21x21 magic square

 

Explanation 21x21 magic square

The 21x21 magic square is odd and a multiple of 3 (3 x 7 is simular as 3x5, 3x11, 3x13, 3x17, ... that is the 15x15, 33x33, 39x39, 51x51, ... magic square). It is possible to use 3x3 a proportional 7x7 panmagic square to construct a composite 21x21 magic square. Use a 7x3 (= 3x7) magic rectangle to construct an ultra magic 21x21 square.

 

Construction methods:

Methods to construct the 21x21 magic square are:

 

The first five methods lead to simple symmetric magic 21x21 squares.

 

Composite (3), (3a) and (4) give 21x21 magic squares with special magic features.

 

With the shift methods you can construct panmagic 21x21 squares. Shift method (2) gives a more tight structure.

 

It is also possible to use (as first grid) 9x the same 7x7 panmagic vierkant and 2 ternary grids (the third grid is a reflection of the second grid) to construct a panmagic and 7x7 compact 21x21 magic square.

 

The 21x21 magic square consists of 9 proportional panmagic 7x7 squares, so each 1/3 row/column/diagonal gives 1/3 of the magic sum. This 21x21 magic square is also panmagic and 7x7 compact (but not symmetric).

 

Use a 7x3 (= 3x7) magic rectangle to construct an ultra magic 21x21 square, which is panmagic, symmetric, 3x3 and 7x7 compact.

 

Concentric and pan 19x19 in 21x21 are specials.