Explanation 27x27 magic square

The 27x27 magic square is odd and a multiple of 3 (3x9). Use 3x3 an ultra magic 9x9 square to construct an ultra magic 27x27 square.

Construction methods:

Methods to construct the 27x27 magic square are:

• Composite, Simple a
• Composite, Simple b
• Composite, Prop. (1) a
• Composite, Prop. (1) b
• Diagonal method
• Siamese method
• Knight movement method
• Symmetric transformation
• Lozenge method
• Shift method (1)
• Shift method (2)
• Ultra magic
• Ultra 25x25 in 27x27 (1)
• Ultra 25x25 in 27x27 (2)
• Concentric
• Concentric with diamond

Composite simple a and b lead to simple magic 27x27 squares.

Composite proportional (1)a leads to a magic square consisting of 81 (semi)magic 3x3 squares, so each 1/9 row/column gives 1/9 of the magic sum. The 27x27 magic square is 3x3 compact too.

Composite proportional (1)b leads to a magic square consisting of 9 ultramagic 9x9 squares, so each 1/3 row/column/diagonal gives 1/3 of the magic sum. The 27x27 magic square is also panmagic, 3x3 compact and symmetric, and that is ultramagic.

Diagonal method,  Siamese method, knight movement method, symmetric transformation and Lozenge method lead to simple, symmetric magic 27x27 squares.

Use shift method (1) to construct panmagic 27x27 squares. And use shift method (2) to construct panmagic 27x27 squares with extra special magic futures.

The ultra magic 27x27 square is panmagic, 3x3 compact, symmetric and each 1/9 row/column gives 1/9 of the magic sum and each 1/3 diagonal gives 1/3 of the magic sum.

Ultra 25x25 in 27x27 and concentric [with diamond] are specials.