Most perfect magic square for each order

 

[3x3]  The 3x3 magic square is the only magic square, which has no extra magic features. Though the 3x3 magic square is symmetric (= one extra magic feature).

 

[4x4]  The most magic 4x4 square is the panmagic 4x4 square. The panmagic 4x4 square is also the smallest most perfect magic square. The square is panmagic and (2x2) compact.

 

[5x5]  The most magic 5x5 square is the ultra magic 5x5 square. The ultra magic 5x5 square is panmagic and symmetric.

 

[6x6]  The most magic 6x6 square is a 6x6 square with a 4x4 panmagic inlay.

 

[7x7]  The most magic 7x7 square is the ultra magic 7x7 square. The ultra magic 7x7 square is panmagic and symmetric.

 

[8x8]  The most magic 8x8 square is a most perfect (Franklin pan)magic 8x8 square (with the extra tight 'Willem Barink' structure)

 

[9x9]  The most magic 9x9 square is an ultra magic 9x9 square. The ultra magic 9x9 square is panmagic, symmetric, (3x3) compact and each 1/3 row and 1/3 column gives 1/3 of the magic sum.

 

[10x10]  The most magic 10x10 square is a concentric magic 10x10 square.

 

[11x11]  The most magic 11x11 square is the ultra magic 11x11 square. The ultra magic 11x11 square is panmagic and symmetric.

 

[12x12]  The most magic 12x12 square is a most perfect magic 12x12 square (with the extra tight 'Willem Barink' structure) or an ultra magic 12x12 square. The ultra magic 12x12 square is panmagic, symmetric inside each 4x4 sub/square, (2x2) compact and each 1/2 row, 1/2 column and each 1/3 diagonal gives a proportional part of the magic sum. 

 

[13x13]  The most magic 13x13 square is the ultra magic 13x13 square. The ultra magic 13x13 square is panmagic and symmetric.


[14x14]
 
The most magic 14x14 square is a 14x14 inlaid square.

 

[15x15]  The most magic 15x15 square is an ultra magic 15x15 square. The ultra magic 15x15 square is panmagic, symmetric and double (3x3 and 5x5) compact.

 

[16x16]  The most magic 16x16 square is a most perfect (Franklin pan)magic 16x16 square (with the extra tight 'Willem Barink' structure)

 

[17x17]  The most magic 17x17 square is the ultra magic 17x17 square. The ultra magic 17x17 square is panmagic and symmetric.

 

[18x18]  The most magic 18x18 (composite) square consists of 3x3 proportional 4x4 in 6x6 magic squares. Each 1/3 row/column/diagonal of the magic 18x18 square gives 1/3 of the magic sum.

 

[19x19]  The most magic 19x19 square is the ultra magic 19x19 square. The ultra magic 19x19 square is panmagic and symmetric.

 

[20x20]  The most magic 20x20 square is a most perfect magic 20x20 square (with the extra tight 'Willem Barink' structure), or an ultra magic 20x20 square. The ultra magic 20x20 square is panmagic, symmetric inside each 4x4 sub-square, (2x2) compact and each 1/2 row, each 1/2 column and each 1/5 diagonal gives a proportional part of the magic sum.

 

[21x21]  The most magic 21x21 square is an ultra magic 21x21 square. The ultra magic 21x21 square is panmagic, symmetric and double (3x3 and 7x7) compact.

 

[22x22]  The most magic 22x22 square is a 22x22 inlaid square with four panmagic 7x7 inlays and five panmagic 4x4 inlays.

 

[23x23]  The most magic 23x23 square is the ultra magic 23x23 square. The ultra magic 23x23 square is panmagic and symmetric.

 

[24x24]  The most magic 24x24 square is a most perfect (Franklin pan)magic 24x24 square (with the extra tight 'Willem Barink' structure).  

 

[25x25]  The most magic 25x25 square is an ultra magic 25x25 square. The ultra magic 25x25 square is panmagic, symmetric, (5x5) compact and each 1/5 row/column/diagonal gives 1/5 of the magic sum.

 

[26x26]  The most magic 26x26 square is a concentric magic 26x26 square.  

 

[27x27]  The most magic 27x27 square is an ultra magic 27x27 square. The ultra magic 27x27 square is panmagic, symmetric, (3x3) compact and each 1/9 row, 1/9 column and 1/3 diagonal gives a proportional part of the magic sum.

 

[28x28] The most magic 28x28 square is a most perfect magic 28x28 square (with the extra tight 'Willem Barink' structure), or an ultra magic 28x28 square. The ultra magic 28x28 square is panmagic, symmetric inside each 4x4 sub-square, (2x2) compact and each 1/2 row, each 1/2 column and each 1/7 diagonal gives a proportional part of the magic sum.

 

[29x29]  The most magic 29x29 square is the ultra magic 29x29 square. The ultra magic 29x29 square is panmagic and symmetric.

 

[30x30]  The most magic 30x30 (composite) square consists of 3x3 proportional 4x4 in 6x6 in 8x8 in 10x10 magic squares. Each 1/3 row/column/diagonal of the magic 30x30 square gives 1/3 of the magic sum.

 

[31x31]   The most magic 29x29 square is the ultra magic 31x31 square. The ultra magic 31x31 square is panmagic and symmetric.
 

[32x32]  The most magic 32x32 square is a most perfect (Franklin pan)magic 32x32 square (with the extra tight 'Willem Barink' structure).