (Natalia Makarova)

Definitions

Definition 1: Magic Cubes

A magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in an n x n x n pattern such that the sum of the numbers on each row, each column, each pillar and the four main space diagonals is equal to a single number, the so-called magic constant (S) of the cube.

For example – the classic magic cube of order 3:

18 23 1
22 3 17
2 16 24

20 7 15
9 14 19
13 21 8

4 12 26
11 25 6
27 5 10

S=42

Definitions 2

The magic cube is associative (central symmetric) if the sum of any 2 numbers, symmetrically located relative to the center of the cube, is equal to a single number, the so-called constant of associativity of the cube.

For example – an associative magic cube of order 5 of arbitrary natural numbers:

43357 31873 31741 38041 43423
43567 29593 15685 47515 52075
6547  47647 75373 50713 8155
43513 39643 31723 31687 41869
51451 39679 33913 20479 42913

34933 34051 41611 34297 43543
35317 37327 42247 37423 36121
36277 43123 32311 42211 34513
40807 41401 32143 34183 39901
41101 32533 40123 40321 34357

36667 32563 38371 37561 43273
40573 36637 43621 33793 33811
37531 39841 37687 35533 37843
41563 41581 31753 38737 34801
32101 37813 37003 42811 38707

41017 35053 35251 42841 34273
35473 41191 43231 33973 34567
40861 33163 43063 32251 39097
39253 37951 33127 38047 40057
31831 41077 33763 41323 40441

32461 54895 41461 35695 23923
33505 43687 43651 35731 31861
67219 24661 1     27727 68827
23299 27859 59689 45781 31807
31951 37333 43633 43501 32017

S=188435

Magic cubes of order 3 are simple magic cubes.
All magic cubes of order 3 are associative (see [3], [7]).

The Contest

This competition requires to make magic cubes of distinct primes:

  1. magic cubes of orders n = 4, 5, 6, 7 (by definition 1) –  task #1
  2. associative magic cubes of orders n = 4, 5, 6, 7 (by definition 2) – task #2

For example – magic cube of order n = 3 of prime numbers:

1061 3167 863
2243 431 2417
1787 1493 1811

2447 23 2621
1871 1697 1523
773 3371 947

1583 1901 1607
977 2963 1151
2531 227 2333

S = 5091

There are thus 8 distinct problems.

All prime numbers must be less than 2*10^9 (exceptions are made for n = 7).
Solutions can have magic constants S1> S2> S3> …> Smin.

Rule:

Two solutions of task #1 with equal magic constants S are considered equal, even if they  are not equivalent.

This rule also applies to the solutions of the task #2.
Note:
Solutions are called equivalent if they are obtained by rotations and reflections.

Format of solution

The first line is written task number (1 or 2) and the order of the cube (4, 5, 6, 7), separated by commas. For example, in the task 1 for n = 4 in the first row must be written: 1,4

The second line is recorded magic cube elements, separated by commas, for example:

1061,3167,863,2243,431,2417,1787,1493,1811,2447,23,2621,1871,1697,1523,773,3371,947,1583,1901,1607,977,2963,1151,2531,227,2333

In this line you can insert a “new line” in any place, for example:

1061,3167,863,2243,431,2417,1787,1493,
1811,2447,23,2621,1871,1697,1523,773,
3371,947,1583,1901,1607,977,2963,1151,2531,227,2333

Earning points for solutions

Policies is the same for both tasks. Scores received by the participant for solving task #1 and task #2 are summarized.
Let solutions contestants A, B, C, D, E for n = 3 (although the order is not included in the contest problem, but the real values ​​are known magic constants [7]).

contest2

The Prize

Established a prize to the participant who has won first place – $100 (U.S.)

Notes:

  1. Prize to the participant who has won first place in the number of points will not be awarded if he would submit to the contest only known solutions, posted on the Internet. See above Rule. In this case consider the following results in the standings participants.
  2. Prize contestant from Russia will be paid in rubles at the official rate for the end of the competition day.

Links

  1. Theme “Magic Cubes” forum dxdy.ru: http://dxdy.ru/topic27852.html
  2. Theme “Magic Cubes” forum Portal Natural Sciences: http://e-science.ru/forum/index.php?showtopic=31432&st=0
  3. N. Makarova. Magic cubes third order: http://www.natalimak1.narod.ru/kub3.htm
  4. N. Makarova. Magic cubes fourth order: http://www.natalimak1.narod.ru/kub4.htm
  5. The Magic Encyclopedia: http://www.magichypercubes.com/Encyclopedia/DataBase/DataBase.html
  6. Website Christian Boyer (France): http://www.multimagie.com/
  7. Magic constants of the magic cubes 3 x 3 x 3 made of primes: https://oeis.org/A239671
  8. Wikipedia article “Magic cube”: http://en.wikipedia.org/wiki/Magic_cube
  9. Magic Cubes – Order 5: http://www.magic-squares.net/c-t-htm/c_cube-5.htm#Hugel
  10. K. Y. Lin. Magic Cubes and Hypercubes of order 3: http://yadi.sk/d/1EGOJXcyKExTi
  11. Andrews W. S. Magic Squares& Cubes, Dover Publ, 1960 (original publication Open Court, 1917): http://yadi.sk/d/mYEzZjdR5pX8Y
  12. Prime Number Magic Cubes: http://www.magic-squares.net/c-t-htm/c_prime.htm