Perfect magic squaresContact / guestbook3x3 magic squareSudoku method (1)Sudoku method (2)Sudoku method (3)ExplanationPan magic 4x4 squarePanmagic 5x5 square6x6 magic squareKhajuraho methodBasic pattern method (1)Basic pattern method (2)Basic pattern method (3)Analysis Franklin panm. 8x8Basic key method (1)Basic key method (2)pan magic 9x9 squarePan magic 15x15 squarePan magic 27x27 squareBordered squaresEach magic sumWater retention challengeFavorite Links
Perfect magic squares
Perfect magic squares
Do you like Sudoku puzzles and/or don't you know anything (or just a little) about magic squares:
  Read page 'Explanation' and/or download my free e-book (see link, yellow marked on this page).
 
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 squares.
 
 

*** 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).
 

 
 
 
ABOUT THE SITE
 
Aim
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: 3x3, panmagic 4x4, panmagic 5x5 (= odd,
but no multiple of 3), 6x6 (= even, but no multiple of 4), panmagic 7x7 (see 5x5), Franklin panmagic 8x8 (see
Basic pattern method), panmagic 9x9 (= odd number squared, like 25x25), 10x10 (see 6x6), panmagic 11x11
(see 5x5), most magic 12x12 (= odd multiple of 4; see Basic key method 2), panmagic 13x13 (see 5x5), 14x14
(see 6x6), panmagic 15x15 (= odd multiple of 3, but no multiple of 9), perfect Franklin Panmagic 16x16 (= mul-
tiple of 8; see Basic key method 1), ..., panmagic 27x27, ...
 
3x3 magic square
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 panmagic square to produce a 8x8 Franklin panmagic square or a perfect
16x16 Franklin panmagic square or a perfect 32x32 Franklin panmagic 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).


Explanation magic squares
A short explanation is then given on magic squares and their magic features. I also reveal 'secrets' behind
structures and methods of construction of magic squares.
 
Pan magic 4x4 square
I present a method of construction to produce 4x4 panmagic 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 panmagic squares (including rotation
and/or mirroring).
 
Pan magic 5x5 square
I present a method of construction to produce 5x5 panmagic 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
Panmagic (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 panmagic 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
panmagic square. This method is the underlying method of the basic pattern method.
 
Basic pattern method (1)
The basic pattern method of construction uses (the pattern of) each 4x4 panmagic square to produce an 8x8
Franklin panmagic square. It is even possible to use the pattern of a 4x4 panmagic square to produce a perfect
16x16 Franklin panmagic square or a perfect 32x32 Franklin panmagic square.
 
Basic pattern method (2)
It is also possible to use the pattern of a 4x4 panmagic square to produce a 24x24 panmagic square.

Basic pattern method (3)
It is even possible to use a 2x2 carpet of one of the 384 panmagic 4x4 squares to produce a Franklin panmagic
8x8 square. And yes (!), it is possible to use a 2x2 carpet of each Franklin panmagic 8x8 square to produce a
perfect Franklin panmagic 16x16 square.
 
Analysis Franklin pan magic 8x8 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 panmagic squares can be produced directly by the basic pattern
method; probably all 8x8 Franklin panmagic squares can [if you use alternative digits swaps] be traced back to
the pattern of a 4x4 panmagic square).
 
Basic key method (1)
The basic key method of construction produces perfect Franklin panmagic 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).
 
Basic key method (2)
A presentation of the basic key method of construction for squares that are odd multiples of 4 (12x12, 20x20,
28x28, ...).

Pan magic 9x9 square
I present a method to produce panmagic squares which have the size of odd number squared (9x9, 25x25,
49x49, 81x81, ...).

Pan magic 15x15 square
15x15 is a 'difficult' size. The method of construction to produce 15x15 panmagic squares is a bit more compli-
cated than the method to produce 5x5 panmagic or 9x9 panmagic squares. You can use this method for squa-
res that are odd multiples of 3, but no multiple of 9 (= 15x15, 21x21, 33x33, 39x39, ...).

Pan magic 27x27 square
A 27x27 square is an odd multiple of 3 (and no odd number squared), but still panmagic 27x27 squares exist.
By analyzing an existing panmagic 27x27 square I have discovered a method of construction.

Bordered magic squares
A bordered square is an (impure) magic square inside a bigger (pure) magic square. I present a method of con-
struction to produce even bordered squares, I give an example of an odd bordered square and I give a link
to a website with (more complicated) inlaid squares.
 
Each magic sum
I present the key to produce a 4x4 magic square for each random chosen magic sum.

Water retention challenge
I give an introduction (and link to the website about) the water retention challenge, an idea on the website of
Craig Knecht


[Translation Dutch website www.magischvierkant.nl into English is corrected by Kieran Behan]


For a better printable version of the website, download my free
e-book:


www.lulu.com/content/e-boek/how-to-produce-perfect-magic-squares/8308796




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Learn step by
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