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What is the most magic square per order?
Does a method exist to produce magic squares of all orders? I give the answer in chapter 1. Odd, double
odd and multiple of four.
It is also important to know that not all magic squares are equally magic. For each order, with exception
of the 3x3 magic square, exist magic squares with extra (= more than the minimum) magic features. On
this website you find for each order a magic square with the maximum number of (extra) magic features.
In chapter 2. Magic features, you find an explanation of the (extra) magic features. In chapter 3. Most
magic squares per order, you find per order a magic square with the maximum number of (extra) magic
features.
1. Odd, double odd and multiple of four
Does a method exist to produce magic squares of all orders? According to Wikipedia you need three different
methods to produce magic squares of all orders. You need a method to produce odd (choose one of the three
classic methods to produce an odd [7x7] magic square on page '14x14 magic square'), a method to produce
double odd (see the medig method on page '6x6 magic square') and a method to produce multiples of four (see
the classic method to produce a multiple of four [i.e. the 16x16 inlay] on page '3x extra magic 18x18 square').
Does a method exist to produce magic squares of all orders? My answer is yes! Use the method to produce
concentric magic squares (see page 'bordered magic squares'). N.B.: In the execution of this method a distinc-
tion is made between odd and even orders.
Do you want to produce the most magic squares, you need more methods.
2. Magic features
I give an explanation of the (extra) magic features. You need this information to understand, what I mean by the
most magic square per order.
[pure] A magic square is pure if it consists of the digits 1 up to n x n. A pure magic 3x3 square consists of the
digits 1 up to (3 x 3 =) 9. On this website you find only pure magic squares with exception of 'Each magic sum'.
[minimal magic features] Minimal the addition of the digits of each row/column/diagonal must give (the same)
magic sum.
[magic sum] For each pure magic square you can calculate the magic sum. The magic sum is [(1 + n x n) / 2] x n.
For example the sum of the 3x3 magic square is: [(1 + 3 x 3) / 2) x 3 = 15.
[concentric] An odd concentric magic square consists of a centre of one cell and an even concentric magic square
consists of a centre of 2x2 cells, and you can put borders around it again and again. For example a concentric
magic 14x14 square consists of a (each time proportional) 4x4 in 6x6 in 8x8 in 10x10 in 12x12 in 14x14 magic
square.
[panmagic] A magic square is panmagic, if addition of the digits of each pandiagonal gives the magic sum. A
pandiagonal is a broken diagonal, which consists of two parts. The first part is a line, which starts from the outside
row or outside column (but not from a corner) of the magic square. The second part is a line or a dot (and the dot
ends in one of the corners of the magic squares). See for example the pandiagonals of the panmagic 4x4 square
on page 'panmagic 4x4 square, explanation'.
[symmetric] In a symmetric magic squares each time addition of two digits, which can be connected with a straight
line through the centre of the magic square and which are at the same distance to the centre, gives the same sum.
The sum is 1 + n x n (for example the sum in a symmetric 5x5 magic square is: 1 + 5 x 5 = 26). It is also possible
that the magic square is not symmetric as a whole, but the magic square is symmetric in each sub-square (see for
example on page 'Basic key method (2)')
[centre] The centre of an odd magic square is the middle cell (n.b.: in the middle cell of an odd symmetric magic
square you find allways the middle digit; for example in a symmetric 5x5 magic square you find the digit 13 in the
middle cell). The centre of an even magic square is the crosspoint of the middle 2x2 cells.
[compact] If a magic square is a multiple of 2, 3, 5, 7, … than compact means, that each random chosen 2x2, 3x3,
5x5, 7x7, … sub- square gives the same (proportional part of the) magic sum. A magic square can be double compact.
For example the ultra magic 15x15 square on this website gives a proportional part of the magic sum for each 3x3 sub-
square and for each 5x5 sub-square.
[ultramagic] For an odd order (with execption of the 3x3 magic square) is ultra magic the most magic square. An odd
ultra magic square is always panmagic and symmetric and (if the order of the square is not a prime number) compact.
If possible in the ultra magic square also a part of each row, column and/or diagonal gives a proportional part of the
magic sum. For example in the ultra panmagic 27x27 square on this website gives each 1/9 row, 1/9 column and 1/3
diagonal a proportional part of the magic sum.
[most perfect] For order is multiple of four the most perfect magic square is the most magic square. Willem Barink
teached us, that a little part of the most perfect magic squares has the extra magic feature X. See for detailed expla-
nation (including the similarities and differences with the Franklin panmagic square), page 'Most perfect magic squares,
explanation´.
[Scope] Scope means, the (absolute or relative) number of different magic squares you can produce by using a method
of construction. A scope of 100% means, that you can produce all possible magic squares by using a method of construc-
tion. See a method of construction with 100% scope on page ´8x8 most perfect magic squares, binary´.
[prime number A prime number can only be devided by one or by the number itself.
3. Most magic square per order
See below the most magic square (with the maximum number of magic features) per order.
[4x4] The most magic 4x4 square is the panmagic 4x4 square. The panmagic 4x4 square is also the smallest most per-
fect magic square. The square is panmagic and (2x2) compact. See how you can produce all (100% scope) panmagic
4x4 squares on page ‘Panmagic 4x4 square’.
[5x5] The most magic 5x5 square is the ultra magic 5x5 square. The ultra magic 5x5 square is panmagic and symmetric.
See the key to produce one (so the scope is 1) ultra magic 5x5 square, on page ‘[ultra] panmagic 5x5 square’.
[6x6] The most magic 6x6 square is a 6x6 square with a 4x4 panmagic inlay. See how you can use each panmagic 4x4
square to produce a 4x4 in 6x6 square, on page ‘bordered magic squares’.
[7x7] The most magic 7x7 square is the ultra magic 7x7 square. The ultra magic 7x7 square is panmagic and symmetric.
See the key to produce one (so the scope is 1) ultra magic 7x7 square, on page ‘[ultra] panmagic 5x5 square’.
[8x8] The most magic 8x8 square is a most perfect 8x8 magic square (with the extra magic feature X). See how you can
produce all (is 100% scope) most perfect magic 8x8 squares (with the extra magic feature X), on page ‘most perfect
magic square binary’.
[9x9] The most magic 9x9 square is an ultra magic 9x9 square. The ultra magic 9x9 square is panmagic, symmetric,
(3x3) compact and each 1/3 row and 1/3 column gives 1/3 of the magic sum. See how to produce one (scope of 1)
ultra magic 9x9 square, on page ‘[ultra] panmagic 9x9 square (1)’.
[10x10] The most magic 10x10 square is a concentric magic 10x10 square. See how to produce one (scope is 1) con-
centric magic 10x10 square, on page ‘Bordered magic squares’.
[11x11] The most magic 11x11 square is the ultra magic 11x11 square. The ultra magic 11x11 square is panmagic and
symmetric. See the key to produce one (so the scope is 1) ultra magic 11x11 square, on page ‘[ultra] panmagic 5x5
square’.
[12x12] The most magic 12x12 square is a most perfect magic 12x12 square (with the extra magic feature X; see page
‘Basic pattern method (2)’), or an ultra panmagic 12x12 square (see page ‘Basic key method (2)’). The ultra panmagic
12x12 square is panmagic, symmetric inside each 4x4 sub/square, (2x2) compact and each 1/2 row, 1/2 column and
each 1/3 diagonal gives a proportional part of the magic sum.
[13x13] The most magic 13x13 square is the ultra magic 13x13 square. The ultra magic 13x13 square is panmagic and
symmetric. See the key to produce one (so the scope is 1) ultra magic 13x13 square, on page ‘[ultra] panmagic 5x5 square’.
[14x14] The most magic 14x14 square is a 14x14 inlaid square. See how you can use each most perfect magic 8x8 square
or each magic 4x4 in 6x6 square to produce a 14x14 inlaid square on page ‘Inlaid square (1)’.
[15x15] The most magic 15x15 square is an ultra magic 15x15 square. The ultra magic 15x15 square is panmagic, sym-
metric and double (3x3 and 5x5) compact. See how to produce one (scope is 1) ultra magic 15x15 square on page ‘[ultra]
panmagic 15x15 square’.
[16x16] The most magic 16x16 square is a most perfect 16x16 magic square (with the extra magic feauture X). See
how to use each panmagic 4x4 square to produce a most perfect magic 16x16 square (with the extra magic feature X),
on page ‘Basic pattern method (3c)’.
[17x17] The most magic 17x17 square is the ultra magic 17x17 square. The ultra magic 17x17 square is panmagic and
symmetric. See the key to produce one (so the scope is 1) ultra magic 17x17 square, on page ‘[ultra] panmagic 5x5 square’.
[18x18] The most magic 18x18 square consists of 3x3 proportional 4x4 in 6x6 squares (see the third method on page
‘3x extra magic 18x18 square’. Each 1/3 row/column/diagonal of the magic 18x18 square gives 1/3 of the magic sum.
[19x19] The most magic 17x17 square is the ultra magic 17x17 square. The ultra magic 17x17 square is panmagic and
symmetric. See the key to produce one (so the scope is 1) ultra magic 17x17 square, on page ‘[ultra] panmagic 5x5 square’.
[20x20] The most magic 20x20 square is a most perfect magic 20x20 square (with the extra magic feature X; see page
‘basic pattern method (4)’), or an ultra panmagic 20x20 square (see page ‘basic key method (2)’). The ultra panmagic
20x20 square is panmagic, symmetric inside each 4x4 sub-square, (2x2) compact and each 1/2 row, each 1/2 column
and each 1/5 diagonal gives a proportional part of the magic sum.
[21x21] The most magic 21x21 square is an ultra magic 21x21 square. The ultra magic 21x21 square is panmagic, sym-
metric and double (3x3 and 7x7) compact. See how to produce ultra magic 21x21 squares on page ‘[ultra] panmagic
15x15 square’.
[22x22] The most magic 22x22 square is an inlaid square with four panmagic 7x7 inlays and five panmagic 4x4 inlays.
See page ‘Inlaid square (2)’.
[23x23] The most magic 23x23 square is the ultra magic 23x23 square. The ultra magic 23x23 square is panmagic and
symmetric. See the key to produce one (so the scope is 1) ultra magic 23x23 square, on page ‘[ultra] panmagic 5x5 square’.
[24x24] The most magic 24x24 square is a most perfect 24x24 magic square (with the extra magic feauture X). See
how to use each panmagic 4x4 square to produce a most perfect magic 24x24 square (with the extra magic feature X),
on page ‘Basic pattern method (5)’.
[25x25] The most magic 25x25 square is an ultra magic 25x25 square. The ultra magic 25x25 square is panmagic, sym-
metric, (5x5) compact and each 1/5 row/column/diagonal gives 1/5 of the magic sum. See how to use each ultra magic
5x5 square to produce an ultra magic 25x25 square, on page ‘[ultra] panmagic 25x25 square’.
[26x26] The most magic 26x26 square is a concentric magic 26x26 square. See how to produce one (scope is 1) con-
centric magic 26x26 square, on page ‘Bordered magic squares’.
[27x27] The most magic 27x27 square is an ultra magic 27x27 square. The ultra magic 27x27 square is panmagic, sym-
metric, (3x3) compact and each 1/9 row, 1/9 column and 1/3 diagonal gives a proportional part of the magic sum. See
how to use each ultra magic 9x9 square to produce an ultra magic 27x27 square, on page ‘[ultra] panmagic 27x27 square’.
[28x28] The most magic 28x28 square is a most perfect magic 28x28 square (with the extra magic feature X; see page
‘basic pattern method (6)’), or an ultra panmagic 28x28 square (see page ‘basic key method (2)’). The ultra panmagic
28x28 square is panmagic, symmetric inside each 4x4 sub-square, (2x2) compact and each 1/2 row, each 1/2 column
and each 1/7 diagonal gives a proportional part of the magic sum.
[29x29] The most magic 29x29 square is the ultra magic 29x29 square. The ultra magic 29x29 square is panmagic and
symmetric. See the key to produce one (so the scope is 1) ultra magic 29x29 square, on page ‘[ultra] panmagic 5x5 square’.
[30x30] The most magic 30x30 square is a concentric magic 30x30 square. See how to produce one (scope is 1) con-
centric magic 30x30 square, on page ‘Bordered magic squares’.
[31x31] The most magic 29x29 square is the ultra magic 31x31 square. The ultra magic 31x31 square is panmagic and
symmetric. See the key to produce one (so the scope is 1) ultra magic 31x31 square, on page ‘[ultra] panmagic 5x5 square’.
[32x32] The most magic 32x32 square is a most perfect 32x32 magic square (with the extra magic feauture X). See
how to use each panmagic 4x4 square to produce a most perfect magic 32x32 square (with the extra magic feature X),
on page ‘Basic pattern method (7b)’.
[33x33] The most magic 33x33 square is an ultra magic 33x33 square. The ultra magic 33x33 square is panmagic, sym-
metric and double (3x3 and 11x11) compact. See how to produce ultra magic 33x33 squares on page ‘[ultra] panmagic 15x15
square’.
[34x34] The most magic 34x34 square is a concentric magic 34x34 square. See how to produce one (scope is 1) con-
centric magic 34x34 square, on page ‘Bordered magic squares’.
[35x35] The most magic 35x35 square is an ultra magic 35x35 square. The ultra magic 35x35 square is panmagic, sym-
metric and double (5x5 and 7x7) compact. See how to produce one (scope is 1) ultra magic 35x35 square, on page ‘[ultra]
panmagic 35x35 square’.
[…] etcetera
LEGENDA:
[ ] The 3x3 magic square is the only magic square, which has no extra magic features.
[ ] For order is multiple of 4 is the most perfect magic square (with the extra magic feature X of Willem Barink) the
most magic square. With the basic pattern method you use 2x2, 3x3, 4x4, 5x5, ... the same panmagic 4x4 square to pro-
duce a most perfect (with the extra magic feature of Willem Barink) 8x8, 12x12, 16x16, 20x20, ... magic square.
N.B.: For odd multiples of 4, is the ultra magic (= panmagic, symmetric, compact and a part of a row/column/diagonal
gives a proportional part of the magic sum) an similar magic alternative.
[ ] For order is prime number, there is one method to produce all (= 100% scope) panmagic squares. Per order I
give one simple key to produce one (scope is 1) ultra magic (= panmagic and symmetric) square.
[ ] For order is double odd, extra magic squares are bordered, concentric or inlaid squares. N.B.: For order is odd
multiple of 6 it is also possible that a part of the row/column/diagonal gives a proportional part of the magic sum. For
example the 18x18 magic square on this website consists of 9 proportional 4x4 in 6x6 squares and each 1/3 row/column/
diagonal gives 1/3 of the magic sum.
[ ] For order is tripartite (for example: order 9 = 3x3 or order 27 = 3x3x3) is ultra magic (panmagic, symmetric,
3x3 compact) and each 3/n row/column and each 9/n diagonal gives a proportinal part of the magic sum (for example
1/9 row/column and 1/3 diagonal of the 27x27 magic square gives a proportional part of the magic sum) the most magic
square.
[ ] For order is a multiple of two different prime numbers, is ultra magic (panmagic, symmetric and double compact)
the most magic square.
[ ] For order is prime number x prime number (with exception of the 9x9 magic square) is ultra magic (panmagic,
symmetric and compact) and each 1/prime number row/column/diagonal gives 1/prime number of the magic sum (for
example 1/5 row/column/diagonal of the 25x25 square gives 1/5 of the magic sum) the most magic square.
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