Archivi tag: Magic Squares

Primes k-tuple

K-Tuple, Magic Squares,

Competition #4

This competition is organized by Макарова Наталия (Natalia Makarova)

Definitions

Definition 1:

A prime k-tuple is a finite collection of values (p + a1, p + a2, p + a3, …, p + ak), where p, p + a1, p + a2, p + a3, …, p + ak are prime numbers, (a1, a2, a3, …, ak) are pattern. Typically the first value in the pattern is 0 and the rest are distinct positive even numbers.

We consider the k-tuple, where p + a1, p + a2, p + a3, …, p + ak are consecutive primes.

Definition 2:

k-tuple (p + a1, p + a2, p + a3, …, p + a [k / 2], p + a [k / 2+1], …, p + a [k-2], p + a [k-1], p + ak) for k even, is called symmetric, if the following condition is satisfied:
a1 + ak = a2 + a[k-1] = a3 + a[k-2] = … = a[k/2] + a[k/2+1]

Example: symmetric 8-tuple

(17 + 0, 17 + 2, 17 + 6, 17 + 12, 17 + 14, 17 + 20, 17 + 24, 17 + 26)

Shortened we write this:

17: 0, 2, 6, 12, 14, 20, 24, 26

Definition 3:

k-tuple (p + a1, p + a2, p + a3, …, p + a [(k-1) / 2], p + a [(k-1) / 2 + 1], p + a [(k-1) / 2 + 2], …, p + a [k-2], p + a [k-1], p + ak) for k odd called symmetric, if the following condition is satisfied:
a1 + ak = a2 + a[k-1] = a3 +a [k-2] =…= a[(k-1)/2] + a[(k-1)/2+2] = 2 a[(k-1)/2+1]

Example: symmetric 5-tuple

18713: 0, 6, 18, 30, 36

Definition 4:

The diameter d of k-tuple is the difference of its largest and smallest elements.

Example: 8-tuple

17: 0, 2, 6, 12, 14, 20, 24, 26

It has a diameter d = 26.

Definition 5:

A pandiagonal magic square is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.

The Contest

In the contest is required to compete for those tasks:

Task 1
Required to find k-tuples with the minimal value p:
for an even k > 24; for odd k > 15.

Example

15-tuple, p=3945769040698829 (minimal)
3945769040698829: 0, 12, 18, 42, 102, 138, 180, 210, 240, 282, 318, 378, 402, 408, 420

Task 2
Required to find k-tuples with a minimal diameter d:
for an even k > 10; for odd k > 13.

Example
8-tuple with a minimal diameter d = 26

17: 0, 2, 6, 12, 14, 20, 24, 26

Task 3
Required to find the 16-tuple, the elements of which it is possible to make pandiagonal magic square of order 4 with magic constant S as: 94615738903617540 < S < 29643562211780078520 Example 16-tuple 

23653934725904299: 0, 12, 22, 34, 48, 60, 70, 82, 90, 102, 112, 124, 138, 150, 160, 172

pandiagonal magic square

23653934725904299+
0  160   60  124
82  102   22  138
112   48  172   12
150   34   90   70

S=94615738903617540

Rule:

Prime numbers can contain no more than 100 digits.

In tasks 1 and 2 k <= 50.

Format of solution

For every task the first line is the number of task you are entering: so 1, 2 or 3. After there are those lines according to the tasks you are inserting.

In tasks 1 and 2

k-tuple is represented as

p: a1, a2, a3, …, ak

Example

18713: 0, 6, 18, 30, 36

In task 3
it is 16-tuple and pandiagonal magic square of order 4, composed of the elements of the tuple.

Example

23653934725904299: 0, 12, 22, 34, 48, 60, 70, 82, 90, 102, 112, 124, 138, 150, 160, 172
0, 160, 60, 124, 82, 102, 22, 138, 112, 48, 172, 12, 150, 34, 90, 70

Scoring

The contestant receives one point for every solution to tasks 1 and 3.
For each k in task 2 one point counted towards only those participants, who will have a minimum diameter d.

The Prize

If two or more contestants have equal number of points, the winner will be the entrant who submitted solutions ahead of other contestants.
The winner will receive a prize of 5,000 rubles.
If the winner is not from Russia, the prize will be paid in US dollars at the official exchange rate on the day ending of the contest.

Thanks

We thanks  Wolfram Alpha as we use their API for testing the primality of the given big numbers.

Links

[1] https://en.wikipedia.org/wiki/Prime_k-tuple
[2] https://en.wikipedia.org/wiki/Pandiagonal_magic_square
[3] http://oeis.org/A256234
[4] http://oeis.org/A081235
[5] http://oeis.org/A055380
[6] http://oeis.org/A055382
[7] http://oeis.org/A175309
[8] http://www.primepuzzles.net/problems/prob_060.htm
[9] http://dxdy.ru/topic93581.html
[10] http://dxdy.ru/topic87170.html

Result for Ultra Magic Squares of prime numbers

Magic Squares, Result, ,

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

Ultra Magic Squares of prime numbers

Magic Squares

Competition #3

This competition is organized by Макарова Наталия (Natalia Makarova)

Definitions

Definition 1:

Magic square is called ultra magic square, if it is both associative (center symmetric) and pandiagonal.

Ultra magic squares exist for orders n > 4.

The Contest

In the contest is required to built ultra magic squares of order 7 – 16 of distinct primes.

Example

n = 5
 
 113 1151 1229  911  101
 839  521   41 1013 1091
 941  953  701  449  461
 311  389 1361  881  563
1301  491  173  251 1289

S=3505

This is the minimal solution for n = 5.

Known as a minimal solution for n = 6 (author M. Alekseyev)

103  59 163 233 139 293
229 257 307 131  13  53
283  17  67 173 181 269
 61 149 157 263 313  47
277 317 199  23  73 101
 37 191  97 167 271 227

S=990

Rule:

For each order n = 7 – 16 you can imagine several solutions with magic constants S1 <S2 <S3 …

Known solutions:

n = 7, S = 4613 (author N. Makarova)
n = 8, S = 2640 (author N. Makarova)
n = 9, S = 24237 (author A. Chernov)

Contestant shall not be deemed a winner, if he would submit only solutions with known magic constants

Format of solution

The solution is represented in the form:

n: a (1), a (2), a (3), …, a (n^2)

Example:

5:113,1151,1229,911,101,839,521,41,1013,1091,941,953,701,449,461,311,389,1361,881,563,1301,491,173,251,1289

Scoring

Contestant receives for every n the score: Smin/S, where

Smin – the minimal magic constant of solution in the contest;

S –  the minimal magic constant of solution by contestant.

 

The Prize

Winner receives a prize of 3000 rubles.

If the winner is not from Russia, the prize will be paid in $USA at the official rate on the day of end the contest.

Organizers N. Makarova and S. Tognon can participate in the contest, but does not receive the prize in case of winning.

 

Pandiagonal Squares of Consecutive Primes

Magic Squares,

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