21x21 magic square

 

Explanation 21x21 magic square

The 21x21 magic square is odd and 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 three methods lead to simple symmetric magic 21x21 squares.

 

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.

 

Construct a 3x3 in 5x5 in 7x7 in 9x9 in 11x11 in 13x13 in 15x15 in 17x17 in 19x19 in 21x21 (concentric) magic square.