Explanation 15x15 magic square

The 15x15 magic square is odd and a multiple of 3 (3 x 5). It is possible to use a 3x3 and a 5x5 magic square to construct 15x15 composite squares and I show you different options. Use a 5x3 (of 3x5) magic rectangle to construct an ultra magic 15x15 square.

Construction methods:

The methods to construct a 15x15 magic square are:

• Diagonal method
• Siamese method
• Knight movement method
• Symmetric transformation
• Lozenge method
• Composite, Simple a
• Composite, Simple b
• Composite, AxB compact
• Composite, Prop. (1) a
• Composite, Prop. (1) b
• Shift method (1)
• Shift method (2)
• Panmagic & 5x5 compact
• Ultra magic
• Pan 13x13 in 15x15 (1)
• Pan 13x13 in 15x15 (2)
• Concentric
• Al Antaakii
• Al Antaakii, improved

Use the first five methods to construct simple, symmetric magic 15x15 squares.

Composite, simple leads to (simple) 15x15 magic squares consisting of (not proportional) 5x5 respectively 3x3 magic squares.

Composite, AxB compact leads to (simple) 15x15 magic squares with the extra magic feature that each 3x5 or 5x3 rectangle gives the magic sum.

With composite, proportional (1)a each 1/5 row/column of the 15x15 magic square gives 1/5 of the magic sum and the 15x15 magic square is 3x3 compact. Alternative is symmetric instead of 3x3 compact.

With composite, proportional (1)b you can construct a 15x15 magic square consisting of 9 proportional 5x5 panmagic squares. This 15x15 magic square is panmagic, 5x5 compact (but not symmetric) and each 1/3 row/column/diagonal gives 1/3 of the magic sum.

Use the shift methods to construct panmagic 15x15 squares (shift method (2) gives a more tight structure).

It is also possible to use 9x a 5x5 panmagic square and 2 ternary grids to construct a panmagic and 5x5 compact 15x15 magic square.

Use a 5x3 (or 3x5) magic rectangle to construct a 15x15 ultra magic square, which is panmagic, symmetric, 3x3 compact and 5x5 compact.

Concentric, pan 13x13 in 15x15 and Al Antaakii [improved] are specials.