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Ultra Magic Squares of prime numbers

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.

 

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