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● 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.
Aim of the website
The aim of the website 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.
Look at the webpage with the 'most magic square per order'.
Explanation and reference to all webpages
3x3 magic square
For kids turtle Lowi tells his story and explains how to produce 3x3 magic squares.
3x3 magic square, explanation
Read the detailed explanation about the 3x3 magic square.
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 most perfect (Franklin pan)magic square
or a 16x16 most perfect (Franklin pan)magic square or a 32x32 most perfect (Franklin pan)magic square (=
alternate presentation of the basic pattern method).
Sudoku method (2)
It is also possible to use the Sudoku pattern of a 4x4 panmagic square to produce a most perfect magic
12x12 square. With this method you can produce for each multiple of 4 most perfect (for each multiple of 8
also Franklin pan)magic squares.
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 per-
fect 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).
Pan magic 4x4 square, explanation
Read the detailed explanation about the pan magic 4x4 square (= smallest most perfect magic square).
Pan magic 4x4 square, binary
It is possible to use binary patterns to produce all 384 panmagic 4x4 squares (including rotation and/or
mirroring).
Dürer and Franklin transformation
Learn how to use transformation to produce the famous magic square of Albecht Dürer and the famous
magic square of Benjamin Franklin.
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).
Transformation method, analysis
It is also possible to transform a most perfect magic square backwards to its starting position. All starting
positions have the same features (= boundary conditions). Are you able to create a starting positiion with
the right boundary conditions and to produce a correct most perfect magic square?
[Ultra] pan magic 5x5 square
I present a method of construction to produce all possible 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). I give also the keys to produce ultra panmagic
5x5, 7x7, 11x11, ... magic squares.
Pan magic 5x5 square, explanation
Read the detailed explanation about the pan magic 5x5 square.
6x6 magic square
Pan magic (or extra magic) 6x6 squares don't exist. I present firstly the Medjig method of Willem Barink
(source: Wikipedia, Dutch language version). You can use this method to produce 'double odd' (= 6x6,
10x10, 14x14, 18x18, ...) as well as 'double even' (8x8, 12x12, 16x16, 20x20, ...) magic squares. I present
secondly the method of Strachey and thirdly Paul Michelet's trick.
Ultra (pan)magic 8x8 square
The Franklin panmagic (= most perfect magic) 8x8 square has the most magic features. The most magic odd
squares are ultra (pan)magic squares. But also (even, for example) 8x8 ultra (pan)magic squares exist. Look
how I have produced a 8x8 ultra (pan)magic square.
Most perfect magic squares, explanation
Read the detailed explanation about most perfect magic squares.
8x8 most perfect magic squares, binary
It is possible to use binary patterns to produce all possible most perfect magic 8x8 squares.
Khajuraho method
From a pan magic 4x4 square, for example the famous Khajuraho square, you can produce bigger pan magic
squares (8x8, 12x12, 16x16, 20x20, ...). If you swap digits it is even possible to produce an 8x8 most perfect
(Franklin pan)magic square. This method is the underlying method of the basic pattern method.
Khajuraho method, explanation
Read the detailed information how I have discovered the Khajuraho method (and 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 most perfect (Franklin pan)magic square. It is even possible to use the pattern of a 4x4
pan magic square to produce a 16x16 most perfect (Franklin pan)magic square or a 32x32 most perfect
(Franklin pan)magic square.
Basic pattern method (1b)
Use 4x the same panmagic 4x4 square (and 2 fixed grids) to produce a 8x8 most perfect (Franklin pan)magic
square with the extra magic feature X (discoverded by Willem Barink).
Basic pattern method (2)
Use 9x the same 4x4 pan magic square (and 2 fixed grids) to produce a 12x12 most perfect magic square
with the extra magic feature X (discovered by Willem Barink).
Basic pattern method (3a)
It is even possible to use two 2x2 carpets of a pan magic 4x4 square to produce a most perfect (Franklin
pan)magic 16x16 square.
Basic pattern method (3b)
Use 4x4 the same panmagic 4x4 square to produce a most perfect (Franklin pan)magic 16x16 square with
the extra magic feature X (discovered by Willem Barink).
Basic pattern method (3c)
Use 16x the same 4x4 pan magic square (and 2 fixed grids) to produce a 16x16 most perfect (Franklin pan)
magic square with the extra magic feature X (discovered by Willem Barink).
Basic pattern method (4)
Use 25x the same 4x4 pan magic square (and 2 fixed grids) to produce a 20x20 most perfect magic square
with the extra magic feature X (discovered by Willem Barink).
Basic pattern method (5)
Use 36x the same 4x4 pan magic square (and 2 fixed grids) to produce a 24x24 most perfect (Franklin pan)
magic square with the extra magic feature X (discovered by Willem Barink).
Basic pattern method (6)
Use 49x the same 4x4 pan magic square (and 2 fixed grids) to produce a 28x28 most perfect magic square
with the extra magic feature X (discovered by Willem Barink).
Basic pattern method (7a)
Use 8x8 the same panmagic 4x4 square to produce a most perfect (Franklin pan)magic 32x32 square.
Basic pattern method (7b)
Use 64x the same 4x4 pan magic square (and 2 fixed grids) to produce a 32x32 most perfect (Franklin
pan)magic square with the extra magic feature X (discovered by Willem Barink).
This contains an analysis of 8x8 most perfect (Franklin pan)magic squares. I proof that all the 8x8 Franklin
panmagic squares can be traced back to the pattern of a 4x4 pan magic square.
Analysis Franklin pan magic 8x8 square (2)
I have determined a complete classification of all 368640 most perfect 8x8 magic squares in 6 groups.
Basic key method (1)
With the basic key method of construction you can produce most 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 [by 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, ...). N.B.: It is even possible to produce a ultra panmagic 12x12 square. Half of each row/
column gives half of the magic sum. One third of each diagonal and each 2x2 subsquare gives 1/3 of the
magic sum.
Quadrant method (Willem Barink)
The quadrant method is suited to construct most perfect panmagic 8x8 squares, but in adapted form the
method can also be used for the construction of higher order most perfect panmagic squares. See for
some panmagic constructions of order 12 and 16 the website http://wba.novaloka.nl/magic-squares.html. This
paper deals only with panmagic 8x8 squares, and confines to squares starting with the number 1 top left.
[Ultra] pan magic 9x9 square (1)
I present a method to produce pan magic squares which have the size of odd number squared (9x9, 25x25,
49x49, 81x81, ...).
N.B.: It is even possible to use a similar method to produce an ultra panmagic 9x9 square (and each 1/3 of
a row and 1/3 of a column gives the same sum).
Pan magic 9x9 square (2)
Disadvantage of the method pan magic 9x9 square (1) is that it gives only a few possibilities. With the method
pan magic 9x9 square (2) we use 4 instead of 2 grids to produce a pan magic 9x9 square and that gives more
possibilities.
Pan magic 9x9 square (3)
Discover the millions of possibilities to produce a pan magic 9x9 square.
3x extra magic 9x9 square
I present 3 methods to produce 9x9 magic squares, which are not panmagic, but the squares have other extra
magic features.
10x10 magic square
I present the improved method of Strachey and the Medjig method without puzzling (= LUX method) to produce
a 10x10 magic square.
Composite 12x12 magic square
I show you how to produce the composite 12x12 magic square of Royal Vale Heath from "Scripta Mathematica"
of 1938.
14x14 magic square
Use the method of Strachey to produce a 14x14 magic square (learn how to swap digits). You need four 7x7
magic squares to produce a 14x14 magic square. I present three popular methods to produce a 7x7 magic
square (that you need to produce a 14x14 magic square). As extra I show you how to produce a 14x14 magic
square with a 12x12 Bree/Ollerenshaw magic square as inlay.
[Ultra] pan magic 15x15 square
15x15 is a 'difficult' size. The method of construction to produce 15x15 pan magic squares is a bit more compli-
cated than the method to produce 5x5 panmagic or 9x9 pan magic squares. You can use this method for squares
that are odd multiples of 3, but no multiple of 9 (= 15x15, 21x21, 33x33, 39x39, ...).
N.B.: It is also possible to use 3x3 the same panmagic 5x5 square (and two fixed grids) to produce a panmagic
15x15 square. And after the contribution of George Chen, there is even a simple method to produce an ultra
panmagic 15x15 square.
3x extra magic 15x15 square
Did I say that 15x15 is a difficult size? I show you three different methods to produce extra magic 15x15 squares.
The perfect magic square
The perfect magic square is perfect because of its perfect magic features as well as its perfect composition.
3x extra magic 18x18 square
I present 3 methods to produce an 18x18 magic square. It is even possible to produce an 18x18 magic square,
which consists of 9 proportional 4x4 in 6x6 squares and each 1/3 row, each 1/3 column and each 1/3 diagonal
gives 1/3 of the magic sum !!!
Ultra pan magic 25x25 square
It is possible to use 5x5 the same ultra panmagic 5x5 square to produce an ultra panmagic 25x25 square.
[Ultra] pan magic 27x27 square
A 27x27 square is an odd multiple of 3 (and no odd number squared), but still pan magic 27x27 squares exist.
By analyzing an existing pan magic 27x27 square I have discovered a method of construction.
N.B.: It is even possible to use 3x3 the same ultra panmagic 9x9 square (and two fixed grids) to produce an
ultra (pan)magic 27x27 square. The square is panmagic and each 1/9 row and 1/9 column gives 1/9 of the
magic sum and each 1/3 diagonal and each (random chosen) 3x3 sub- square gives 1/3 of the magic sum.
[ultra] pan magic 35x35 square
It is possible to use a pan magic 5x5 square and a pan magic 7x7 square to produce a pan magic 35x35 square.
And I give the key, a symmetric magic 5x7 rectangle, to produce an ultra panmagic 35x35 square, that is sym-
metric and each 5x5 and 7x7 sub-square gives the same (proportional part of the magic) sum.
Extra magic 35x35 square
The extra magic square is not panmagic, but each 1/7 row, 1/7 column and 1/7 diagonal gives 1/7 of the magic
sum!!!
Bordered magic squares
A bordered square is an (impure) magic square inside a bigger (pure) magic square. I present a method of
construction to produce even bordered squares and a method of construction to produce odd bordered squares,
and I give a link to a website with a method of construction of concentric squares.
N.B.: Look also at the amazing al-Antaakii concentric magic 15x15 square with extra magic feature dated from 987!!!
Inlaid square (1)
An inlaid square is a magic square with a several magic squares inside. I present a method of construction
to produce a 12x12 magic square consisting of four proportional 6x6 magic squares with in each 6x6 square
a 4x4 panmagic inlaid square.
Inlaid square (2)
The challenge is to produce an even magic squares with odd as well as even (not diamond) inlays. Look at the
production of a 22x22 magic square with a 20x20 inlay, four 7x7 panmagic inlays and five 4x4 panmagic inlays.
Each magic sum
I present the key to produce a 4x4 magic square for each random chosen magic sum. And I present an alterna-
tive, which is probably even better.
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.
'most magic' 4x4x4 cube
You can use each 4x4 panmagic square to produce a 'most magic' 4x4x4 cube.
Magic 8x8x8 cube
You can use each 4x4 panmagic square to produce an (simple, symmetric) magic 8x8x8 cube.
[Almost] perfect magic 9x9x9 cube
You can use an ultramagic 9x9 square to produce an almost perfect magic 9x9x9 cube. The cube is sym-
metric, pandiagonal magic, 3x3 compact, but only 3/4 pantriagonal magic. And I reveal the secret behind
Frost's perfect (but not compact) 9x9x9.
Trick with 8x8 bimagic square
It is possible to transform a 8x8 bimagic square into another (not rotated and/or mirrored version of the) bi-
magic 8x8 square.
N.B.: This is not just a trick. If you swap the highest digit with the lowest digit, the second highest digit with
the second lowest digit, etc. you create an inverse magic square. The inverse magic square has the same
magic features as the original magic square (even if the original is a pan magic, most magic, bimagic, tri-
magic, concentric, bordered or inlaid square)!!!
[Translation Dutch website www.magischvierkant.nl into English is corrected by Kieran Behan]
Free e-book
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/10923112
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