Problems & Puzzles: Puzzles

Puzzle 671 Most Perfect Magic Squares

Natalia Makarova sent the following hard puzzle, on November 2012:

Definition: Pandiagonal classic magic square of order n = 4k, k = 1, 2, 3, ... this is the most perfect magic square if the following properties:

1. Every 2 x 2 block of cells (including wrap-around) sum to 2T (where T= n^2 + 1) (i.e. compact);
2. Any pair of integers distant n/2 along a diagonal sum to T (i.e. complete). 

For example the most perfect classic magic square of order 4:

1 8 11 14
12 13 2 7
6 3 16 9
15 10 5

Magical sum = 34

Definition: Non-traditional pandiagonal magic square of order n = 2k, k = 2, 3, ... this is the most perfect magic square if the following properties:

1. Every 2 x 2 block of cells (including wrap-around) sum to 2T (where T= 2S/n, S magic constant) (i.e. compact);
2. Any pair of integers distant n/2 along a diagonal sum to T (i.e. complete). 

For example the non-traditional most perfect magic square of order 4 of prime numbers:

7 107 23 103
89 37 73 41
97 17 113 13
47 79 31 83 

Magic constant of the square is minimal 240.

I found most perfect magic square of prime numbers of order 6, 8:

149 9161 2309 6701 2609 8861
9067 1483 6907 3943 6607 1783
4139 5171 6299 2711 6599 4871
3229 7321 1069 9781 769 7621
5987 3323 8147 863 8447 3023
7219 3331 5059 5791 4759 3631

Magic constant 29790.

19 5923 1019 4423 4793 1277 3793 2777
4877 1193 3877 2693 103 5839 1103 4339
499 5443 1499 3943 5273 797 4273 2297
5297 773 4297 2273 523 5419 1523 3919
1213 4729 2213 3229 5987 83 4987 1583
5903 167 4903 1667 1129 4813 2129 3313
733 5209 1733 3709 5507 563 4507 2063
5483 587 4483 2087 709 5233 1709 3733

Magic constant 24024.

Q1. It is possible to make similar squares with a smaller magic constant?

Q2. Can you make a non-traditional most perfect magic square of order 10 of prime numbers?

 


On Jun-1-2015, Natalia sent the following message:

I proved that the solutions of the puzzle # 671 for n = 6, n = 8 are minimal.

See https://oeis.org/A258082

Now we need to find the solutions for n > 8.

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