## Magic Cubes of Prime Numbers

# Competition #1

This competition is organized by Макарова Наталия (Natalia Makarova)

## Definitions

### Definition 1: Magic Cubes

A

magic cubeis the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in ann x n x npattern such that thesum of the numbers on each row, each column, each pillar and the four main space diagonalsis 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 thesum 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:

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

## The Prize

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

Notes:

- 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.
- Prize contestant from Russia will be paid in rubles at the official rate for the end of the competition day.

## Links

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

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