● Do you want to know how to produce 3x3, 4x4, 5x5, 6x6, ... squares: Find on this website for
  each size (order) an easy method to produce the most magic squares.

 

● Are you a whiz kid and do you want to produce your own magic squares: Find on this website
  (as input) all kinds of patterns to produce the most magic (bordered) squares and magic cubes.

 
 

*** Most magic method ***

On page Sudoku method (3) I show you that it is possible (in 9 steps) to use only one
4x4 Sudoku (with 4x4 the same digits) to produce a most perfect 1024x1024 magic
square (with more than a million different digits). 


Discover also the most magic square on page the perfect magic square. 
 

 

 

The aim is to present as simple as possible the best information on methods of construction in order to produce
perfect magic squares. To out it simply, squares that have as many magic features as possible.

On this website you find a method of construction for every size (= order): 3x3, pan magic 4x4, pan magic 5x5
(= odd, but no multiple of 3), 6x6 (= even, but no multiple of 4), pan magic 7x7 (see 5x5), Franklin pan magic 8x8
(see Basic pattern method), pan magic 9x9 (= odd number squared, like 25x25), 10x10 (see 6x6), pan magic 11x11
(see 5x5), most perfect 12x12 (= odd multiple of 4; see Basic key method 2), pan magic 13x13 (see 5x5), 14x14
(see 6x6), pan magic 15x15 (= odd multiple of 3, but no multiple of 9), perfect Franklin pan magic 16x16 (= mul-
tiple of 8; see Basic key method 1), ..., ultra pan magic 25x25, ..., pan magic 27x27, ..., pan magic 35x35, ...

 
 

For kids turtle Lowi tells his story and explains how to produce 3x3 magic squares.

 
 

Sudoku method(1)
It is possible to produce 4x4 magic squares with the ’Sudoku’ method of construction. It is even possible to
use the Sudoku patterns of a 4x4 pan magic square to produce a 8x8 Franklin pan magic square or a perfect
16x16 Franklin pan magic square or a perfect 32x32 Franklin pan magic square (see also basic pattern method).

 

Sudoku method (2)
It is also possible to use the Sudoku patterns of a 4x4 panmagic square to produce a perfect 'Franklin' panmagic
12x12 square, not for 1/2 but for 1/3 of the rows, columns and [parallel][mirrored][bent] diagonals. With this
method you can produce perfect Franklin (or for odd multiples of 4: 'Franklin') panmagic squares for each multiple
of 4.


Sudoku method (3)
I show you (in 9 steps) how to use only one 4x4 Sudoku (with 4x4 the same digits) to produce
a most perfect 1024x1024 square (with more than a million different digits).


Pan magic 4x4 square
I present a method of construction to produce 4x4 pan magic squares. You only need to know 3 squares to
produce the (3x16=) 48 possible 4x4 panmagic squares (excluding rotation and/or mirroring). On the 2x2
carpet of one of the three 4x4 squares you can find 16 different 4x4 squares. If you rotate and/or mirror
each of the 48 squares you can produce the (48x8=) 384 possible 4x4 pan magic squares (including rotation
and/or mirroring).


Transformation method
I show you that it is possible (in 5 steps) to transform a (4x4, 8x8, 12x12, or 16x16) square with sequential
digits to a most-perfect magic square (you can use this method to produce squares that are multiples
of 4).

 

Pan magic 5x5 square
I present a method of construction to produce 5x5 pan magic squares and give a link to the website where
you will find the 'mother method'. You can use this method to produce odd squares that are no multiples of 3
(= 5x5, 7x7, 11x11, 13x13, 17x17).


6x6 magic square
Pan magic (or extra magic) 6x6 squares don't exist. I present the medjig method of Willem Barink (source:
Wikipedia, Dutch language version). You can use this method to produce even squares that are no multiples
of 4 (= 6x6, 10x10, 14x14, 18x18, ...).


Khajuraho method
From a pan magic 4x4 square, for example the famous Khajuraho square, you can produce bigger panmagic
squares (8x8, 12x12, 16x16, 20x20, ...). If you swap digits it is even possible to produce an 8x8 Franklin
pan magic square. This method is the underlying method of the basic pattern method.
 

The basic pattern method of construction uses the pattern (splitted up) of each 4x4 pan magic square to produce
an 8x8 Franklin pan magic square. It is even possible to use the pattern of a 4x4 pan magic square to produce a
perfect 16x16 Franklin pan magic square or a perfect 32x32 Franklin pan magic square.

 

Basic pattern method (2)
It is also possible to use the pattern (splitted up) of a 4x4 pan magic square to produce a 24x24 pan magic square.


Basic pattern method (3)
It is even possible to use a 2x2 carpet of a pan magic 4x4 square (not splitted up) to produce a perfect Franklin
pan magic 16x16 square.
 

This contains an analysis of 8x8 Franklin panmagic squares. I have discovered an alternative basic pattern
(probably 26.67% of all possible 8x8 Franklin pan magic squares can be produced directly by the basic pattern
method; probably all 8x8 Franklin pan magic squares can [if you use alternative digits swaps] be traced back to
the pattern of a 4x4 pan magic square).

 

 Basic key method (1)
The basic key method of construction produces perfect Franklin pan magic squares that are multiples of 8 (8x8,
16x16, 24x24, …). This method of construction has already been discovered by Donald Morris. I have improved
the method of construction a little by using the so called orthogonal (i.e. to produce the magic square you use
two squares and the second square is a rotation [a quarter turn] of the first square; page Basic key method (2)
contains an explanation regarding the conditions for this).